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# Stability of the Global Weak Axisymmetric Solution to the Quantum Euler System with Vorticity in Dimension $d = 2$

Boris Haspot<fnref n="1" /> Ⓒ · Marc-Antoine Vassenet<fnref n="1" />

Received: 17 May 2023 / Accepted: 28 May 2024 / Published online: 22 July 2024
© The Author(s) 2024

## Abstract

We consider the stability of the global weak solution of the Quantum Euler system in two space dimensions. More precisely, we establish compactness properties of global finite energy weak solution for large initial data provided that these are axisymmetric. The main novelty is that the initial velocity is not necessary irrotational when the density is not vanishing, our main argument is based on the Madelung transform which enables us to prove new Kato estimates on the irrotational part of the velocity.

**Keywords** Fluids mechanics · Compressible Euler system with quantum pressure · Stability of the global weak solution · Axisymmetric solutions

## 1 Introduction

The quantum Euler system that we are considering is described by the following system:

$$ egin{cases} 
\partial_t \rho + \text{div}(\rho u) = 0, \\
\partial_t (\rho u) + \text{div}(\rho u \otimes u) + \nabla P(\rho) = \kappa \rho \nabla \left( \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} \right), \quad (x, t) \in \mathbb{R}^d \times \mathbb{R}_+, \\
(\rho, u)|_{t=0} = (\rho_0, u_0), \quad x \in \mathbb{R}^d.
\end{cases} 
\quad (1.1) $$

Here $u = u(t, x) \in \mathbb{R}^d$ with $d \ge 2$ stands for the velocity field, $\rho = \rho(t, x) \in \mathbb{R}^+$ is the density, $\kappa \ge 0$ is the capillary coefficient and $P$ the pressure. In the sequel we will only consider the case of the dimension $d = 2$ and the pressure is simply a $\gamma$ law $P(\rho) = \rho^\gamma$ with $\gamma > 1$. This system appears as a relevant model in various areas of physics: superfluids theory (see e.g. [14, 23]), weakly interacting Bose gases (see e.g. [18]), quantum semiconductors (see e.g. [15]). . . We would like also to point out that the above system belongs to a wider class

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B. Haspot
haspot@ceremade.dauphine.fr

M.-A. Vassenet
vassenet@ceremade.dauphine.fr

<footnote n="1">¹ Université Paris Dauphine, PSL Research University, Ceremade, Umr Cnrs 7534, Place du Maréchal De Lattre De Tassigny, 75775 Paris cedex 16, France</footnote>
of fluid dynamic equations, the so called Euler-Korteweg systems modeling in particular liquid-vapor mixture. In order to describe the behavior of the mixture at the interface between the fluid and the vapor, the system is endowed with the Korteweg tensor, which is intended to take into account capillary effects. The system reads as follows:

$$ egin{cases} 
\partial_t \rho + \text{div}(\rho u) = 0, \quad (x, t) \in \mathbb{R}^d \times I, \\
\partial_t (\rho u) + \text{div}(\rho u \otimes u) + \nabla P(\rho) = \text{div } K, \quad (x, t) \in \mathbb{R}^d \times \mathbb{R}^+, \\
(\rho, u)_{/t=0} = (\rho_0, u_0), \quad x \in \mathbb{R}^d.
\end{cases} 
\quad (1.2) $$

The general Korteweg tensor can be written as follows:

$$ \operatorname{div} K = \operatorname{div} \left( (\rho \kappa(\rho) \Delta \rho + \frac{1}{2}(\kappa(\rho) + \rho \kappa'(\rho)) |\nabla \rho|^2) I d - \kappa(\rho) \nabla \rho \otimes \nabla \rho \right). \quad (1.3) $$

The capillary coefficient $\kappa$ is a smooth function $\mathbb{R}^{+*} \to \mathbb{R}^{+*}$. We can notice that the quantum Euler system (1.1) corresponds to the Euler-Korteweg system (1.2) with $\kappa(\rho) = \frac{\kappa}{2\rho}$ with $\kappa > 0$.

One can recall now the physical energy bounds associated to the Euler-Korteweg system. Let $\bar{\rho} > 0$ be a constant reference density, and let $\Pi$ be defined by:

$$ \Pi(s) = s \left( \int_{\bar{\rho}}^{s} \frac{P(z)}{z^2} dz - \frac{P(\bar{\rho})}{\bar{\rho}} \right), $$

so that $P(s) = s\Pi'(s) - \Pi(s)$, $\Pi'(\bar{\rho}) = 0$ and:

$$ \partial_t \Pi(\rho) + \text{div}(u\Pi(\rho)) + P(\rho)\text{div}(u) = 0 \quad \text{in } \mathcal{D}'((0, T) \times \mathbb{R}^d). $$

Notice that $\Pi$ is convex as far as $P$ is non decreasing (since $P'(s) = s\Pi''(s)$), which is the case for $\gamma$-type pressure laws. We refer to [17] for non-monotone pressures. Multiplying the equation of momentum conservation in the system (1.2) by $u$ and integrating by parts over $\mathbb{R}^d$, we obtain at least formally:

$$ \begin{aligned} & \int_{\mathbb{R}^d} \left( \frac{1}{2} \rho |u|^2 + (\Pi(\rho) - \Pi(\bar{\rho})) + \frac{1}{2} \kappa(\rho) |\nabla \rho|^2 \right) (t) dx \\ & = \int_{\mathbb{R}^d} \left( \frac{1}{2} \rho_0 |u_0|^2 + (\Pi(\rho_0) - \Pi(\bar{\rho})) + \frac{\kappa(\rho_0)}{2} |\nabla \rho_0|^2 \right) dx. \end{aligned} \quad (1.4) $$

It follows that assuming that the initial total energy is finite:

$$ \mathcal{E}_0 = \int_{\mathbb{R}^d} \left( \rho_0 \frac{|u_0|^2}{2} + (\Pi(\rho_0) - \Pi(\bar{\rho})) + \frac{\kappa(\rho_0)}{2} |\nabla \rho_0|^2 \right) dx < +\infty, $$

then we have the a priori following bounds:

$$ \begin{aligned} & \Pi(\rho) - \Pi(\bar{\rho}), \quad \rho|u|^2 \in L^\infty(0, \infty, L^1(\mathbb{R}^d)), \\ & \text{and } \kappa(\rho)|\nabla\rho|^2 \in L^\infty(0, \infty, L^1(\mathbb{R}^d)). \end{aligned} \quad (1.5) $$

In the sequel for simplifying the notation, we will only consider a far field $\bar{\rho} = 1$. Before stating our main result, we propose for the convenience of the reader a short review on
the results concerning both the existence of strong solution for the Euler-Korteweg system and the existence of global weak solution with large initial data. The system (1.2) can be recasted as a quasilinear degenerate Schrödinger equation by using the classical effective velocity $v = u + i \sqrt{\frac{\kappa(\rho)}{\rho}} \nabla \rho$, the momentum equation takes then the following form:

$$ \partial_t v + u \cdot \nabla v + i(\nabla v) \cdot w + i \nabla (a(\rho) \text{div} v) + \nabla F(\rho) = 0, \quad (1.6) $$

with $F'(\rho) = \frac{P'(\rho)}{\rho}$, $w = \sqrt{\frac{\kappa(\rho)}{\rho}} \nabla \rho$ and $a(\rho) = \sqrt{\rho \kappa(\rho)}$. In other words we can expect dispersive estimates on the solutions of the Euler Korteweg system provided that we consider an irrotational velocity, indeed in this framework $\nabla(a(\rho)\text{div}v)$ is strongly elliptic. By working on the equation (1.6) Benzoni, Danchin and Descombes in [10] have proved the local existence of strong solutions for the Euler Korteweg system with large initial data provided that $(\rho_0 - 1, u_0)$ belong to $H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d)$ with $s > \frac{d}{2} + 1$ and the initial density is far away from the vacuum $\rho_0 \ge c > 0$. The proof is based on tricky energy and commutator estimates in $H^s$ Sobolev space of hyperbolic type combined with a gauge method which allows to avoid some loss of derivative on the convective terms. In particular in [10] the authors show that the $H^s$ regularity of the solution is preserved all along the time provided that they control the following quantity $\int_0^T \|\nabla u(s)\|_{L^\infty} ds$ which corresponds to the Lipschitz norm of the velocity in norm $L^1$ in time. It seems now reasonable to think that the existence of global strong solution can only be deduced by obtaining global dispersive estimates in time on the solution in order to control the norm of the gradient of the velocity in $L^1(\mathbb{R}^+, L^\infty(\mathbb{R}^d))$. It would allow in particular via the energy estimate of [10] to prove that the $H^s$ norm is preserved for any time $t > 0$. This is done in [7, 8] where the authors prove the existence of global strong solution which scatter in dimension $d \ge 3$, for small initial data with irrotational flow by using the theory of non space time resonance. It is relevant to mention that the case $d \le 2$ seems less favorable, indeed in [5] Audiard showed in dimension $d = 2$ the existence of travelling wave of arbitrary small energy (see [9] for the one-dimensional case). In particular since these solutions do not scatter, it may prevent some result of global strong scattering solution for small initial data for $d \le 2$. We refer also to [6] for the existence of multi-soliton of the Euler-Korteweg system in dimension one.

The question of the existence of global weak solution with large initial data for the Euler-Korteweg system remains actually largely open in dimension $d \ge 1$, we would like to explain what are the main difficulties related to the proof of such result. Classically the method for showing the existence of global weak solution consists in constructing in a first step a sequence of global approximate regular solution $(\rho_n, u_n)_{n \in \mathbb{N}}$ (via a Fadeo-Galerkin argument) satisfying uniformly in $n$ the energy estimate (1.4). The second step requires to prove that we can extract a subsequence $(\rho_{\varphi(n)}, u_{\varphi(n)})_{n \in \mathbb{N}}$ which converges weakly to a global weak solution $(\rho, u)$ of the Euler-Korteweg system (1.1). It corresponds to the notion of stability of the global weak solution, however it needs in general to have enough compactness informations on the sequel $(\rho_n, u_n)_{n \in \mathbb{N}}$ in order to pass to the limit in the nonlinear terms involved in the system (1.2). One can then easily notice that in the case of the Euler-Korteweg system the main difficulty consists in passing to the limit in the quadratic terms $\kappa(\rho_n) \nabla \rho_n \otimes \nabla \rho_n$ and $\rho_n u_n \otimes u_n$ which are only bounded via the energy estimate uniformly in $n$ in $L^\infty(L^1(\mathbb{R}^d))$. It implies that these terms converges up to extraction only to some measure $\nu$ and $v_1$, the natural question which arises is to prove that $\nu = \kappa(\rho) \nabla \rho \otimes \nabla \rho$ and $v_1 = \rho u \otimes u$ where $\rho$ and $\sqrt{\rho}u$ are strong limit of the sequences $(\rho_{\varphi(n)})_{n \in \mathbb{N}}$ and $(\sqrt{\rho_{\varphi(n)}} u_{\varphi(n)})_{n \in \mathbb{N}}$ in appropriate functional space.

In the case of the quantum Euler system (1.1), Antonelli and Marcati in [1, 2] have obtained the existence of global weak solution in dimension $d = 2, 3$ for “well-prepared”
initial data ($\rho_0$, $\sqrt{\rho_0}u_0$) in the sense that there exists some wave function $\psi_0 \in H^1$ such that $\rho_0 = |\psi_0|^2$ and $\rho_0 u_0 = J_0 = \text{Im}(\bar{\psi}_0 \nabla \psi_0)$. Under these conditions there exists a correspondence between the system (1.1) and the nonlinear Schrödinger equation via the notion of polar factorization where we have $\rho = |\psi|^2$ and $\rho u = \text{Im}(\bar{\Psi} \nabla \Psi)$ (see [1, 2]) with $\Psi$ solution of the following (NLS) equation:

$$i \partial_t \Psi + \frac{1}{2} \Delta \Psi = g(|\Psi|^2) \Psi, \quad \Psi(0, \cdot) = \Psi_0. \qquad (\text{NLS})$$

It is of course reminiscent of the so called *Madelung transform* which ensures that when $u = \nabla \phi$ is irrotational we have:

$$(\rho, \nabla \phi) \mapsto \Psi := \sqrt{\rho} e^{i\phi}. \qquad (1.7)$$

In [1, 2] the authors begin by constructing a sequence of exact solutions ($\rho_n$, $\sqrt{\rho_n}u_n$)$_{n \in \mathbb{N}}$ of the system (1.1) using polar factorization associated to $\Psi_n$ solution of the equation (NLS) provided that the sequence $(\Psi_{0,n})_{n \in \mathbb{N}}$ of initial data belong to $H^s$ Sobolev space with $s > 0$ sufficiently large (in addition this sequence converges strongly to $\Psi_0$ in $H^1$). We should however point out that this procedure must be realized carefully since $\psi_n$ can vanish, and so that $\sqrt{\rho_n}u_n = \text{Im}(\bar{\psi}_n \nabla \psi_n / |\psi_n|)$ is only defined for $|\psi_n| \neq 0$ and $\sqrt{\rho_n}u_n = 0$ otherwise using a Sard theorem. The key point of the proof is now the strong $L^1_{\text{loc}}$ convergence of the nonlinear terms $\sqrt{\rho_n}u_n \otimes \sqrt{\rho_n}u_n$ and $|\nabla \sqrt{\rho_n}|^2$, this is a consequence of standard stability estimates (based on Strichartz inequalities) which claim that $\Psi_n$ converges to $\Psi$ in $L^\infty_T(H^1)$ and of the fact that via the formula of polar factorization we have $|\nabla \Psi_n|^2 = |\nabla \sqrt{\rho_n}|^2 + |\sqrt{\rho_n}u_n|^2$. It suffices then now to prove the strong $L^2$ convergence by combining weak $L^2$ convergence and convergence of the $L^2$ norms $\|\sqrt{\rho_n}u_n\|_{L^2}$ and $\|\nabla \sqrt{\rho_n}\|_{L^2}$. Concerning the existence of global weak solution in one dimension for the system (1.1), we mention also the work of Antonelli et al [3] where the authors give in particular some sufficient and necessary condition for an initial data ($\rho_0$, $\sqrt{\rho_0}u_0$) to admit an associated wave function $\Psi_0$ in $H^1$ and even in $H^2$. It allows then to construct more regular global weak solution ($\rho$, $\sqrt{\rho}u$) provided that the initial associated wave function is in $H^2$, in addition they obtain also dispersive estimates on these solutions by adapting standard dispersive estimates issue of the NLS equation. In [4], the authors extend the previous result to the case of the dimension $d = 2$ for initial data endowed with point vortices. More precisely the velocity $u_0 = \frac{\partial_0}{\rho_0}$ where $J_0$ is the momentum is a distribution and satisfies curl $u_0 = 2\pi \sum_j k_j \delta_{x_j}$ where $\delta_{x_j}$ is the Dirac-delta function in $x_j$ and $(x_j)_{j \in \mathbb{N}}$ is a set of isolated points. Furthermore $\rho_0$ is continuous and positive everywhere except in the points $x_j$. Under these assumptions the authors develop a wave function lifting argument to ensure that the initial data are “well prepared” as in [1, 2] which implies the existence of global weak solution.

We can observe that the previous results are concerned with the far field $\bar{\rho} = 0$, in other words the following boundary condition $\lim_{|x| \to +\infty} \rho_0(x) = 0$ is satisfied (we refer to [19] for non trivial far field). Furthermore the initial velocity $u_0$ in [4] is irrotational except in the vacuum region where curl $u_0$ corresponds to Dirac masses. In particular it implies that in dimension $d \ge 2$ the problem of the existence of global weak solution for initial data which are generically irrotational remains open. We will discuss about this issue later.

To finish this short review, we mention also the global existence of dissipative solutions for the Euler-Korteweg system (1.2) which are even weaker than the weak solutions (see [11]). We also would like to refer to [12] for a very interesting survey on the well-posedness of the Euler-Korteweg system and Gross-Pitaevskii equations.
In the sequel, we aim at solving the problem of the stability of the global weak solution of the quantum Euler system (1.1) in dimension $d = 2$ for an initial velocity which is not necessary irrotational. In other words, if we assume that there exists a sequence of global smooth approximate solutions $(\rho_n, u_n)_{n \in \mathbb{N}}$ such that the initial data $(\rho_n^0, u_n^0)$ satisfy in $n$ the bounds (1.5) issue of the energy estimate (1.4), we wish to prove that up to a subsequence the sequel $(\rho_n, u_n)_{n \in \mathbb{N}}$ converges weakly to a global weak solution $(\rho, u)$ of the system (1.1).

**Remark 1.1** Naturally the stability of weak solutions does not imply the global existence of weak solutions. Indeed, it remains to show the global existence of approximate solutions $(\rho_n, u_n)$ satisfying uniformly in $n$ all the required energy estimates. This work is in general very delicate.

Dividing now the momentum equation of (1.1) by $\rho$, applying the operator curl in dimension $d = 2$ (with curl $u = \partial_1 u_2 - \partial_2 u_1$) and using the mass equation, we get at least formally the following equation:

$$ \partial_t \left( \frac{\text{curl } u}{\rho} \right) + u \cdot \nabla \left( \frac{\text{curl } u}{\rho} \right) = 0. \qquad (1.8) $$

This remarkable relation on $\frac{\text{curl } u}{\rho}$ has been identified in particular by Serre in [24] for the compressible Euler system in dimension $d = 2$. It implies that we can bound the $L^\infty(\mathbb{R}^2)$ and $L^1(\mathbb{R}^2)$ norm of respectively $\frac{\text{curl } u}{\rho}$ and curl $u$ all along the time by using a maximum principle. That is why in the sequel we shall work with initial data verifying:

$$ \frac{1}{\rho_0} \in L^\infty(\mathbb{R}^2) \quad \text{and} \quad \text{curl } u_0 \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2). \qquad (1.9) $$

In addition the initial data have finite energy, it implies from the estimate (1.4) that:

$$ \nabla \sqrt{\rho_0} \in L^2(\mathbb{R}^2), \quad \sqrt{\rho_0} u_0 \in L^2(\mathbb{R}^2) \quad \text{and} \quad (\rho_0 - 1) \in L_2^\gamma(\mathbb{R}^2), \qquad (1.10) $$

$L_2^\gamma(\mathbb{R}^2)$ is here an Orlicz space (we refer to the Sect. 2.1 for the definition). We assume now that there exists a sequence $(\rho_n, u_n)_{n \in \mathbb{N}}$ of global regular solutions of (1.1) with $\rho_n$ positive and satisfying the following properties:

$$ \| (\rho_n - 1, u_n) \|_{C(\mathbb{R}^+, H^s)} \le C_n^s, \quad \text{with} \quad C_n^s \underset{n \to +\infty}{\longrightarrow} +\infty, \forall s \ge 0, \qquad (1.11) $$

$$ \left\| \frac{1}{\rho_n} \right\|_{L^\infty(\mathbb{R}^+ \times \mathbb{R}^n)} \le C_n, \quad \text{with} \quad C_n \underset{n \to +\infty}{\longrightarrow} +\infty. $$

We expect also that $(\rho_n, u_n)_{n \in \mathbb{N}}$ verify for $C > 0$ independently on $n$ and any $n \in \mathbb{N}$ the energy estimates (1.4) and the maximum principle issue of the equation (1.8):

$$ \begin{align} \| \nabla \sqrt{\rho_n} \|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} &\le C, \quad \| \rho_n - 1 \|_{L^\infty(\mathbb{R}^+, L_2^\gamma(\mathbb{R}^2))} \le C, \nonumber \\ \| \sqrt{\rho_n} u_n \|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} &\le C, \quad \| \text{curl } u_n \|_{L^\infty(\mathbb{R}^+, L^1(\mathbb{R}^2))} \le C, \tag{1.12} \end{align} $$

$$ \left\| \frac{\text{curl } u_n}{\rho_n} \right\|_{L^\infty(\mathbb{R}^+, L^\infty(\mathbb{R}^2))} \le C. $$
Furthermore the initial data $(\rho_n^0, u_n^0)_{n \in \mathbb{N}}$ verify for $C > 0$ independent on $n$:

$$ \left\{ \begin{array}{l} \|\nabla \sqrt{\rho_n^0}\|_{L^2(\mathbb{R}^2)} \le C, \quad \|\sqrt{\rho_n^0} u_n^0\|_{L^2(\mathbb{R}^2)} \le C, \quad \|(\rho_n^0 - 1)\|_{L^2(\mathbb{R}^2)} \le C, \\ \| \mathrm{curl} \, u_n^0 \|_{L^1(\mathbb{R}^2)} \le C, \quad \| \mathrm{curl} \, u_n^0 \|_{L^\infty(\mathbb{R}^2)} \le C, \end{array} \right. \qquad (1.13) $$

and

$$ \begin{gathered} \|(\rho_n^0 - 1, u_n^0)\|_{H^s(\mathbb{R}^2)} \le C_n^s, \quad \text{with } C_n^s \underset{n \to +\infty}{\longrightarrow} +\infty, \forall s \ge 0, \\ \left\| \frac{1}{\rho_n^0} \right\|_{L^\infty(\mathbb{R}^2)} \le C. \end{gathered} \qquad (1.14) $$

To finish we assume that:

$$ \left\{ \begin{array}{l} \rho_n^0 \underset{n \to +\infty}{\longrightarrow} \rho_0 \quad \text{strongly in } L_{\mathrm{loc}}^1(\mathbb{R}^2), \\ \rho_n^0 u_n^0 \underset{n \to +\infty}{\longrightarrow} \rho_0 u_0 \quad \text{strongly in } L_{\mathrm{loc}}^1(\mathbb{R}^2), \end{array} \right. \qquad (1.15) $$

and

$$ \left\{ \begin{array}{l} \rho_n^0 - 1 \underset{n \to +\infty}{\longrightarrow} \rho_0 - 1 \quad \text{strongly in } L_2^\gamma(\mathbb{R}^2), \\ \nabla \sqrt{\rho_n^0} \underset{n \to +\infty}{\longrightarrow} \nabla \sqrt{\rho_0} \quad \text{strongly in } L^2(\mathbb{R}^2), \\ \sqrt{\rho_n^0} u_n^0 \underset{n \to +\infty}{\longrightarrow} \sqrt{\rho_0} u_0 \quad \text{strongly in } L^2(\mathbb{R}^2). \end{array} \right. \qquad (1.16) $$

**Remark 1.2** It is important to mention again that the question of the existence of global regular solution with large initial data for the Quantum Euler system in dimension $d = 2$ remains actually an open problem. That is why the sequence of global solution $(\rho_n, u_n)_{n \in \mathbb{N}}$ must be understood as a sequence of global regular solution of an approximate system of the Quantum Euler system which satisfies uniformly in $n$ all the estimates issue of the energy inequality (1.4). A classical method for getting such sequence of solution is to use a Fadego-Galerkine method, we will explain however why in our framework this type of construction seems particularly delicate.

In order to distinguish in a simple way the irrotational and the rotational part of the convective term $u \cdot \nabla u$ in (1.1) we are going to consider axisymmetric initial data, more precisely the initial data satisfy the following relation :<fnref n="1" />

$$ \left\{ \begin{array}{l} \rho_0(x) = \rho_0(|x|), \\ u_0(x) = \nabla\theta_0(|x|) + \nabla^\perp(\theta_1)_0(|x|), \end{array} \right. \qquad (1.17) $$

with $\rho_0$, $\theta_0$ and $(\theta_1)_0$ radial functions. In the sequel, we assume now that the initial data satisfy (1.17), (1.9) and (1.10). Since we are interested in looking at solutions of the form:

$$ \rho(t, x) = \rho_1(t, |x|), \quad u(t, x) = \nabla\theta(t, |x|) + \nabla^\perp\theta_1(t, |x|) = u^1(t, x) + u^2(t, x), \qquad (1.18) $$

<footnote n="1">¹We will note in dimension $d = 2$, $x^\perp = (-x_2, x_1)^t$ for $x = (x_1, x_2)^t$ and $\nabla^\perp = (-\partial_2, \partial_1)$.</footnote>
with $x \in \mathbb{R}^2 \setminus \{0\}$, we can then rewrite the system (1.1) at least formally as follows:

$$ egin{cases} 
\partial_t \rho + \text{div}(\rho u^1) = 0, \\
\rho \partial_t u^1 + \rho u^1 \cdot \nabla u^1 + \rho u^2 \cdot \nabla u^2 + \nabla P(\rho) = \kappa \rho \nabla \left( \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} \right), \\
\rho \partial_t u^2 + \rho u^1 \cdot \nabla u^2 + \rho u^2 \cdot \nabla u^1 = 0.
\end{cases} 
\quad (1.19) $$

Due to the assumption (1.18) on the initial data, we will assume that the initial data $(\rho_n^0, u_n^0)_{n \in \mathbb{N}}$ are also axisymmetric and verify:

$$ egin{cases} 
\rho_n^0(x) = \rho_n(0, |x|), \\
u_n^0(x) = \nabla \theta_0^n(|x|) + \nabla^\perp (\theta_1)^n_0(|x|).
\end{cases} 
\quad (1.20) $$

We can observe now that the rotational part of the velocity $u^1$ satisfies essentially the momentum equation of the Euler-Korteweg system (1.2) except that this equation is coupling with the convective term $\rho u^2 \cdot \nabla u^2$. It is then tempting at this level to consider this term as a source term and to apply a Madelung transformation to the equation on $u^1$ by considering the new unknown $\Psi_1 = \sqrt{\rho} e^{i\epsilon\theta_1}$ with $c \in \mathbb{R}$ a constant well chosen. We will see later that this is possible and that the unknown $\Psi_1$ satisfies the following equation:

$$ \partial_t \Psi_1 + i c_1 \Delta \Psi_1 + i c_2 \Psi_1 (F(|\Psi_1|^2) - F(1) + G) = 0, \quad (1.21) $$

with $c_1, c_2 \in \mathbb{R}$, $F$ a regular function depending on the pressure $P$ and $G$ a source term depending on the convective term $\rho u^2 \cdot \nabla u^2$.

Our main feature now consists in proving that up to a subsequence the sequence $(\rho_n, u_n)_{n \in \mathbb{N}}$ converges strongly to a global weak solution $(\rho, u)$ of the Quantum Euler system. As we mentioned previously the main difficulty consists in proving that the nonlinear terms $\rho_n u_n \otimes u_n$ and $\nabla \sqrt{\rho_n} \otimes \nabla \sqrt{\rho_n}$ converge in the sense of the distribution respectively to $\rho u \otimes u$ and $\nabla \sqrt{\rho} \otimes \nabla \sqrt{\rho}$ where $\rho$ and $u$ are strong limit of the sequences $(\rho_n)_{n \in \mathbb{N}}$ and $(u_n)_{n \in \mathbb{N}}$. Since curl $u_n$ should satisfy the equation (1.8), it is natural to think that that curl $u_n$ verifies the estimates (1.12). It implies in particular that the rotational part $Pu_n$ must belongs to Sobolev spaces $W^{1,p}(\mathbb{R}^2)$ with $p > 2$ via Sobolev embedding, using now compact embedding it is sufficient to show that the term $\rho_n Pu_n \otimes Pu_n$ converges strongly to $\rho Pu \otimes Pu$. The term the most delicate to treat is then the convective term $\rho_n Qu_n \otimes Qu_n$. At this level the use of axisymmetric solution is crucial since it enables to write a momentum equation on $Qu_n$ as in the system (1.19) where $u^1$ takes the place of $Qu_n$. We can then extend the so called Madelung transform by considering the unknown $\Psi_n = \sqrt{\rho_n} e^{i\epsilon\theta_n}$ with $Qu_n = \nabla\theta_n$ and $c \in \mathbb{R}$. Since $\Psi_n$ satisfies a nonlinear Schrödinger equation, we can use the Kato local smoothing effects on

$$ \nabla \Psi_n = (\nabla \sqrt{\rho_n} - \frac{i}{2c_2} (Qu_n) \sqrt{\rho_n}) e^{i\epsilon\theta_n}, $$

provided that we estimate $G_n$ in (1.21) in a suitable way. Then we use now the Aubin-Lions lemma to prove that the sequence $\rho_n Qu_n \otimes Qu_n$ converges to $\rho Qu \otimes Qu$. We recall now what is a global weak solution for the Quantum Euler system.

**Definition 1.1** We say that $(\rho, \Lambda, \Lambda_2)$ satisfying $\rho \ge 0$ and $\text{div}(\rho\Lambda_2) = 0$ in the sense of the distributions is said to be a global weak solution of the Quantum Euler system (1.1) with initial data $(\rho_0, u_0)$ with $\Lambda_0 = \sqrt{\rho_0} Qu_0$, $(\Lambda_2)_0 = Pu_0$ satisfying (1.9), (1.17), and (1.10) if:
(1) Integrability conditions:

$$ 
\begin{gathered}
\rho - 1 \in L^\infty(\mathbb{R}^+, L_2^\gamma(\mathbb{R}^2)), \quad \nabla\sqrt{\rho} \in L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2)), \\
\Lambda \in L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2)), \quad \sqrt{\rho}\Lambda_2 \in L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2)).
\end{gathered}
\quad (1.22) $$

(2) Kato's effect: For any $T > 0$ we have:

$$ \nabla\sqrt{\rho} \in L_T^2(H_{loc}^{\frac{1}{2}}), \quad \Lambda \in L_T^2(H_{loc}^{\frac{1}{2}}). \quad (1.23) $$

(3) Mass equation:

$$ \int_{\mathbb{R}^2} \rho_0 \varphi(0, \cdot) dx + \int_{\mathbb{R}_+ \times \mathbb{R}^2} \rho \partial_t \varphi dt dx + \int_{\mathbb{R}_+ \times \mathbb{R}^2} \sqrt{\rho} \Lambda \cdot \nabla \varphi dt dx = 0, \quad (1.24) $$

for any $\varphi \in C_c^\infty(\mathbb{R}^+ \times \mathbb{R}^2)$.

(4) Momentum equation:

$$ 
\begin{aligned}
& \int_{\mathbb{R}^2} \rho_0 u_0 \cdot \varphi(0, \cdot) dx + \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\rho \Lambda_2 + \sqrt{\rho} \Lambda) \cdot \partial_t \varphi dx dt \\
& + \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\Lambda \otimes \Lambda + \sqrt{\rho} \Lambda_2 \otimes \Lambda + \Lambda \otimes \sqrt{\rho} \Lambda_2 + \rho \Lambda_2 \otimes \Lambda_2) : \nabla \varphi dx dt \\
& + \int_{\mathbb{R}_+ \times \mathbb{R}^2} P(\rho) \operatorname{div}(\varphi) dx dt - \frac{\kappa}{2} \int_{\mathbb{R}_+ \times \mathbb{R}^2} \nabla \rho \cdot \nabla (\operatorname{div}(\varphi)) dx dt \\
& - 2\kappa \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\nabla \sqrt{\rho} \otimes \nabla \sqrt{\rho}) : \nabla \varphi dx dt = 0,
\end{aligned}
\quad (1.25) $$

holds for any test function $\varphi \in C_c^\infty(\mathbb{R}_+ \times \mathbb{R}^2)^2$.

(5) Energy inequality: if

$$ E(t) = \int_{\mathbb{R}^2} \left( \frac{1}{2} \rho |u|^2(t, x) + \kappa |\nabla \sqrt{\rho}(t, x)|^2 + j_\gamma(\rho(t, x)) \right) dx, $$

with $j_\gamma(\rho) = \frac{1}{\gamma-1}(\rho^\gamma - 1 - \gamma(\rho - 1))$ for $\gamma > 1$ then the following energy inequality is satisfied for any $t > 0$:

$$ E(t) \le E(0). $$

**Remark 1.3** It is important to point out that heuristically $\Lambda$ takes the role of $\sqrt{\rho}Qu$ whereas $\Lambda_2$ is the rotational part $Pu$.

We can now state our main result.

**Theorem 1** Let $d = 2$ and $(\rho_0, u_0)$ satisfying (1.9), (1.10), (1.17) and under the hypotheses (1.13), (1.14), (1.15), (1.16), (1.20) the sequence $(\rho_n, u_n)_{n \in \mathbb{N}}$ of global regular solution of the system (1.1) satisfying (1.11), (1.12) converges in the sense of distribution, up to a subsequence, to a global weak solution $(\rho, u)$ of system (1.1) in the sense specified in Definition 1.1, for initial data

$$ \rho(0, \cdot) = \rho_0, \quad \Lambda(0, \cdot) = \sqrt{\rho_0}Qu_0 \quad \text{and} \quad (\Lambda_2)_0 = Pu_0. $$
**Remark 1.4** This result of stability of global weak solution generalizes the works [1–4] to the case of initial data not necessarily irrotational when the density is not vanishing.

**Organization of the Article** First of all in Sect. 2.1 we present the a priori estimates. In Sect. 2.2 we show that the geometry of the solutions is conserved all along the time. In Sect. 2.3 we use an extended version of the Madelung transform when the velocity is not necessary irrotational in order to use in 2.4 Kato's local smoothing effect on the rotational part of the solution. In Sect. 2.5 we pass to the limit in the different nonlinear terms of the equations by using Aubin Lions lemma. Finally, in the last section we prove the Theorem 1.

## 2 Proof of the Theorem 1

We assume now that $(\rho_n, u_n)_{n \in \mathbb{N}}$ is a sequence of global regular solution of the system (1.1) satisfying (1.11), (1.12) with initial data verifying (1.13), (1.14), (1.15), (1.16) and (1.20). Our aim is to prove that this sequence converges up to a subsequence to a global solution $(\rho, u)$ of the Quantum Euler system with initial data $(\rho_0, u_0)$ satisfying the assumption of Theorem 1. We are going to start by proving uniform estimate.

### 2.1 Uniform Estimate

This section is devoted to the proof of the following propositions which give uniform estimates in $n$ on the sequence $(\rho_n, u_n)_{n \in \mathbb{N}}$.

**Proposition 2.1** There exist some positive constant $C > 0$, not depending on $n$, such that for all $t \ge 0$

$$ \|\nabla \sqrt{\rho_n}(t, \cdot)\|_{L^2} \le C, \quad \|\sqrt{\rho_n} u_n(t, \cdot)\|_{L^2} \le C, \qquad (2.1) $$

$$ \|\rho_n(t, \cdot) - 1\|_{L_2^\gamma} \le C, \qquad (2.2) $$

$$ \| \mathrm{curl} \, u_n(t, \cdot) \|_{L^1} \le C, \quad \| \frac{\mathrm{curl} \, u_n}{\rho_n}(t, \cdot) \|_{L^\infty} \le C. \qquad (2.3) $$

Next we deduce the following Corollary.

**Corollary 2.1** There exists some constant $C_p > 0$ depending only on $p$, such that for all $t \ge 0$

$$ \|\sqrt{\rho_n}(t, \cdot) - 1\|_{H^1} \le C, \qquad (2.4) $$

$$ \|\rho_n(t, \cdot) - 1\|_{L^p} \le C_p, \quad \forall p \in [2, +\infty[, \qquad (2.5) $$

$$ \| \mathrm{curl} \, u_n(t, \cdot) \|_{L^p} \le C_p, \quad \forall p \in [1, +\infty[, \qquad (2.6) $$

$$ \|\nabla \mathbb{P} u_n(t, \cdot)\|_{L^p} \le C_p, \quad \forall p \in ]1, +\infty[, \qquad (2.7) $$

$$ \| \mathbb{P} u_n(t, \cdot) \|_{W^{1,p}} \le C_p, \quad \forall p \in ]2, +\infty[. \qquad (2.8) $$
**Proof of Proposition 2.1 and Corollary 2.1** Multiplying the momentum equation of (1.1) by $u_n$ and integrating by parts on $(0, t) \times \mathbb{R}^2$ with $t > 0$ we obtain easily:

$$ 
\begin{aligned}
& \int_{\mathbb{R}^2} \left( \frac{1}{2} \rho_n |u_n|^2(t, x) + \kappa |\nabla \sqrt{\rho_n}(t, x)|^2 \right) dx + \int_0^t \int_{\mathbb{R}^2} \nabla P(\rho_n)(s, x) \cdot u_n(s, x) ds dx \\
& = \int_{\mathbb{R}^2} \left( \frac{1}{2} \rho_n |u_n|^2(0, x) + \kappa |\nabla \sqrt{\rho_n}(0, x)|^2 \right) dx.
\end{aligned}
\quad (2.9) $$

Using now the same computation as in [22], we can show that for a pressure $P(\rho) = \rho^\gamma$ we get:

$$ \int_{\mathbb{R}^2} \nabla P(\rho_n(t, x)) \cdot u_n(t, x) dx = \frac{1}{\gamma - 1} \int_{\mathbb{R}^2} \partial_t j_\gamma(\rho_n)(t, x) dx. \quad (2.10) $$

From the convexity of the function $x \to x^\gamma$ we can observe that $j_\gamma(\rho_n) \ge 0$ for $\rho_n \ge 0$. Combining (2.9) and (2.10) we have:

$$ 
\begin{aligned}
& \int_{\mathbb{R}^2} \left( \frac{1}{2} \rho_n |u_n|^2(t, x) + \kappa |\nabla \sqrt{\rho_n}(t, x)|^2 + j_\gamma(\rho_n(t, x)) \right) dx \\
& = \int_{\mathbb{R}^2} \left( \frac{1}{2} \rho_n |u_n|^2(0, x) + \kappa |\nabla \sqrt{\rho_n}(0, x)|^2 + j_\gamma(\rho_n(0, x)) \right) dx.
\end{aligned}
\quad (2.11) $$

According to [22] it is well known that the $L^1$ norm of $j_\gamma(\rho_n)$ is equivalent to $L_2^{\gamma 2}$ norm of $(\rho_n - 1)$, we deduce then from (2.11) and (1.13) that there exists $C > 0$ such that we have $\forall t \in \mathbb{R}$ and any $n \in \mathbb{N}$:

$$ 
\begin{aligned}
& \|\nabla \sqrt{\rho_n}(t, \cdot)\|_{L^2(\mathbb{R}^2)} \le C, \quad \|\sqrt{\rho_n} u_n(t, \cdot)\|_{L^2(\mathbb{R}^2)} \le C, \\
& \|(\rho_n - 1)(t, \cdot)\|_{L_2^2(\mathbb{R}^2)} \le C.
\end{aligned}
\quad (2.12) $$

For $\gamma \ge 2$ according to (2.12) it yields that $(\rho_n - 1)1_{\{|\rho_n - 1| \ge 1\}}$ is uniformly bounded in $n$ in $L^\infty(\mathbb{R}^+, L^\gamma(\mathbb{R}^2))$ and that $\{|[\rho_n(t, \cdot) - 1] \ge 1\}$ is uniformly bounded in $n$ in $L^\infty(\mathbb{R}^+)$. It implies in particular that $(\rho_n - 1)1_{\{|\rho_n - 1| \ge 1\}}$ is uniformly bounded in $n$ in $L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))$, using now the fact that $(\rho_n - 1)1_{\{|\rho_n - 1| \le 1\}}$ is in $L^\infty(\mathbb{R}^+, L^2)$ from (2.12) and the definition of Orlicz space we deduce that there exist $C > 0$ such that for any $t \in \mathbb{R}^+$ and any $n \in \mathbb{N}$:

$$ \int_{\mathbb{R}^2} (\rho_n(t, x) - 1)^2 dx \le C. \quad (2.13) $$

Writing $\sqrt{\rho_n} - 1 = \frac{\rho_n - 1}{\sqrt{\rho_n + 1}}$, we have using (2.13) and (2.12) for any $t \in \mathbb{R}^+$ and any $n \in \mathbb{N}$:

$$ \|\sqrt{\rho_n}(t, \cdot) - 1\|_{H^1(\mathbb{R}^2)} \le C. \quad (2.14) $$

For $1 < \gamma < 2$, we have also for all $t > 0$:

$$ \|\sqrt{\rho_n}(t, \cdot) - 1\|_{L^2(\mathbb{R}^2)} \le C. \quad (2.15) $$

Indeed from the definition of Orlicz space and (2.12), we know that $\rho_n - 1 = f_n + g_n$ with $f_n$ and $g_n$ are respectively uniformly bounded in $L^\infty(\mathbb{R}^+, L^\gamma(\mathbb{R}^2))$ and $L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))$,

<footnote n="2">²We refer to [22] for the definition of $L_2^\gamma$.</footnote>
now since $\sqrt{\rho_n} - 1 = \frac{\rho_n - 1}{\sqrt{\rho_n + 1}}$ it implies in a similar way that $\sqrt{\rho_n} - 1 = f'_n + g'_n$ with $f'_n$ and $g'_n$ respectively uniformly bounded in $L^\infty(\mathbb{R}^+, L^\gamma(\mathbb{R}^2))$ and $L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))$. From the Plancherel Theorem it suffices to bounded the $L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))$ norm of $\sqrt{\rho_n} - 1 = \widehat{f'_n} + \widehat{g'_n}$ to prove that $\sqrt{\rho_n} - 1$ is uniformly bounded in $L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))$. Using Riesz-Thorin, Plancherel Theorem and (2.12) we have for $C, C_1, C_2 > 0$ independent on $n$ and $t > 0$

$$ \begin{aligned} \| \widehat{f'_n(t, \cdot)}\|_{L^2(\mathbb{R}^2)} &\le C \| f'_n(t, \cdot) \|_{L^\gamma(\mathbb{R}^2)} \le C_1, \\ \| \widehat{g'_n(t, \cdot)}\|_{L^2(\mathbb{R}^2)} &\le C \| g'_n(t, \cdot) \|_{L^2(\mathbb{R}^2)} \le C_1. \end{aligned} \quad (2.16) $$

Moreover using (2.12) and Plancherel Theorem there exists $C > 0$ such that for any $n \in \mathbb{N}$ and $t > 0$:

$$ \| \mathcal{F}(\sqrt{\rho_n}(t, \cdot) - 1)\|_{L^2(\mathbb{R}^2)} \le \| \nabla \sqrt{\rho_n}(t, \cdot) \|_{L^2(\mathbb{R}^2)} \le C, \quad (2.17) $$

where $\mathcal{F}(f) = \hat{f}$. Using (2.16) and (2.17), we obtain that when $1 < \gamma < 2$ (2.15) is satisfied and in particular from (2.12) for any $n \in \mathbb{N}$ and $t > 0$:

$$ \| (\sqrt{\rho_n} - 1)(t, \cdot) \|_{L^2(\mathbb{R}^2)} \le C. \quad (2.18) $$

Combining (2.18) and (2.14), we have for all $\gamma > 1$

$$ \| \sqrt{\rho_n} - 1 \|_{L^\infty(\mathbb{R}^+, H^1(\mathbb{R}^2))} \le C. \quad (2.19) $$

Finally using Sobolev embedding, we have

$$ \| \sqrt{\rho_n} - 1 \|_{L^\infty(\mathbb{R}^+, L^p(\mathbb{R}^2))} \le C_p, \quad \forall p \ge 2. \quad (2.20) $$

Now since there exist $M > 0$ independent on $n$ such that

$$ |\rho_n - 1| \mathbb{1}_{|\rho_n - 1| \le 1} \le M|\sqrt{\rho_n} - 1|, \quad |\rho_n - 1| \mathbb{1}_{|\rho_n - 1| \ge 1} \le M|\sqrt{\rho_n} - 1|^2, $$

then using (2.20), we have:

$$ \| \rho_n - 1 \|_{L^\infty(\mathbb{R}^+, L^p(\mathbb{R}^2))} \le C_p, \quad \forall p \ge 2. \quad (2.21) $$

We wish now to prove the estimates on $\text{curl } u_n$ in Proposition 2.1 and in Corollary 2.1.

*Curl estimate:* We have seen that $\text{curl } u_n$ satisfies the two following equations:

$$ \partial_t \text{curl } u_n + \text{div}(\text{curl } u_n u_n) = 0, \quad (2.22) $$

and:

$$ \partial_t \left( \frac{\text{curl } u_n}{\rho_n} \right) + u_n \cdot \nabla \left( \frac{\text{curl } u_n}{\rho_n} \right) = 0. \quad (2.23) $$

According to the transport equation (2.23) and the conditions on the initial data (1.13) we obtain using classical maximum principle that:

$$ \left\| \frac{\text{curl } u_n}{\rho_n} \right\|_{L^\infty(\mathbb{R}^+, L^\infty(\mathbb{R}^2))} \le C. \quad (2.24) $$
Now using (2.22) we have

$$ \partial_t \mathrm{curl} u_n + u_n \cdot \nabla \mathrm{curl} u_n + \mathrm{curl} u_n \mathrm{div} u_n = 0. \quad (2.25) $$

Setting the function $\varphi_k(x) = \frac{x^2}{\sqrt{x^2+\frac{1}{k}}}$ with $k \in \mathbb{N}^*$ which approximate the absolute value and multiplying (2.23) by $\varphi'_k(\mathrm{curl} u_n)$ we obtain:

$$ \partial_t \varphi_k(\mathrm{curl} u_n) + u_n \cdot \nabla \varphi_k(\mathrm{curl} u_n) + \varphi'_k(\mathrm{curl} u_n) \mathrm{div} u_n \mathrm{curl} u_n = 0. \quad (2.26) $$

Let $f: \mathbb{R}_+ \to \mathbb{R}$ satisfy

$$ f(s) = \begin{cases} 1, & s \in [0, \frac{1}{2}], \\ \text{non-negative polynomial}, & s \in [\frac{1}{2}, 1], \\ 0, & s \in [1, \infty] \end{cases} $$

such that $f \in C^2$. We observe that

$$ \|f\|_{\infty} + \|f'\|_{\infty} < \infty, $$

and we define, for any $R > 0$, $f_R(x) = f(\frac{|x|}{R})$. Multiplying (2.26) by $f_R$ we obtain

$$ \partial_t \varphi_k(\mathrm{curl} u_n) f_R + u_n \cdot \nabla \varphi_k(\mathrm{curl} u_n) f_R + \varphi'_k(\mathrm{curl} u_n) \mathrm{div} u_n \mathrm{curl} u_n f_R = 0. \quad (2.27) $$

Integrating this equation we have:

$$ \int_{\mathbb{R}^2} (\partial_t \varphi_k(\mathrm{curl} u_n) f_R + u_n \cdot \nabla \varphi_k(\mathrm{curl} u_n) f_R + \varphi'_k(\mathrm{curl} u_n) \mathrm{div} u_n \mathrm{curl} u_n f_R) dx = 0. $$

After integrating by parts and using (1.11) we have

$$ \int_{\mathbb{R}^2} u_n \cdot \nabla \varphi_k(\mathrm{curl} u_n) f_R dx = - \int_{\mathbb{R}^2} \mathrm{div} u_n \varphi_k(\mathrm{curl} u_n) f_R dx - \int_{\mathbb{R}^2} u_n \cdot \nabla f_R \varphi_k(\mathrm{curl} u_n) dx. $$

So we have

$$ \int_{\mathbb{R}^2} \partial_t \varphi_k(\mathrm{curl} u_n) f_R dx + \int_{\mathbb{R}^2} \mathrm{div} u_n f_R (-\varphi_k(\mathrm{curl} u_n) + \varphi'_k(\mathrm{curl} u_n) \mathrm{curl} u_n) dx \\ - \int_{\mathbb{R}^2} u_n \cdot \nabla f_R \varphi_k(\mathrm{curl} u_n) dx = 0. \quad (2.28) $$

We have then obtained by integrating the previous estimate on $(0, t)$:

$$ \begin{aligned} & \int_{\mathbb{R}^2} \left( \frac{(\mathrm{curl} u_n)^2}{\sqrt{(\mathrm{curl} u_n)^2 + \frac{1}{k}}} \right) (t, x) f_R(x) dx + \frac{1}{k} \int_0^t \int_{\mathbb{R}^2} \mathrm{div} u_n f_R \frac{\mathrm{curl}^2 u_n}{(\mathrm{curl}^2 u_n + \frac{1}{k})^{\frac{3}{2}}} dx dt \\ & - \int_0^t \int_{\mathbb{R}^2} u_n \cdot \nabla f_R \left( \frac{(\mathrm{curl} u_n)^2}{\sqrt{(\mathrm{curl} u_n)^2 + \frac{1}{k}}} \right) dx dt = \int_{\mathbb{R}^2} \left( \frac{(\mathrm{curl} u_n)^2}{\sqrt{(\mathrm{curl} u_n)^2 + \frac{1}{k}}} \right) (0, x) f_R(x) dx. \end{aligned} \quad (2.29) $$
Using Lebesgue theorem and since $f_R$ is in $L^1(\mathbb{R}^2)$, we observe that

$$ \lim_{k \to \infty} \int_{\mathbb{R}^2} \left( \frac{(\mathrm{curl} u_n)^2}{\sqrt{(\mathrm{curl} u_n)^2 + \frac{1}{k}}} \right) (t, x) f_R(x) dx = \int_{\mathbb{R}^2} |\mathrm{curl} u_n|(t, x) f_R(x) dx. \quad (2.30) $$

Now since $\frac{1}{k} \frac{\mathrm{curl}^2 u_n}{(\mathrm{curl}^2 u_n + \frac{1}{k})^{\frac{3}{2}}} \le \frac{1}{\sqrt{k}}$ and $\mathrm{div} u_n f_R$ is in $L^1([0, t] \times \mathbb{R}^2)$ using (1.11) and the fact that $f_R$ is in $L^1(\mathbb{R}^2)$, we deduce that:

$$ \frac{1}{k} \int_0^t \int_{\mathbb{R}^2} \mathrm{div} u_n f_R \frac{\mathrm{curl}^2 u_n}{(\mathrm{curl}^2 u_n + \frac{1}{k})^{\frac{3}{2}}} dx dt \underset{k \to +\infty}{\longrightarrow} 0. \quad (2.31) $$

Similarly since $\frac{|\mathrm{curl} u_n|}{\sqrt{(\mathrm{curl} u_n)^2 + \frac{1}{k}}} \le 1$ and $u_n \cdot \nabla f_R \mathrm{curl} u_n \in L^1([0, t] \times \mathbb{R}^2)$ from (1.11) and because $\nabla f_R \in L^1(\mathbb{R}^2)$, we have by Lebesgue theorem

$$ \int_0^t \int_{\mathbb{R}^2} u_n \cdot \nabla f_R \left( \frac{(\mathrm{curl} u_n)^2}{\sqrt{(\mathrm{curl} u_n)^2 + \frac{1}{k}}} \right) dx dt \underset{k \to +\infty}{\longrightarrow} \int_0^t \int_{\mathbb{R}^2} u_n \cdot \nabla f_R |\mathrm{curl} u_n| dx dt. \quad (2.32) $$

From (1.13) we have $|\mathrm{curl} u_n|(0, \cdot) f_R(\cdot) \in L^1(\mathbb{R}^2)$ and from Lebesgue Theorem, we deduce that

$$ \int_{\mathbb{R}^2} \left( \frac{(\mathrm{curl} u_n)^2}{\sqrt{(\mathrm{curl} u_n)^2 + \frac{1}{k}}} \right) (0, x) f_R(x) dx \underset{k \to +\infty}{\longrightarrow} \int_{\mathbb{R}^2} |\mathrm{curl} u_n|(0, x) f_R(x) dx. \quad (2.33) $$

Thus passing to the limit when $k \to \infty$ in (2.29) we have

$$ \int_{\mathbb{R}^2} |\mathrm{curl} u_n|(t, x) f_R(x) dx = \int_{\mathbb{R}^2} |\mathrm{curl} u_n|(0, x) f_R(x) dx \\ + \int_0^t \int_{\mathbb{R}^2} u_n \cdot \nabla f_R |\mathrm{curl} u_n| dx dt. \quad (2.34) $$

Then since there exists a constant $C > 0$ independent on $R$ such that $\|\nabla f_R\|_{L^\infty(\mathbb{R}^2)} \le \frac{C}{R}$ and from (1.11) we deduce that:

$$ \lim_{R \to +\infty} \int_0^t \int_{\mathbb{R}^2} u_n \cdot \nabla f_R |\mathrm{curl} u_n| dx dt = 0. \quad (2.35) $$

Passing to the limit when $R \to \infty$ in (2.34) and using the Fatou Lemma and the fact that $|\mathrm{curl} u_n|(0, \cdot) \in L^1(\mathbb{R}^2)$ we obtain:

$$ \int_{\mathbb{R}^2} |\mathrm{curl} u_n|(t, x) dx \le \int_{\mathbb{R}^2} |\mathrm{curl} u_n|(0, x) dx. $$

We obtain using (1.13) that there exists $C > 0$ such that for any $t > 0$ and $n \in \mathbb{N}$:

$$ \int_{\mathbb{R}^2} |(\mathrm{curl} u_n)|(t, \cdot) dx \le C. $$
So it implies that:

$$ \| \mathrm{curl} \, u_n \|_{L^\infty(\mathbb{R}^+, L^1(\mathbb{R}^2))} \le C. \qquad (2.36) $$

Now, we can prove that for any $t > 0$ we have:

$$ \forall p \ge 1, \quad \| \mathrm{curl} \, u_n(t, \cdot) \|_{L^p} \le C_p. \qquad (2.37) $$

Indeed we have for $p \ge 1$

$$ \int_{\mathbb{R}^2} |\mathrm{curl} \, u_n(t, x)|^p dx \le \int_{\mathbb{R}^2} |\mathrm{curl} \, u_n(t, x)| 1_{\{|\mathrm{curl} \, u_n(t, x)| \le 1\}} dx \\ + \int_{\{|\mathrm{curl} \, u_n(t, y)| \ge 1\}} \frac{|\mathrm{curl} \, u_n(t, x)|^p}{\rho_n(t, x)^{p-\frac{1}{2}}} \rho_n(t, x)^{p-\frac{1}{2}} dx. \qquad (2.38) $$

From (2.24), (2.36) and by interpolation it yields that $\frac{\mathrm{curl} \, u_n}{\rho_n^{\frac{p-1}{p}}}$ is uniformly bounded in $n$ in $L^\infty(\mathbb{R}^+, L^p(\mathbb{R}^2))$ for any $p \ge 1$. It implies that there exists $C > 0$ such that for any $t > 0$ we have:

$$ \left\| \frac{|\mathrm{curl} \, u_n|^p}{\rho_n^{p-\frac{1}{2}}} \right\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} \le C. \qquad (2.39) $$

Now we observe again that for $p > \frac{3}{2}$ there exists $M_p \in \mathbb{R}^+$ independent on $n$ such that:

$$ |\rho_n^{p-\frac{1}{2}} - 1| \le M_p |\rho_n - 1| + M_p |\rho_n - 1|^{p-\frac{1}{2}}. \qquad (2.40) $$

Combining (2.40) and (2.21), we deduce that for $p > \frac{3}{2}$, $(\rho_n^{p-\frac{1}{2}} - 1)_{n \in \mathbb{N}}$ is uniformly bounded in $n$ in $L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))$. From (2.39) and the fact that $|\{|\mathrm{curl} \, u_n(t, \cdot)| \ge 1\}|$ is uniformly bounded in $n$ in $L^\infty(\mathbb{R}^+)$ using (2.36), it implies that for $p > \frac{3}{2}$ and any $t > 0$:

$$ \begin{aligned} & \int_{\{|\mathrm{curl} \, u_n(t, y)| \ge 1\}} \frac{|\mathrm{curl} \, u_n(t, x)|^p}{\rho_n(t, x)^{p-\frac{1}{2}}} \rho_n(t, x)^{p-\frac{1}{2}} dx \\ \le & \left\| \frac{|\mathrm{curl} \, u_n|^p}{\rho_n^{p-\frac{1}{2}}} \right\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} \|\rho_n^{p-\frac{1}{2}} - 1\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} \\ & + \int_{\{|\mathrm{curl} \, u_n(t, y)| \ge 1\}} \frac{|\mathrm{curl} \, u_n(t, x)|^p}{\rho_n(t, x)^{p-\frac{1}{2}}} \rho_n(t, x)^{p-\frac{1}{2}} dx \\ \le & C. \end{aligned} \qquad (2.41) $$

So we have, using (2.36), (2.38), (2.41) and by interpolation for any $t > 0$

$$ \forall p \ge 1, \quad \| \mathrm{curl} \, u_n(\cdot, \cdot) \|_{L^\infty(L^p)} \le C_p. \qquad (2.42) $$

Moreover, we have

$$ \nabla P u_n = \nabla \Delta^{-1} \nabla^{\perp} \mathrm{curl} \, u_n, \qquad (2.43) $$
then using (2.42) and the continuity of the Riez transform in $L^p$ for $p > 1$ we have:

$$ \|\nabla P u_n\|_{L^\infty(\mathbb{R}^+, L^p(\mathbb{R}^\pm))} \le C_p, \quad \forall p \in [1, +\infty]. \qquad (2.44) $$

Finally using Sobolev embedding we deduce from (2.44) that:

$$ \|P u_n\|_{L^\infty(\mathbb{R}^+, W^{1,p})} \le C_p, \quad \forall p \in [2, +\infty]. \qquad (2.45) $$

□

## 2.2 Symmetry of the Solutions

We wish now to prove that for any $n \in \mathbb{N}$, $(\rho_n, u_n)$ have the form

$$ \begin{cases} \rho_n(t, x) = \rho_n(t, |x|), \\ u_n(t, x) = \nabla[\theta_n(t, |x|)] + \nabla^\perp[\theta_{1,n}(t, |x|)], \quad t \in \mathbb{R}^+. \end{cases} \qquad (2.46) $$

We can verify that for any rotation matrix $A$ satisfying $A A' = A' A = I$ and $\det A = 1$, we have since $(\rho_n, u_n)$ satisfies (1.1):

$$ \left\{ \begin{aligned} & \partial_t \rho_n(t, A^t x) + \operatorname{div}(\rho_n(t, A^t x)Au_n(t, A^t x)) = 0, \\ & \rho_n(t, A^t x)\partial_t(Au_n(t, A^t x)) + \rho_n(t, A^t x)Au_n(t, A^t x)) \cdot \nabla(Au_n(t, A^t x)) \\ & + \nabla P(\rho_n(t, A^t x)) = \kappa \rho_n(t, A^t x) \nabla \left( \frac{\Delta \sqrt{\rho_n(t, A^t x)}}{\sqrt{\rho_n(t, A^t x)}} \right). \end{aligned} \right. \qquad (2.47) $$

From (1.20), we can observe that:

$$ \begin{cases} \rho_n(0, x) = \rho_n(0, A^t x), \\ u_n(0, x) = Au_n(0, A^t x), \end{cases} \qquad (2.48) $$

then since $(\rho_n, u_n)$ are strong solution of an approximate system of (1.1) we get by uniqueness that for all $t \in \mathbb{R}_+$ we have for any $x \in \mathbb{R}^2$:

$$ \begin{cases} \rho_n(t, x) = \rho_n(t, A^t x), \\ u_n(t, x) = Au_n(t, A^t x). \end{cases} \qquad (2.49) $$

**Remark 2.1** We recall that for the Euler-Korteweg system we have uniqueness of the solution $(\rho_n, u_n)$ in $C([0, T]; H^{s+1}(\mathbb{R}^2) \times H^s(\mathbb{R}^2)) \cap C^1([0, T]; H^{s-1}(\mathbb{R}^2) \times H^{s-2}(\mathbb{R}^2))$ for any $T > 0$ and with $s > \frac{d}{2} + 1$ (see [10]).

From (2.49) we deduce that $\rho_n$ is radial or that for any $t \ge 0$:

$$ \rho_n(t, x) = \rho_n(t, |x|). \qquad (2.50) $$

Let $x \in \mathbb{R}^2 \setminus \{0\}$ we write $u_n(t, x)$ in the basis $(\frac{x}{|x|}, \frac{x^\perp}{|x|})$ and we have:

$$ u_n(t, x) = f^{1,n}(t, x) \frac{x}{|x|} + f^{2,n}(t, x) \frac{x^\perp}{|x|}. $$
Then using (2.49), we have (see [16] Sect. 3)

$$f^{1,n}(t, x) \frac{x}{|x|} + f^{2,n}(t, x) \frac{x^{\perp}}{|x|} = f^{1,n}(t, A^t x) \frac{AA^t x}{|A^t x|} + f^{2,n}(t, A^t x) \frac{A A^t x^{\perp}}{|A^t x|}.$$

Since A is a rotation we deduce that $f^{1,n}(t, x) = f^{1,n}(t, A^t x)$ and $f^{2,n}(t, x) = f^{2,n}(t, A^t x)$. So, we have for $x \neq 0$:

$$u_n(t, x) = f^{1,n}(t, |x|) \frac{x}{|x|} + f^{2,n}(t, |x|) \frac{x^{\perp}}{|x|}. \quad (2.51)$$

Since for any $n \in \mathbb{N}$, $u_n(t, \cdot)$ is continuous for any $t \ge 0$, we can prove that $u_n(t, 0) = 0$. Indeed it suffices to consider the sequences $x_k = t(\frac{1}{k}, 0)$ and $y_k = t(-\frac{1}{k}, 0)$ for $k \in \mathbb{N}^*$, and using (2.51), we deduce that:

$$\begin{gathered} t(f^{1,n}(t, \frac{1}{k}), f^{2,n}(t, \frac{1}{k})) \to_{k \to +\infty} u_n(t, 0) \\ \text{and } t(-f^{1,n}(t, \frac{1}{k}), -f^{2,n}(t, \frac{1}{k})) \to_{k \to +\infty} u_n(t, 0). \end{gathered}$$

We deduce that $u_n(t, 0) = 0$ for any $n \in \mathbb{N}$ and $t \ge 0$. It implies also that $f^{1,n}(t, \cdot)$ and $f^{2,n}(t, \cdot)$ have zero for limit when $r$ goes to 0 and in addition we can prove that $f^{1,n}(t, \cdot)$ and $f^{2,n}(t, \cdot)$ are continuous for any $r > 0$ (to see this we simple use the following formula $f^{1,n}(t, r) = \langle u_n(t, x), \frac{x}{|x|} \rangle, f^{2,n}(t, r) = \langle u_n(t, x), \frac{x^+}{|x|} \rangle$ for $x \in \mathbb{R}^2 \setminus \{0\}$ such that $|x| = r$). We have finally prove that $f^{1,n}(t, \cdot)$ and $f^{2,n}(t, \cdot)$ are continuous on $\mathbb{R}^+$. It implies that $(\rho_n, u_n)$ have the form (2.46) for $t \in \mathbb{R}_+$ with:

$$\theta_n(t, r) = \int_0^r f^{1,n}(t, r')dr' \quad \text{and} \quad \theta_{1,n}(t, r) = \int_0^r f^{2,n}(t, r')dr'. \quad (2.52)$$

In addition since $f^{1,n}(t, |x|) = \langle u_n(t, x), \frac{x}{|x|} \rangle = \psi_n(t, x)$ for $x \in \mathbb{R}^2 \setminus \{0\}$ is $C^\infty$ on $\mathbb{R}^2 \setminus \{0\}$ by using the fact that $u_n(t, \cdot)$ is $C^\infty$ on $\mathbb{R}^2$, we deduce that $f^{1,n}$ is $C^\infty$ on $\mathbb{R}^+$. We have in particular for $r > 0$:

$$(f^{1,n})'(t, r) = \lim_{h \to 0} \frac{\psi_n(t, (r+h)e_1) - \psi_n(t, re_1)}{h} = \partial_1 \psi_n(t, re_1).$$

We deduce then that $\theta_n(t, \cdot)$ and $\theta_{1,n}(t, \cdot)$ are $C^\infty$ on $\mathbb{R}^{+*}$ for any $t \ge 0$.

## 2.3 Extended Madelung Transform

In this section we forget the subscript $n$ to simplify the notations. We set $c_2 = \sqrt{\frac{K}{2}}$ and considering as in [7, 8, 10] the following effective velocity $z = u + ic_2 \nabla \ln \rho$ we can rewrite the momentum equation in (1.1) as follows

$$\rho \partial_t z + \rho u \cdot \nabla z + \nabla P(\rho) + ic_2 \rho \nabla \text{div} z + ic_2 \nabla \rho \cdot \nabla z = 0.$$

Dividing by $\rho$ (which has sense from the estimate (1.11)) we obtain with $F'(\rho) = \frac{P'(\rho)}{\rho}$:

$$\partial_t z + u \cdot \nabla z + \nabla F(\rho) + ic_2 \nabla \text{div} z + z^t \nabla z - u^t \nabla z = 0.$$
Let $z^2 = (z_1)^2 + (z_2)^2$ with $z = (z_1, z_2)'$ and curl $u = \partial_1 u_2 - \partial_2 u_1$ we can rewrite the previous equation as follows:

$$ \partial_t z + u^{\perp} \mathrm{curl} u + \frac{1}{2} \nabla z^2 + \nabla F(\rho) + ic_2 \nabla \mathrm{div} z = 0. \quad (2.53) $$

We wish now to apply the operator $\mathbb{Q}$ to the equation (2.53), to do this we need to know what is $\mathbb{Q}(u^{\perp} \mathrm{curl} u)$. We are going to take advantage of the fact that the solutions are axisymmetric, indeed using (2.46) we have for any $t \ge 0$ and $x \in \mathbb{R}^2 \setminus \{0\}$:

$$ \left\{ \begin{array}{l} \rho(t, x) = \rho(t, |x|), \\ u(t, x) = \nabla\theta(t, |x|) + \nabla^{\perp}\theta_1(t, |x|), \\ z(t, x) = \nabla\psi(t, |x|) + \nabla^{\perp}\theta_1(t, |x|), \end{array} \right. $$

with $\psi(t, |x|) = \theta(t, |x|) + ic_2 \ln \rho(t, |x|)$. We can observe now that:

$$ u^{\perp} \mathrm{curl} u = (\nabla^{\perp}\theta - \nabla\theta_1) \Delta\theta_1. $$

Using again (2.46) we have for any $t \ge 0$, $x \in \mathbb{R}^2 \setminus \{0\}$:

$$ \mathbb{Q}(u^{\perp} \mathrm{curl} u) = -\Delta\theta_1 \nabla\theta_1 = \mathrm{curl} u (\mathbb{P}u)^{\perp} = \nabla\theta_2(t, |x|). \quad (2.54) $$

Indeed we have simply used the fact that $\Delta\theta_1\theta_1'$ admits a primitive on $\mathbb{R}^{+*}$. In particular we have:

$$ \theta_2 = (\Delta)^{-1} \mathrm{div}(\mathrm{curl} u (\mathbb{P}u)^{\perp}). \quad (2.55) $$

Moreover, we have using the fact that $\psi$ and $\theta_1$ are radial, we have for $x \in \mathbb{R}^2 \setminus \{0\}$:

$$ \begin{aligned} z^2 &= (\nabla\psi)^2 + (\nabla\theta_1)^2 + 2\partial_2\psi\partial_1\theta_1 - 2\partial_1\psi\partial_2\theta_1 \\ &= (\nabla\psi)^2 + (\nabla\theta_1)^2. \end{aligned} \quad (2.56) $$

Applying $\mathbb{Q}$ to the equation (2.53) and using (2.54), (2.56), we have in $\mathbb{R}^2 \setminus \{0\}$:

$$ \nabla \left( \partial_t \psi + \theta_2 + \frac{1}{2} (\nabla \psi)^2 + \frac{1}{2} (\nabla \theta_1)^2 + F(\rho) - F(1) + ic_2 \Delta \psi \right) = 0. \quad (2.57) $$

Thus, there exists a function $B$, such that for any $t \ge 0$ we have:

$$ (\partial_t \psi + \theta_2 + \frac{1}{2}(\nabla \psi)^2 + \frac{1}{2}(\nabla \theta_1)^2 + F(\rho) - F(1) + ic_2 \Delta \psi)(t, \cdot) = B(t). \quad (2.58) $$

At this level since $\rho$, $\mathbb{P}u$ and $\mathbb{Q}u$ are regular solutions which satisfy:

$$ \lim_{|x| \to +\infty} \rho(t, x) = 1, \quad \lim_{|x| \to +\infty} \partial^\alpha \mathbb{P}u(t, x) = \lim_{|x| \to +\infty} \partial^\alpha \mathbb{Q}u(t, x) = \lim_{|x| \to +\infty} \partial^\beta \rho(t, x) = 0, $$

for any $\alpha \in \mathbb{N}^3$ and $\beta \in \mathbb{N}^3$ with $|\beta| \ge 1$ where the first derivative of $\partial^{\alpha_1}$, $\partial^{\beta_1}$ are derivative on time. To see this it suffices to observe that $\mathbb{Q}u(t, x) = \langle u(t, x) \frac{x}{|x|} \rangle$, $\frac{x}{|x|}$ when $x \ne 0$ and to use the mass and the momentum equation of (1.1). In particular it implies that:

$$ \lim_{|x| \to +\infty} \left( \frac{1}{2} (\nabla \theta_1)^2 + \frac{1}{2} (\nabla \psi)^2 + F(\rho) - F(1) + ic_2 \Delta \psi \right) (t, x) = 0, $$

$$ \lim_{|x| \to +\infty} \mathrm{Im}(\partial_t \psi + \theta_2)(t, x) = \lim_{|x| \to +\infty} \mathrm{Im}(\partial_t \psi)(t, x) = \lim_{|x| \to +\infty} c_2 \ln \rho(t, |x|) = 0. $$
It implies that Im((∂_tψ + θ_2 + 1/2(∇ψ)^2 + 1/2(∇θ_1)^2 + F(ρ) - F(1) + ic_2Δψ)(t, ·)) = 0 for t > 0, we deduce then that B(t) is a real.

**Remark 2.2** It is interesting to note that we can assume that $B(t) = 0$ for any $t > 0$ provided that we assume that $u_n$ is sufficiently decreasing at infinity and by setting $θ_n(t, r) = - \int_r^{+\infty} f^{1,n}(t, r')dr'$.

We set $ψ_1(t) = ψ(t) - a(t)$ with $a'(t) = B(t)$ (we can observe that $B$ is continuous for $t > 0$), then for $x \in \mathbb{R}^2 \setminus \{0\}$

$$ \partial_t \psi_1 + \theta_2 + \frac{1}{2}(\nabla \psi_1)^2 + \frac{1}{2}(\nabla \theta_1)^2 + F(\rho) - F(1) + ic_2 \Delta \psi_1 = 0. \quad (2.59) $$

Using Cole-Hopf Transform, we obtain, multiplying the previous equation by $f'$ with $f$ a function that we will chose later:

$$ \begin{aligned} \partial_t f(\psi_1) + ic_2 \Delta f(\psi_1) + \left(\frac{1}{2} f'(\psi_1) - ic_2 f''(\psi_1)\right) (\nabla \psi_1)^2 + \frac{1}{2} f'(\psi_1) (\nabla \theta_1)^2 \\ + (F(\rho) - F(1) + \theta_2) f'(\psi_1) = 0. \end{aligned} $$

We set now $w = f(\psi_1) = e^{-\frac{i}{2c_2} \psi_1} = e^{-\frac{i}{2c_2} \theta + \frac{1}{2} \ln \rho + \frac{i}{2c_2} a(t)} = \sqrt{\rho} e^{-\frac{i}{2c_2} \theta + \frac{i}{2c_2} a}$, and we obtain:

$$ \partial_t w + ic_2 \Delta w - \frac{i}{4c_2} w |\nabla \theta_1|^2 - (F(|w|^2) - F(1) + \theta_2) \frac{i}{2c_2} w = 0. $$

Finally we have a modified Gross-Pitaevskii equation using that $\nabla^\perp \theta_1 = \mathbb{P}u$ for $x \in \mathbb{R}^2 \setminus \{0\}$:

$$ \partial_t w + ic_2 \Delta w - \frac{i}{2c_2} w(F(|w|^2) - F(1) + \frac{1}{2}\mathbb{P}u|^2 + \theta_2) = 0, \quad (2.60) $$

with

$$ \theta_2 = (\Delta)^{-1} \operatorname{div}(\operatorname{curl} u (\mathbb{P}u)^\perp). $$

## 2.4 Kato's Local Smoothing Effect

In this section we are going to use Kato's local smoothing effect. It will be the key point to prove strong convergence of the sequence $(\nabla \sqrt{\rho_n}, \sqrt{\rho_n}u_n)_{n \in \mathbb{N}}$ up to a subsequence in $L^2_{\text{loc}}(\mathbb{R}^+ \times \mathbb{R}^2)$. More precisely, we prove the following Proposition.

**Proposition 2.2** let $\chi \in C_c^\infty(\mathbb{R}^2)$ and $T > 0$ then there exists a constant $C_T$ not depending on $n$, such that $\forall n \in \mathbb{N}$ and for any $j \in \{1, 2\}$:

$$ \| \chi \partial_j w_n \|_{L^2_T(H^{1/2})} \le C_T. \quad (2.61) $$

First we need to prove the following lemma:

**Lemma 2.1** Let $\chi \in C_c^\infty(\mathbb{R}^2)$, for any $T > 0$ there exists $C_T > 0$ such that for all $f \in L^2(\mathbb{R}^2)$ we have for $0 < t \le T$

$$ \| \chi e^{it\Delta} f \|_{L^2_T(H^{1/2})} \le C_T \| f \|_{L^2}. \quad (2.62) $$
There exists $C'_T > 0$ such that for all $f \in L_T^{p'}(L^{q'}(\mathbb{R}^2))$ with $(p, q)$ an admissible Strichartz pair (i.e. $\frac{1}{p} + \frac{1}{q} = \frac{1}{2}$ and $2 \le q < +\infty$) we have for $0 < t \le T$:

$$ \left\| \int_0^t \chi e^{i(t-s)\Delta} f(s) ds \right\|_{L_T^2(H^{1/2})} \le C'_T \|f\|_{L_T^{p'}(L^{q'}(\mathbb{R}^2))} \quad (2.63) $$

**Proof of Lemma 2.1** Let $\chi \in C_c^\infty(\mathbb{R}^2)$ and $T > 0$ using [21] corollary 4.2 there exists $C > 0$ such that

$$ \int_{\mathbb{R}} \int_{B(0,R)} |D_x^{1/2} e^{it\Delta} f|^2 dxdt \le C R \|f\|_{L^2(\mathbb{R}^2)}^2, \forall f \in L^2(\mathbb{R}^2), \quad (2.64) $$

with $\mathcal{F}D_x^{1/2}f(\xi) = |\xi|^{1/2}\mathcal{F}f(\xi)$. Thus

$$ \int_{\mathbb{R}} \int_{\mathbb{R}^2} |\chi D_x^{1/2} e^{it\Delta} f|^2 dxdt \le C \|f\|_{L^2(\mathbb{R}^2)}^2, \forall f \in L^2(\mathbb{R}^2). \quad (2.65) $$

Then we have from (2.64) and Strichartz estimates

$$ \begin{aligned} \int_{\mathbb{R}} \int_{\mathbb{R}^2} \left| \int_{\mathbb{R}} \chi D_x^{1/2} e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt &\le \int_{\mathbb{R}} \int_{\mathbb{R}^2} \left| \chi D_x^{1/2} e^{it\Delta} \int_{\mathbb{R}} e^{-is\Delta} f(s) ds \right|^2 dxdt \\ &\le C \left\| \int_{\mathbb{R}} e^{-is\Delta} f(s) ds \right\|_{L^2(\mathbb{R}^2)}^2 \quad \text{from (2.64)} \\ &\le C_1 \|f\|_{L^{p'}(\mathbb{R}, L^{q'}(\mathbb{R}^2))}^2, \end{aligned} \quad (2.66) $$

with $(p, q)$ an admissible Strichartz pair i.e. $\frac{1}{p} + \frac{1}{q} = \frac{1}{2}$ and $2 \le q < +\infty$. We wish to use the Christ-Kiselev Theorem that we recall here.

**Theorem 2 (Christ-Kiselev, [13])** Consider a bounded operator

$$ T : L^{p_1}(\mathbb{R}; B_1) \to L^{q_1}(\mathbb{R}; B_2), $$

given by a locally integrable kernel $K(t, s)$ i.e.

$$ T f(t) = \int_{\mathbb{R}} K(t, s) f(s) ds. \quad (2.67) $$

$K(t, s)$ with values in bounded operators from $B_1$ to $B_2$, where $B_1$ and $B_2$ are Banach spaces. Suppose that $p_1 < q_1$. Then the operator

$$ \int_{-\infty}^{t} K(t, s) f(s), $$

is bounded from $L^{p_1}(\mathbb{R}; B_1)$ to $L^{q_1}(\mathbb{R}; B_2)$.

Let us finish now the proof of the Lemma 2.1], setting $K(t, s) = \chi D_x^{1/2} e^{i(t-s)\Delta}$ and taking $p_1 = p'$, $B_1 = L^{q'}$, $q_1 = 2$, $B_2 = L^2$, we can apply the Theorem 2. Indeed the estimate (2.66) ensures that the operator $T$ is bounded from $L^{p'}(\mathbb{R}, L^{q'}(\mathbb{R}^2))$ and $L^2(\mathbb{R} \times \mathbb{R}^2)$. In the proof
of Theorem 2, the fact that $K(t, s)$ is a bounded operator from $B_1$ to $B_2$ is only a technical point in order to ensure that the quantities which are estimates are defined and finite. If we take $f \in C_c^\infty(\mathbb{R} \times \mathbb{R}^2)$ we can apply Christ-Kiselev's proof as it is and we obtain

$$ \int_{\mathbb{R}} \int_{\mathbb{R}^2} \left| \int_{-\infty}^{t} \chi D_x^{1/2} e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt \leq C \|f\|_{L^{p'}(\mathbb{R}, L^{q'}(\mathbb{R}^2))}^2. $$

Let $f \in C_c^\infty(]0, T[ \times \mathbb{R}^2)$ we obtain then for $t \in [0, T]$

$$ \begin{aligned} \int_{\mathbb{R}} \int_{\mathbb{R}^2} \left| \int_0^t \chi D_x^{1/2} e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt &= \int_{\mathbb{R}} \int_{\mathbb{R}^2} \left| \int_{-\infty}^t \chi D_x^{1/2} e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt \\ &\leq C \|f\|_{L^{p'}(\mathbb{R}, L^{q'}(\mathbb{R}^2))}^2 \\ &\leq C \|f\|_{L_T^{p'}(L^{q'}(\mathbb{R}^2))}^2, \end{aligned} $$

with $(p, q)$ an admissible Strichartz pair. Thus by density of $C_c^\infty(]0, T[ \times \mathbb{R}^2)$ in $L_T^{p'}(L^{q'}(\mathbb{R}^2))$, for all $f \in L_T^{p'}(L^{q'}(\mathbb{R}^2))$ and $T > 0$ with $(p, q)$ an admissible Strichartz pair (i.e $\frac{1}{p} + \frac{1}{q} = \frac{1}{2}$ and $2 \leq q < +\infty$) we have for $t \in [0, T]$:

$$ \int_{\mathbb{R}} \int_{\mathbb{R}^2} \left| \int_0^t \chi D_x^{1/2} e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt \leq C \|f\|_{L_T^{p'}(L^{q'}(\mathbb{R}^2))}^2. \quad (2.68) $$

Now we compute

$$ \left| \mathcal{F}(\chi D_x^{\frac{1}{2}} f - D_x^{\frac{1}{2}}(\chi f)) \right| (\xi) = \left| \int_{\mathbb{R}^2} \hat{\chi}(\eta) (|\xi - \eta|^{\frac{1}{2}} - |\xi|^{\frac{1}{2}}) \hat{f}(\xi - \eta) d\eta \right|. $$

Observing that $||\xi - \eta|^{\frac{1}{2}} - |\xi|^{\frac{1}{2}}| \leq 2 + |\eta|$ (indeed when $|\xi - \eta| \geq 1$ or $|\xi| \geq 1$ we have $||\xi - \eta|^{\frac{1}{2}} - |\xi|^{\frac{1}{2}}| = \frac{||\xi - \eta|| - |\xi||}{|\xi - \eta|^{\frac{1}{2}} + |\xi|^{\frac{1}{2}}} \leq |\eta|$) and combining Young inequality and Plancherel theorem we have

$$ \|\mathcal{F}(\chi D_x^{\frac{1}{2}} f - D_x^{\frac{1}{2}} \chi f)\|_{L^2(\mathbb{R}^2)} \leq \|\hat{\chi}(\eta)(2 + |\eta|)\|_{L^1(\mathbb{R}^2)} \|f\|_{L^2(\mathbb{R}^2)} \leq C \|f\|_{L^2(\mathbb{R}^2)}. $$

Thus

$$ \| \chi D^{\frac{1}{2}} f - D^{\frac{1}{2}} \chi f \|_{L^2(\mathbb{R}^2)} \leq C \|f\|_{L^2(\mathbb{R}^2)}. \quad (2.69) $$

Thus using (2.65), (2.69) and Strichartz estimate we deduce that there exists $C > 0$ depending on $\chi$ and $T$ such that:

$$ \| \chi e^{it\Delta} f \|_{L_T^2(H^{\frac{1}{2}})} \leq C \|f\|_{L^2}. \quad (2.70) $$

Finally we have for $f \in C_c^\infty(]0, T[ \times \mathbb{R}^2)$

$$ \begin{aligned} \left\| \int_0^t \chi e^{i(t-s)\Delta} f(s) ds \right\|_{L_T^2(H^{1/2})}^2 &\leq 2 \int_0^T \int_{\mathbb{R}^2} \left| \int_0^t \chi D_x^{1/2} e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt \\ &+ 2 \int_0^T \int_{\mathbb{R}^2} \left| \int_0^t (\chi D_x^{\frac{1}{2}} - D_x^{\frac{1}{2}} \chi) e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt. \end{aligned} \quad (2.71) $$
Using (2.69) and Strichartz estimates we have

$$ 
\begin{aligned}
\int_0^T \int_{\mathbb{R}^2} \left| \int_0^t (\chi D_x^{\frac{1}{2}} - D_x^{\frac{1}{2}} \chi) e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt &\le C \int_0^T \int_{\mathbb{R}^2} \left| \int_0^t e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt \\
&\le C T \|f\|_{L_T^{p'}(L^{q'}(\mathbb{R}^2))}^2.
\end{aligned}
$$

Thus using (2.68) and (2.71) we have

$$ \left\| \int_0^t \chi e^{i(t-s)\Delta} f(s) ds \right\|_{L_T^2(\dot{H}^{1/2})} \le C \|f\|_{L_T^{p'}(L^{q'}(\mathbb{R}^2))}, $$

with C depending on $\chi$ and $T$. Moreover using Strichartz again

$$ 
\begin{aligned}
\left\| \int_0^t \chi e^{i(t-s)\Delta} f(s) ds \right\|_{L_T^2(L^2(\mathbb{R}^2))} &= \left( \int_0^T \int_{\mathbb{R}^2} \left| \int_0^t \chi e^{i(t-s)\Delta} f(s) ds \right|^2 dxdt \right)^{\frac{1}{2}} \\
&\le C \sqrt{T} \|f\|_{L_T^{p'}(L^{q'}(\mathbb{R}^2))}.
\end{aligned}
$$

Thus we have obtained (2.63) by density of $C_c^\infty(\{0, T \times \mathbb{R}^2\})$ in $L_T^{p'}(L^{q'}(\mathbb{R}^2))$ for a Strichartz pair $(p, q)$ such that $\frac{1}{p} + \frac{1}{q} = \frac{1}{2}$ and $2 \le q < +\infty$. This completes the proof of Lemma 2.1. $\square$

<figure><img src="image_3.png" /></figure>

We must also prove the following lemma.

**Lemma 2.2** There exists $C > 0$, not depending on $n$, such that for $j \in \{1, 2\}$

$$ \|\sqrt{\rho_n} \mathbb{P} u_n\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))}^2 + \|\sqrt{\rho_n} \mathbb{Q} u_n\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))}^2 \le C, \quad (2.72) $$

$$ \|\partial_j w_n\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} \le C. \quad (2.73) $$

Furthermore for any $p \ge 2$ there exists $C_p > 0$ such that

$$ \|F(\rho_n) - F(1)\|_{L^\infty(\mathbb{R}^+, L^p(\mathbb{R}^2))} \le C. \quad (2.74) $$

**Proof** Concerning (2.72) we have from (2.46):

$$ 
\begin{aligned}
\|\sqrt{\rho_n} u_n\|_{L^2}^2 &= \|\sqrt{\rho_n} \nabla \theta^n + \sqrt{\rho_n} \nabla^\perp \theta_1^n\|_{L^2}^2 \\
&= \|\sqrt{\rho_n} \nabla \theta^n\|_{L^2}^2 + \|\sqrt{\rho_n} \nabla^\perp \theta_1^n\|_{L^2}^2 \\
&\quad + 2 \int_{\mathbb{R}^2} \sqrt{\rho_n} \nabla \theta^n \cdot \sqrt{\rho_n} \nabla^\perp \theta_1^n dx. \quad (2.75)
\end{aligned}
$$

Using again (2.46) we can observe that $\sqrt{\rho_n} \nabla \theta^n$ and $\sqrt{\rho_n} \nabla^\perp \theta_1^n$ are orthogonal on $\mathbb{R}^2 \setminus \{0\}$ then it implies that:

$$ \|\sqrt{\rho_n} u_n\|_{L^2}^2 = \|\sqrt{\rho_n} \mathbb{P} u_n\|_{L^2}^2 + \|\sqrt{\rho_n} \mathbb{Q} u_n\|_{L^2}^2. \quad (2.76) $$

Finally using (2.1) we have (2.72).

We recall now that:

$$ w_n = \sqrt{\rho_n} e^{-\frac{i}{2c_2} \theta_n + \frac{i}{2c_2} a_n}, \quad (2.77) $$
and

$$ \partial_j w_n = \partial_j \sqrt{\rho_n} e^{-\frac{i}{2c_2} \theta_n + \frac{i}{2c_2} a_n} - \frac{i}{(2c_2)} \partial_j \theta_n \sqrt{\rho_n} e^{\frac{i}{2c_2} \theta_n + \frac{i}{2c_2} a_n}, $$

so since $\partial_j \theta_n = (\mathbb{Q}u_n)_j$ we have:

$$ \partial_j w_n = \partial_j \sqrt{\rho_n} e^{-\frac{i}{2c_2} \theta_n + \frac{i}{2c_2} a_n} - \frac{i}{(2c_2)} (\mathbb{Q}u_n)_j \sqrt{\rho_n} e^{\frac{i}{2c_2} \theta_n + \frac{i}{2c_2} a_n}. $$

Using the previous formula, estimate (2.73) follows from (2.1) and (2.72). For the estimate (2.74) we recall that $F(\rho) = \frac{\gamma}{\gamma-1} \rho^{\gamma-1}$ so that:

$$ \int_{\mathbb{R}^2} |F(\rho_n) - F(1)|^p dx = \int_{\mathbb{R}^2} |F(\rho_n) - F(1)|^p \mathbb{1}_{\{|\rho_n - 1| < \frac{1}{2}\}} dx \\ + \int_{\mathbb{R}^2} |F(\rho_n) - F(1)|^p \mathbb{1}_{\{|\rho_n - 1| \ge \frac{1}{2}\}} dx. $$

There exists now $C_p > 0$ such that:

$$ |F(\rho_n) - F(1)|^p \mathbb{1}_{\{|\rho_n - 1| < \frac{1}{2}\}} \le C_p |\rho_n - 1|^p, $$

and

$$ |F(\rho_n) - F(1)|^p \mathbb{1}_{\{|\rho_n - 1| \ge \frac{1}{2}\}} \le C_p |\rho_n - 1|^{p(\gamma-1)} \mathbb{1}_{\{|\rho_n - 1| \ge \frac{1}{2}\}}. $$

Estimate (2.74) follows from (2.5). This ends the proof of Lemma 2.2. □

**Proof of Proposition 2.2** Using (2.60) we have

$$ \begin{aligned} \partial_t \partial_j w_n + i c_2 \Delta \partial_j w_n = & \frac{i}{2c_2} \partial_j w_n (F(|w_n|^2) - F(1) + \frac{1}{2} |\mathbb{P}u_n|^2 + (\Delta)^{-1} \operatorname{div}(\operatorname{curl} u_n (\mathbb{P}u_n)^\perp)) \\ & + \frac{i}{2c_2} w_n (2F'(|w_n|^2)(\operatorname{Re} w_n \partial_j \operatorname{Re} w_n + \operatorname{Im} w_n \partial_j \operatorname{Im} w_n) \\ & + \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n + \partial_j (\Delta)^{-1} \operatorname{div}(\operatorname{curl} u_n (\mathbb{P}u_n)^\perp)). \end{aligned} \quad (2.78) $$

Then by Duhamel principle, using the equation (2.78) we can express $\chi \partial_j w_n(t)$ for $t \ge 0$ as follows with $\chi \in C_c^\infty(\mathbb{R}^2)$:

$$ \begin{aligned} \chi \partial_j w_n(t) = & \chi e^{-itc_2 \Delta} \partial_j w_n(0) \\ & + \int_0^t \chi e^{-ic_2(t-t')\Delta} \left[ \frac{i}{2c_2} \partial_j w_n(F(|w_n|^2) - F(1) + \frac{1}{2} |\mathbb{P}u_n|^2 + \theta_2^n) \right. \\ & \left. + \frac{i}{2c_2} w_n(2F'(|w_n|^2)(\operatorname{Re} w_n \partial_j \operatorname{Re} w_n + \operatorname{Im} w_n \partial_j \operatorname{Im} w_n) + \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n \right. \\ & \left. + \partial_j (\Delta)^{-1} \operatorname{div}(\operatorname{curl} u_n (\mathbb{P}u_n)^\perp)) \right] (t') dt'. \end{aligned} \quad (2.79) $$
The proof of (2.61) consists now in applying to the expression (2.79) the Proposition 2.1, to do this we will need the uniform estimates obtained in the Sect. 2.1. First of all, using (2.62) and (2.73), there exists $C_T$, $C'_T > 0$ independent on $n$ such that:

$$ \begin{aligned} \|\chi e^{-ic_2t^\Delta} \partial_j w_n(0)\|_{L_T^2(H^{1/2})} &\le C_T \|\partial_j w_n(0)\|_{L^2} \\ &\le C'_T. \end{aligned} \quad (2.80) $$

Then we obtain similarly for $C_T > 0$ large enough:

$$ \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} \frac{i}{2c_2} \partial_j w_n(F(|w_n|^2) - F(1)) dt' \right\|_{L_T^2(H^{1/2})} \le C_T. \quad (2.81) $$

Indeed using the admissible Strichartz pairs (4, 4), Proposition 2.1 and (2.73), (2.74) we have

$$ \begin{aligned} & \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} \frac{i}{2c_2} \partial_j w_n(F(|w_n|^2) - F(1)) dt' \right\|_{L_T^2(H^{1/2})} \\ &\le C_T \|\partial_j w_n(F(|w_n|^2) - F(1))\|_{L_T^{4/3}(L^{4/3})} \\ &\le C_T T^{3/4} \|\partial_j w_n(F(\rho_n) - F(1))\|_{L_T^\infty(L^{4/3})} \\ &\le T^{3/4} C_T \|\partial_j w_n\|_{L_T^\infty(L^2)} \|F(\rho_n) - F(1)\|_{L_T^\infty(L^4)} \\ &\le C'_T. \end{aligned} $$

Next, we observe that we have

$$ \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} \frac{i}{2c_2} \partial_j w_n \left( \frac{1}{2} |\mathbb{P}u_n|^2 \right) dt' \right\|_{L_T^2(H^{1/2})} \le C_T. \quad (2.82) $$

Again using proposition 2.1 with the admissible Strichartz pairs (4, 4), estimates (2.8) and (2.73) we have

$$ \begin{aligned} & \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} \frac{i}{2c_2} \partial_j w_n \left( \frac{1}{2} |\mathbb{P}u_n|^2 \right) dt' \right\|_{L_T^2(H^{1/2})} \le C_T \|\partial_j w_n |\mathbb{P}u_n|^2\|_{L_T^{4/3}(L^{4/3})} \\ &\le C_T T^{3/4} \|\partial_j w_n\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} \| |\mathbb{P}u_n|^2\|_{L^\infty(\mathbb{R}^+, L^4(\mathbb{R}^2))} \\ &\le C'_T. \end{aligned} $$

Now we want to show that:

$$ \left\| \int_0^t \chi e^{i(t-t')\Delta} \frac{i}{2c_2} \partial_j w_n (\theta_2^n) dt' \right\|_{L_T^2(H^{1/2})} \le C_T, \quad (2.83) $$

with $\theta_2^n = (\Delta)^{-1} \text{div}(\text{curl } u_n (\mathbb{P}u_n)^\perp)$. We have, in view of (2.8) and (2.6),

$$ \| \text{curl } u_n(t, \cdot) (\mathbb{P}u_n)^\perp(t, \cdot) \|_{L^\infty(\mathbb{R}^+, L^{4/3}(\mathbb{R}^2))} \le C. $$

By continuity of the Riesz operator in $L^{4/3}(\mathbb{R}^2)$, we deduce that:

$$ \|\nabla(\Delta)^{-1} \text{div}(\text{curl } u_n(t, \cdot)(\mathbb{P}u_n)^\perp(t, \cdot))\|_{L^\infty(\mathbb{R}^+, L^{4/3}(\mathbb{R}^2))} \le C. $$
By Sobolev embedding we get that:

$$ \| (\Delta)^{-1} \operatorname{div}(\operatorname{curl} u_n (\mathbb{P}u_n)^\perp) \|_{L^\infty(\mathbb{R}^+, L^4(\mathbb{R}^2))} \le C. \quad (2.84) $$

Using proposition 2.1 with the admissible Strichartz pairs (4, 4), (2.84) and (2.73), we have:

$$ \begin{aligned} \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} \frac{i}{2c_2} \partial_j w_n (\theta_2^n) dt' \right\|_{L_T^2(H^{1/2})} &\le C_T \|\partial_j w_n \theta_2^n\|_{L_T^{4/3}(L^{4/3})} \\ &\le T^{3/4} C_T \|\partial_j w_n\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} \|\theta_2^n\|_{L^\infty(\mathbb{R}^+, L^4(\mathbb{R}^2))} \\ &\le C_T', \end{aligned} $$

which gives (2.83). Next, we claim that:

$$ \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} w_n F'(|w_n|^2) (\operatorname{Re} w_n \partial_j \operatorname{Re} w_n + \operatorname{Im} w_n \partial_j \operatorname{Im} w_n) dt' \right\|_{L_T^2(H^{1/2})} \le C_T. \quad (2.85) $$

We first observe that

$$ \begin{aligned} w_n F'(|w_n|^2) (\operatorname{Re} w_n \partial_j \operatorname{Re} w_n) \\ &= (\rho_n F'(\rho_n) - F'(1)) e^{-\frac{i}{2c_2}\theta_n + \frac{i}{2c_2}a_n} \cos\left(-\frac{1}{2c_2}\theta_n + \frac{1}{2c_2}a_n\right) \partial_j \operatorname{Re} w_n \\ &\quad + e^{-\frac{i}{2c_2}\theta_n + \frac{i}{2c_2}a_n} \cos\left(-\frac{1}{2c_2}\theta_n + \frac{1}{2c_2}a_n\right) F'(1) \partial_j \operatorname{Re} w_n. \end{aligned} \quad (2.86) $$

In view of (2.73), we have

$$ \left\| e^{-\frac{i}{2c_2}\theta_n + \frac{i}{2c_2}a_n} \cos\left(-\frac{1}{2c_2}\theta_n + \frac{1}{2c_2}a_n\right) F'(1) \partial_j \operatorname{Re} w_n \right\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} \le C. \quad (2.87) $$

Invoking proposition (2.1) with Strichartz pair $(\infty, 2)$, we deduce from (2.87) that

$$ \begin{aligned} \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} e^{-\frac{i}{2c_2}\theta_n + \frac{i}{2c_2}a_n} \cos\left(-\frac{1}{2c_2}\theta_n + \frac{1}{2c_2}a_n\right) 2F'(1) \partial_j \operatorname{Re} w_n dt' \right\|_{L_T^2(H^{1/2})} & \\ \le C_T \|e^{-\frac{i}{2c_2}\theta_n + \frac{i}{2c_2}a_n} \cos\left(-\frac{1}{2c_2}\theta_n + \frac{1}{2c_2}a_n\right) 2F'(1) \partial_j \operatorname{Re} w_n\|_{L_T^1(L^2)} & \\ \le C_T T \|\partial_j \operatorname{Re} w_n\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} & \\ \le C_T'. \end{aligned} \quad (2.88) $$

We observe now that

$$ (\rho_n F'(\rho_n) - F'(1)) = (\gamma - 1)(F(\rho_n) - F(1)), \quad (2.89) $$

so that, in view of (2.74) and (2.73) we get:

$$ \| (\rho_n F'(\rho_n) - F'(1)) e^{-\frac{i}{2c_2}\theta_n + \frac{i}{2c_2}a_n} \cos\left(-\frac{1}{2c_2}\theta_n + \frac{1}{2c_2}a_n\right) \partial_j \operatorname{Re} w_n \|_{L^\infty(\mathbb{R}^+, L^{3/2}(\mathbb{R}^2))} \le C. \quad (2.90) $$
Invoking Proposition 2.1 with Strichartz pair (6, 3), we deduce from (2.90) that

$$ \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} (\rho_n F'(\rho_n) - F'(1)) e^{-\frac{i}{2c_2}(\theta_n - a_n)} \cos\left(-\frac{1}{2c_2}(\theta_n - a_n)\right) \partial_j \operatorname{Re} w_n dt' \right\|_{L_T^2(H^{1/2})} \\ \le C_T \|(\rho_n F'(\rho_n) - F'(1)) e^{-\frac{i}{2c_2}\theta_n + \frac{i}{2c_2}a_n} \cos\left(-\frac{1}{2c_2}\theta_n + \frac{1}{2c_2}a_n\right) \partial_j \operatorname{Re} w_n \|_{L_T^{6/5}(L^{3/2})} \\ \le C_T'. \qquad (2.91) $$

We deduce from (2.86), (2.88) and (2.91) that

$$ \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} w_n F'(|w_n|^2) (\operatorname{Re} w_n \partial_j \operatorname{Re} w_n) dt' \right\|_{L_T^2(H^{1/2})} \le C_T. \qquad (2.92) $$

Similarly we have

$$ \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} 2w_n F'(|w_n|^2) \operatorname{Im} w_n \partial_j \operatorname{Im} w_n dt' \right\|_{L_T^2(H^{1/2})} \le C_T. \qquad (2.93) $$

So that (2.85) holds. We now claim that

$$ \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} w_n \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n dt' \right\|_{L_T^2(H^{1/2})} \le C_T. \qquad (2.94) $$

Indeed applying Proposition 2.1 with Strichartz pair (4, 4), in view of estimates (2.7) and (2.72), we have

$$ \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} w_n \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n dt' \right\|_{L_T^2(H^{1/2})} \le C_T \|w_n \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n\|_{L_T^{4/3}(L^{4/3})} \\ \le C_T T^{3/4} \|w_n \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n\|_{L^\infty(\mathbb{R}^+, L^{4/3}(\mathbb{R}^2))} \\ \le C_T T^{3/4} \|w_n \mathbb{P}u_n\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} \|\partial_j \mathbb{P}u_n\|_{L^\infty(\mathbb{R}^+, L^4(\mathbb{R}^2))} \\ \le C_T', $$

which is estimate (2.94). Now let us demonstrate that for $j \in \{1, 2\}$:

$$ \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} w_n \partial_j (\Delta)^{-1} \operatorname{div}(\operatorname{curl} u_n (\mathbb{P}u_n)^\perp) dt' \right\|_{L_T^2(H^{1/2})} \le C_T. \qquad (2.95) $$

We first observe that

$$ w_n \partial_j (\Delta)^{-1} \operatorname{div}(\operatorname{curl} u_n (\mathbb{P}u_n)^\perp) = e^{-\frac{i\theta_n}{2c_2} + \frac{i\partial_n}{2c_2}} \partial_j (\Delta)^{-1} \operatorname{div}(\operatorname{curl} u_n (\mathbb{P}u_n)^\perp) \\ + (\sqrt{\rho_n} - 1) e^{-\frac{i\theta_n}{2c_2} + \frac{i\partial_n}{2c_2}} \partial_j (\Delta)^{-1} \operatorname{div}(\operatorname{curl} u_n (\mathbb{P}u_n)^\perp). $$
For the second term in view of Proposition 2.1 with Strichartz pair (4, 4) and via the continuity of the Riesz operator in $L^4$, estimates (2.8), (2.4) and (2.6) give:

$$ \begin{aligned} & \left\| \int_0^t \chi e^{-ic_2(t-t')\Delta} (\sqrt{\rho_n} - 1) e^{-\frac{i(\theta_n - \theta_n)}{2c_2}} \partial_j (\Delta)^{-1} \mathrm{div}(\mathrm{curl} u_n (\mathbb{P}u_n)^\perp) dt' \right\|_{L^2(T)(H^{1/2})} \\ & \le C_T \|(\sqrt{\rho_n} - 1) e^{-\frac{i(\theta_n - \theta_n)}{2c_2}} \partial_j (\Delta)^{-1} \mathrm{div}(\mathrm{curl} u_n (\mathbb{P}u_n)^\perp)\|_{L^4(T)(L^{4/3})} \\ & \le C_T T^{3/4} \|(\sqrt{\rho_n} - 1)\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^2))} \|\mathrm{curl} u_n (\mathbb{P}u_n)^\perp\|_{L^\infty(\mathbb{R}^+, L^4(\mathbb{R}^2))} \\ & \le C'_T. \end{aligned} $$

We operate in a similar manner for $e^{-\frac{i(\theta_n - \theta_n)}{2c_2}} (\partial_j (\Delta)^{-1} \mathrm{div}(\mathrm{curl} u_n (\mathbb{P}u_n)^\perp))$, and we obtain (2.95). Finally, in view of (2.80), (2.81), (2.82), (2.83), (2.85), (2.94), (2.95), we have obtained the desired estimates (2.61). This complete the proof of Proposition 2.2. $\square$

## 2.5 Strong Convergence of the Sequence ($\nabla\sqrt{\rho_n}, \sqrt{\rho_n}\mathbb{Q}u_n)_{n\in\mathbb{N}$

First of all, we wish to prove the strong convergence up to a subsequence of the sequence $(\rho_n)_{n\in\mathbb{N}}$ to a limit $\rho$ in $L^\infty(L_{\mathrm{loc}}^q(\mathbb{R}^2))$ for any $q \ge 1$. The key argument will be the use of the Aubin Lions Lemma that we recall briefly.

**Aubin-Lions Lemma** Let $X_0$, $X$ and $X_1$ be three Banach spaces. Suppose that $X_0$ is compactly embedded in $X$ and that $X$ is continuously embedded in $X_1$. For $1 \le p, q \le \infty$, let

$$ W = \{u \in L^p([0, T]; X_0) \mid \partial_t u \in L^q([0, T]; X_1)\}. $$

* If $p < \infty$ then the embedding of $W$ into $L^p([0, T]; X)$ is compact.

* If $p = +\infty$ and $q > 1$, then the embedding of $W$ into $C([0, T], X)$ is compact.

**Proposition 2.3** There exist $\rho \in L_{\mathrm{loc}}^\infty(\mathbb{R}_+, L_{\mathrm{loc}}^q(\mathbb{R}^2))$ for any $1 \le q < \infty$, such that, up to a subsequence,

$$ \rho_n \to \rho \text{ in } L_{\mathrm{loc}}^\infty(\mathbb{R}_+, L_{\mathrm{loc}}^q(\mathbb{R}^2)), \text{ as } n \to \infty. \quad (2.96) $$

*Proof* Using Hölder inequality, (2.20), (2.21) and (2.12) we have for all $1 \le p < 2$ and any compact $K$

$$ \begin{aligned} \|\nabla \rho_n\|_{L^\infty(\mathbb{R}^+, L^p(K))} &\le 2 \|\sqrt{\rho_n}\|_{L^\infty(\mathbb{R}^+, L^q(K))} \|\nabla \sqrt{\rho_n}\|_{L^\infty(\mathbb{R}^+, L^2(\mathbb{R}^K))} \\ &\le C_{p,K}, \end{aligned} \quad (2.97) $$

where $\frac{1}{p} = \frac{1}{2} + \frac{1}{q}$. We deduce immediately from (2.97) and (2.5) that for any $1 \le p < 2$:

$$ \|\rho_n\|_{L^\infty(W^{1,p}(K))} \le C_{p,K}. \quad (2.98) $$

In order to apply Aubin-Lions Lemma, we need a information on the time derivative of $\rho_n$. Observing that

$$ \rho_n u_n = \sqrt{\rho_n} u_n (\sqrt{\rho_n} - 1) + \sqrt{\rho_n} u_n, $$
we obtain, using (2.12) and (2.20) that for any $1 \le p < 2$:

$$ \|\rho_n u_n\|_{L^\infty(\mathbb{R}^+, L^p(K))} \le C_{p,K}. $$

We deduce via the mass equation that:

$$ \|\partial_t(\rho_n)\|_{L^\infty(\mathbb{R}^+, W^{-1,p}(K))} \le C_{p,K}. $$

We fix now $p \in [1, 2]$, using that $W^{1,p}(K)$ is compactly embedded in $L^p(K)$ and $L^p(K)$ is continuously embedded in $W^{-1,p}(K)$, we deduce using Aubin Lions Lemma and the diagonal extraction procedure that up to a subsequence we have for all $T > 0$ and for all $R > 0$:

$$ \rho_n \to \rho \text{ in } L_T^\infty(L^p(B(0, R))), \text{ as } n \to \infty. \quad (2.99) $$

From (2.99), (2.5) and by interpolation we can prove that (2.96) is verified for any $p \in [1, +\infty[$. $\square$

**Corollary 2.2** Again up to a subsequence, we have:

$$ P(\rho_n) \to P(\rho) \text{ in } L_{\text{loc}}^q(\mathbb{R}_+ \times \mathbb{R}^n), \text{ as } n \to \infty, \quad (2.100) $$

for any $1 \le q < \infty$ and:

$$ \sqrt{\rho_n} \to \sqrt{\rho} \text{ in } L_{\text{loc}}^2(\mathbb{R}_+ \times \mathbb{R}^n), \text{ as } n \to \infty. \quad (2.101) $$

**Proof** In view of proposition 2.3, we have, up to a subsequence

$$ \rho_n \to \rho \text{ in } L_{\text{loc}}^q(\mathbb{R}_+ \times \mathbb{R}^n), \text{ as } n \to \infty, \quad (2.102) $$

for any $1 \le q < \infty$. Since we have:

$$ |P(\rho_n) - P(\rho)| = |\rho_n^\gamma - \rho^\gamma| \le \gamma |\rho_n - \rho|(|\rho_n|^{\gamma-1} + |\rho|^{\gamma-1}), $$

we deduce from (2.102) and (2.5) that (2.100) is satisfied. Let $\eta > 0$, we have

$$ \begin{aligned} |\sqrt{\rho_n} - \sqrt{\rho}|^2 \mathbb{1}_{\{|\sqrt{\rho_n} + \sqrt{\rho}| \le \eta\}} &\le \frac{|\rho_n - \rho|^2}{|\sqrt{\rho_n} + \sqrt{\rho}|^2} \mathbb{1}_{\{|\sqrt{\rho_n} + \sqrt{\rho}| \ge \eta\}} \\ &\le \frac{1}{\eta^2} |\rho_n - \rho|^2, \end{aligned} $$

and since $\rho_n \ge 0$ and $\rho \ge 0$ (indeed up to a subsequence $\rho_n$ converges almost everywhere to $\rho$)

$$ |\sqrt{\rho_n} - \sqrt{\rho}|^2 \mathbb{1}_{\{|\sqrt{\rho_n} + \sqrt{\rho}| \le \eta\}} \le 4\eta^2. $$

Then, we deduce that for any compact set $\Omega$ in $\mathbb{R}^+ \times \mathbb{R}^2$ we have for any $\eta > 0$:

$$ \|\sqrt{\rho_n} - \sqrt{\rho}\|_{L^2(\Omega)}^2 \le \frac{1}{\eta^2} \|\rho_n - \rho\|_{L^2(\Omega)}^2 + 4\eta^2 |\Omega|. \quad (2.103) $$

Combining (2.103) and (2.102) completes the proof. $\square$
We are now going to prove the main part of this section on the strong convergence of the sequence $(\nabla\sqrt{\rho_n}, \sqrt{\rho_n}\mathbb{Q}u_n)_{n\in\mathbb{N}}$ in $L_{\text{loc}}^2(\mathbb{R}^+ \times \mathbb{R}^2)$. Using the Proposition 2.2, we have obtained that for any $\chi \in C_c^\infty(\mathbb{R}^2)$ there exists $C_T > 0$ independent on $n$ and depending only on $T > 0$ and $\chi$

$$ \|\chi \partial_j w_n\|_{L_T^2(H^{1/2})} \le C_T. \quad (2.104) $$

We have:

$$ \begin{aligned} \partial_t(\chi \partial_j w_n) + i c_2 \chi \Delta \partial_j w_n ={}& \chi \left( \frac{i}{2c_2} \partial_j w_n (F(|w_n|^2) - F(1) + \frac{1}{2} |\mathbb{P}u_n|^2 + \theta_2^n) \right. \\ & \left. \qquad + \frac{i}{2c_2} w_n (2F'(|w_n|^2)(\text{Re } w_n \partial_j \text{Re } w_n + \text{Im } w_n \partial_j \text{Im } w_n) \right. \\ & \left. \qquad + \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n + \partial_j (\Delta)^{-1} \text{div}(\text{curl} u_n (\mathbb{P}u_n)^\perp) \right). \end{aligned} \quad (2.105) $$

We have seen in the Sect. 2.4 that for any $T > 0$:

$$ \begin{aligned} \| \chi \left( \frac{i}{2c_2} \partial_j w_n (F(|w_n|^2) - F(1) + \frac{1}{2} |\mathbb{P}u_n|^2 + \theta_2^n) \right. \right. \\ \left. + \frac{i}{2c_2} w_n (2F'(|w_n|^2)(\text{Re } w_n \partial_j \text{Re } w_n + \text{Im } w_n \partial_j \text{Im } w_n) \right. \\ \left. + \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n + \partial_j (\Delta)^{-1} \text{div}(\text{curl} u_n (\mathbb{P}u_n)^\perp) \right) \|_{L_T^\infty(L^{4/3}(\mathbb{R}^2))} \le C_T. \end{aligned} \quad (2.106) $$

So, we have using Sobolev embedding:

$$ \begin{aligned} \| \chi \left( \frac{i}{2c_2} \partial_j w_n (F(|w_n|^2) - F(1) + \frac{1}{2} |\mathbb{P}u_n|^2 + \theta_2^n) \right. \right. \\ \left. + \frac{i}{2c_2} w_n (2F'(|w_n|^2)(\text{Re } w_n \partial_j \text{Re } w_n + \text{Im } w_n \partial_j \text{Im } w_n) \right. \\ \left. + \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n + \partial_j (\Delta)^{-1} \text{div}(\text{curl} u_n (\mathbb{P}u_n)^\perp) \right) \|_{L_T^\infty(H^{-\frac{1}{2}})} \le C_T. \end{aligned} \quad (2.107) $$

Using now that the following embedding are continuous $H^{-\frac{1}{2}}(\mathbb{R}^2) \hookrightarrow H^{-2}(\mathbb{R}^2)$, we get:

$$ \begin{aligned} \| \chi \left( \frac{i}{2c_2} \partial_j w_n (F(|w_n|^2) - F(1) + \frac{1}{2} |\mathbb{P}u_n|^2 + \theta_2^n) \right. \right. \\ \left. + w_n (2F'(|w_n|^2)(\text{Re } w_n \partial_j \text{Re } w_n + \text{Im } w_n \partial_j \text{Im } w_n) \right. \\ \left. + \partial_j \mathbb{P}u_n \cdot \mathbb{P}u_n + \partial_j (\Delta)^{-1} \text{div}(\text{curl} u_n (\mathbb{P}u_n)^\perp) \right) \|_{L_T^2(H^{-2})} \le C_T. \end{aligned} \quad (2.108) $$

Moreover using the fact that multiplication by a function of $S(\mathbb{R}^2)$ is a continuous map from $H^s(\mathbb{R}^2)$ into itself and using (2.61) we have

$$ \|i c_2 \chi \Delta \partial_j w_n\|_{L_T^2(H^{-2}(\mathbb{R}^2))} \le C_T. \quad (2.109) $$
So we have using (2.108) and (2.109):

$$ \|\partial_t (\chi \partial_j w_n)\|_{L^2_t(H^{-2}(\mathbb{R}^2))} \le C_T. \qquad (2.110) $$

Let $K$ a compact, we define by $H_K^{\frac{1}{2}}$ the distributions which belongs to $H^{\frac{1}{2}}$ and are supported in $K$. We know that $H_K^{1/2}(\mathbb{R}^2)$ is compactly embedded in $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R}^2)$ is continuously embedded in $H^{-2}(\mathbb{R}^2)$ then from the Aubin Lions Lemma, (2.104), (2.110) and applying the diagonal extraction procedure we deduce the following proposition.

**Proposition 2.4** Up to a subsequence there exists some map $f^j \in L^2_{\text{loc}}(\mathbb{R}^+ \times \mathbb{R}^2)$ with $j \in \{1, 2\}$ such that for all $T > 0$ and all $R > 0$ we have:

$$ \partial_j w_n \to f^j \text{ in } L^2_t(L^2(B(0, R)), \text{as } n \to \infty. \qquad (2.111) $$

We recall now that

$$ \nabla w_n = \nabla \sqrt{\rho_n} e^{-\frac{i}{2c_2} \theta_n + \frac{i}{2c_2} a_n} - \frac{i}{2c_2} \nabla \theta_n \sqrt{\rho_n} e^{\frac{i}{2c_2} \theta_n + \frac{i}{2c_2} a_n}, $$

where $\partial_j \theta_n = (\mathbb{Q}u_n)_j$. It implies that:

$$ \nabla w_n = (\nabla \sqrt{\rho_n} - \frac{i}{2c_2} (\mathbb{Q}u_n) \sqrt{\rho_n}) \phi_n, $$

with $\phi_n = e^{-\frac{i}{2c_2} \theta_n + \frac{i}{2c_2} a_n}$. In other word, we have:

$$ \begin{gathered} \|\phi_n\|_{L^\infty(\mathbb{R}^+ \times \mathbb{R}^2)} \le 1, \\ \nabla \sqrt{\rho_n} = \text{Re}(\bar{\phi}_n \nabla w_n) \quad \text{and} \quad \sqrt{\rho_n} \mathbb{Q}u_n = \text{Im}(\bar{\phi}_n \nabla w_n), \qquad (2.112) \\ |\nabla \sqrt{\rho_n}|^2 + \frac{1}{4c_2^2} |\sqrt{\rho_n} \mathbb{Q}u_n|^2 = |\nabla w_n|^2. \end{gathered} $$

Since $(\phi_n)_{n \in \mathbb{N}}$ is bounded by 1, it converges up to extraction in $L^\infty(\mathbb{R}^+ \times \mathbb{R}^2)$ weak * to some function $\phi$ such that $\|\phi\|_{L^\infty(\mathbb{R}^+ \times \mathbb{R}^2)} \le 1$. In addition according to proposition 2.4, we know that $\nabla w_n$ converges strongly to $f = (f^1, f^2)^t$ in $L^2(K)$ for any compact $K$ of $\mathbb{R}^+ \times \mathbb{R}^2$, this implies that:

$$ \bar{\phi}_n \nabla w_n \to \bar{\phi} f \text{ in } L^2(K). \qquad (2.113) $$

From (2.101) and (2.112) we deduce up to a subsequence and for any compact $K$ of $\mathbb{R}^+ \times \mathbb{R}^2$ that:

$$ \nabla \sqrt{\rho} = \text{Re}(\bar{\phi} f) \quad \text{and} \quad \nabla \sqrt{\rho_n} \to \nabla \sqrt{\rho} \text{ in } L^2(K). \qquad (2.114) $$

Similarly up to an extraction we have for any $K$ compact set of $\mathbb{R}^+ \times \mathbb{R}^2$:

$$ -\frac{1}{2c_2} \sqrt{\rho_n} \mathbb{Q}u_n \to -\frac{1}{2c_2} \Lambda = \text{Im}(\bar{\phi} f) \text{ in } L^2(K). \qquad (2.115) $$
We have seen that up to a subsequence $\sqrt{\rho_n}\mathbb{Q}u_n$ and $\nabla\sqrt{\rho_n}$ converge weakly in $L^2(K)$ for any compact $K$ of $\mathbb{R}^+ \times \mathbb{R}^2$ to some function $\Lambda$ and $\nabla\sqrt{\rho_n}$. We wish now to prove that these convergence are strong, it suffices then to show that:

$$ \|\nabla\sqrt{\rho_n}\|_{L^2(K)} \underset{n \to +\infty}{\longrightarrow} \|\nabla\sqrt{\rho}\|_{L^2(K)} \quad \text{and} \quad \|\sqrt{\rho_n}\mathbb{Q}u_n\|_{L^2(K)} \underset{n \to +\infty}{\longrightarrow} \|\Lambda\|_{L^2(K)}. \quad (2.116) $$

The weak convergence implies that:

$$ \|\nabla\sqrt{\rho}\|_{L^2(K)} \le \liminf_{n \to \infty} \|\nabla\sqrt{\rho_n}\|_{L^2(K)} \quad \text{and} \quad \|\Lambda\|_{L^2(K)} \le \liminf_{n \to \infty} \|\sqrt{\rho_n}\mathbb{Q}u_n\|_{L^2(K)}. \quad (2.117) $$

From Proposition (2.4) and (2.112) we have:

$$ \begin{aligned} & \liminf_{n \to \infty} \|\nabla\sqrt{\rho_n}\|_{L^2(K)}^2 + \frac{1}{4c_2^2} \liminf_{n \to \infty} \|\rho_n\mathbb{Q}u_n\|_{L^2(K)}^2 && (2.118) \\ & \le \lim_{n \to \infty} (\|\nabla\sqrt{\rho_n}\|_{L^2(K)}^2 + \frac{1}{4c_2^2} \|\sqrt{\rho_n}\mathbb{Q}u_n\|_{L^2(K)}^2) = \lim_{n \to \infty} \|\nabla w_n\|_{L^2(K)}^2 = \|f\|_{L^2(K)}^2. \end{aligned} $$

It remains now to prove that:

$$ \|f\|_{L^2(K)}^2 = \int_K |\phi|^2 (|\nabla \sqrt{\rho}|^2 + \frac{1}{4c_2^2} |\Lambda|^2) dt dx = \|\nabla \sqrt{\rho}\|_{L^2(K)}^2 + \frac{1}{4c_2^2} \|\Lambda\|_{L^2(K)}^2. \quad (2.119) $$

To do this we will use the Aubin-Lions lemma on $\chi w_n$. First of all using (2.20) and Lemma 2.2 we have for any $T > 0$

$$ \|\chi w_n\|_{L_T^2(H^1)} \le C_T. $$

We have using (2.60)

$$ \partial_t \chi w_n + ic_2 \chi \Delta w_n - \frac{i}{2c_2} \chi w_n (F(|w_n|^2) - F(1) + \frac{1}{2} |\mathbb{P}u_n|^2 + \theta_2^n) = 0. \quad (2.120) $$

Thus using Hölder's inequality and (2.74), (2.8), (2.84) we have

$$ \left\| \frac{i}{2c_2} \chi w_n (F(|w_n|^2) - F(1) + \frac{1}{2} |\mathbb{P}u_n|^2 + \theta_2^n) \right\|_{L_T^2(L^2)} \le C. $$

Using $\chi \Delta w_n = \Delta(\chi w_n) - \Delta(\chi)w_n - 2\nabla\chi \cdot \nabla w_n$ and (2.20), (2.61) we have

$$ \|\partial_t(\chi w_n)\|_{L_T^2(H^{-2})} \le C. $$

For any compact $K$, we know that $H_K^1(\mathbb{R}^2)$ is compactly embedded in $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R}^2)$ is continuously embedded in $H^{-2}(\mathbb{R}^2)$ then from the Aubin Lions Lemma and applying the diagonal extraction procedure we deduce the following proposition.

**Proposition 2.5** Up to a subsequence there exists some map $w \in L_{loc}^2(\mathbb{R}^+ \times \mathbb{R}^2)$ such that for all $T > 0$ and all $R > 0$ we have:

$$ w_n \to w \text{ in } L_T^2(L^2(B(0, R))), \text{ as } n \to \infty. \quad (2.121) $$
From propositions 2.5 and 2.4 we deduce that $\nabla w = f$. In addition up to a subsequence we get

$$w_n \to w \text{ a.e. as } n \to \infty$$

We have using (1.11) that $|w_n(t, x)| > 0$ everywhere thus

$$\phi_n(t, x) = \frac{w_n(t, x)}{|w_n(t, x)|}.$$

Then

$$w_n(t, x) \mathbb{1}_{\{|w(t,x)|>0\}} \to w(t, x) \mathbb{1}_{\{|w(t,x)|>0\}} \text{ a.e., as } n \to \infty.$$

Then

$$\phi_n(t, x) \mathbb{1}_{\{|w(t,x)|>0\}} = \frac{w_n(t, x)}{|w_n(t, x)|} \mathbb{1}_{\{|w(t,x)|>0\}} \\ \underset{n \to +\infty}{\longrightarrow} \frac{w(t, x)}{|w(t, x)|} \mathbb{1}_{\{|w(t,x)|>0\}} \text{ a.e.}.$$

Thus for any $\varphi \in L^1(\mathbb{R}_+ \times \mathbb{R}^2)$ using Lebesgue's theorem and the definition of the $L^\infty$ weak * limit we have

$$\int_{\mathbb{R} \times \mathbb{R}^2} \phi_n \varphi = \int_{\mathbb{R} \times \mathbb{R}^2} \phi_n \varphi \\ = \int_{\mathbb{R} \times \mathbb{R}^2} \phi_n \varphi \mathbb{1}_{\{|w(t,x)|>0\}} + \int_{\mathbb{R} \times \mathbb{R}^2} \phi_n \varphi \mathbb{1}_{\{|w(t,x)|=0\}} \\ \underset{n \to +\infty}{\longrightarrow} \int_{\mathbb{R} \times \mathbb{R}^2} \frac{w(t, x)}{|w(t, x)|} \mathbb{1}_{\{|w(t,x)|>0\}} \varphi + \int_{\mathbb{R} \times \mathbb{R}^2} \phi_n \mathbb{1}_{\{|w(t,x)|=0\}} \varphi.$$

Then we have

$$\phi(t, x) = \frac{w(t, x)}{|w(t, x)|} \mathbb{1}_{\{|w(t,x)|>0\}} + \phi(t, x) \mathbb{1}_{\{|w(t,x)|=0\}}.$$

Recall (see e.g. Theorem 6.19 in [20]) that

$$\nabla w = 0 \text{ a.e. on } w^{-1}(\{0\}),$$

whenever w is locally in $W^{1,1}$. Thus we have

$$\begin{align*}
|\overline{\phi(t, x)} f(t, x)| &= |\phi(t, x)| |\nabla w(t, x)| \\
&= \left| \frac{w(t, x)}{|w(t, x)|} \mathbb{1}_{\{|w(t, x)|>0\}} \right| |\nabla w(t, x)| \quad \text{a.e.} \tag{2.122} \\
&= \mathbb{1}_{\{|w(t, x)|>0\}} |\nabla w(t, x)| \quad \text{a.e.} \\
&= |\nabla w(t, x)| \quad \text{a.e.} \\
&= |f(t, x)| \quad \text{a.e.}
\end{align*}$$
Thus we can prove (2.119). Indeed we have from (2.122), (2.114) and (2.115)

$$ \begin{aligned} \|f\|_{L^2(K)}^2 &= \|\bar{\phi}f\|_{L^2(K)}^2 = \int_K (|\nabla \sqrt{\rho}|^2 + \frac{1}{4c_2^2} |\Lambda|^2) dt dx \\ &= \|\nabla \sqrt{\rho}\|_{L^2(K)}^2 + \frac{1}{4c_2^2} \|\Lambda\|_{L^2(K)}^2. \end{aligned} $$

Combining now (2.118), (2.119) and (2.117) we deduce (2.116). We have finally proved that:

$$ \sqrt{\rho_n}(Qu_n) \to \Lambda \text{ in } L_{\text{loc}}^2(\mathbb{R}_+ \times \mathbb{R}^n), \text{ as } n \to \infty, \quad (2.123) $$

and

$$ \nabla \sqrt{\rho_n} \to \nabla \sqrt{\rho} \text{ in } L_{\text{loc}}^2(\mathbb{R}_+ \times \mathbb{R}^n), \text{ as } n \to \infty. \quad (2.124) $$

**Proposition 2.6** There exists $\Lambda_2 \in L_{\text{loc}}^{\infty}(\mathbb{R}_+, L_{\text{loc}}^q(\mathbb{R}^2))$ for all $1 \le q < \infty$, such that, up to a subsequence,

$$ \mathbb{P}u_n \to \Lambda_2 \text{ in } L_{\text{loc}}^{\infty}(\mathbb{R}_+, L_{\text{loc}}^q(\mathbb{R}^2)), \text{ as } n \to \infty. \quad (2.125) $$

**Proof** For any compact $K$ in $\mathbb{R}^2$ we deduce from (2.8) that for any $p \in [1, 2[$:

$$ \|\mathbb{P}u_n\|_{L^{\infty}(W^{1,p}(K))} \le C_{p,K}. \quad (2.126) $$

We have, using (2.25)

$$ \partial_t \text{curl } u_n = -\text{div}(u_n \text{curl}(u_n)). \quad (2.127) $$

Observing that

$$ \text{curl}(u_n)u_n = \frac{\text{curl}(u_n)}{\rho_n} \sqrt{\rho_n} u_n (\sqrt{\rho_n} - 1) + \frac{\text{curl} u_n}{\rho_n} \sqrt{\rho_n} u_n. $$

We have using Hölder's inequality and (2.12), (2.24), (2.4) that:

$$ \left\| \frac{\text{curl}(u_n)}{\rho_n} \sqrt{\rho_n} u_n (\sqrt{\rho_n} - 1) \right\|_{L^\infty(L^p(\mathbb{R}^\infty))} \le C_p, \quad (\forall p \in [1, 2]), $$

and

$$ \left\| \frac{\text{curl}(u_n)}{\rho_n} \sqrt{\rho_n} u_n \right\|_{L^\infty(L^2(\mathbb{R}^\infty))} \le C. $$

Then using the continuity of the Riesz operator in $L^p$ with $1 < p < +\infty$ and the identity

$$ \partial_t \mathbb{P}u_n = -(\Delta)^{-1} \nabla^\perp \text{div}(u_n \text{curl} u_n). $$

We deduce that $(\partial_t \mathbb{P}u_n)_{n \in \mathbb{N}}$ is uniformly bounded in $L^\infty(\mathbb{R}^+, L^p(K))$ for $p \in [1, 2[$. So using that $W^{1,p}(K)$ is compactly embedded in $L^p(K)$ and applying Aubin Lions Lemma and the diagonal extraction procedure we deduce the Proposition 2.6 for $q = p$. The other case are obtained by interpolation. $\square$
## 2.6 Convergence of the Sequence $(\rho_n, u_n)_{n \in \mathbb{N}}$ to a Global Weak Solution

We have for all $n \in \mathbb{N}$ since $(\rho_n, u_n)_{n \in \mathbb{N}}$ is a regular approximate solution of the system (1.1) with $\varphi \in C_c^\infty(\mathbb{R}^+ \times \mathbb{R}^2)$ and using the fact that $\mathrm{div}(\rho_n P u_n) = 0$:

$$ \int_{\mathbb{R}^2} (\rho_n \varphi)(0, \cdot) dx + \int_{\mathbb{R}_+ \times \mathbb{R}^2} \rho_n \partial_t \varphi dt dx + \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\sqrt{\rho_n}(\sqrt{\rho_n} \mathbb{Q} u_n)) \cdot \nabla \varphi dt dx = 0, \quad (2.128) $$

$$ \begin{aligned} & \int_{\mathbb{R}^2} (\rho_n u_n) \varphi(0, \cdot) dx + \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\rho_n P u_n + \sqrt{\rho_n}(\sqrt{\rho_n} \mathbb{Q} u_n)) \partial_t \varphi dx dt \\ & + \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\sqrt{\rho_n} \mathbb{Q} u_n \otimes \sqrt{\rho_n} \mathbb{Q} u_n + \sqrt{\rho_n} P u_n \otimes \sqrt{\rho_n} \mathbb{Q} u_n + \sqrt{\rho_n} \mathbb{Q} u_n \otimes \sqrt{\rho_n} P u_n \\ & + \rho_n P u_n \otimes P u_n) : \nabla \varphi dx dt + \int_{\mathbb{R}_+ \times \mathbb{R}^2} P(\rho_n) \mathrm{div}(\varphi) dx dt \\ & - \frac{\kappa}{2} \int_{\mathbb{R}_+ \times \mathbb{R}^2} \nabla \rho_n \cdot \nabla(\mathrm{div}(\varphi)) dx dt \\ & - 2\kappa \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\nabla \sqrt{\rho_n} \otimes \nabla \sqrt{\rho_n}) : \nabla \varphi dx dt = 0. \end{aligned} \quad (2.129) $$

Then passing to the limit in $n$ we have using (1.15), (2.123), (2.124), (2.96), (2.100), (2.101), (2.125)

$$ \int_{\mathbb{R}^2} \rho_0 \varphi(0, \cdot) dx + \int_{\mathbb{R}_+ \times \mathbb{R}^2} \rho \partial_t \varphi dt dx + \int_{\mathbb{R}_+ \times \mathbb{R}^2} \sqrt{\rho} \Lambda \cdot \nabla \varphi dt dx = 0, \quad (2.130) $$

and

$$ \begin{aligned} & \int_{\mathbb{R}^2} \rho_0 u_0 \cdot \varphi(0, \cdot) dx + \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\rho \Lambda_2 + \sqrt{\rho} \Lambda) \cdot \partial_t \varphi dx dt \\ & + \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\Lambda \otimes \Lambda + \sqrt{\rho} \Lambda_2 \otimes \Lambda + \Lambda \otimes \sqrt{\rho} \Lambda_2 + \rho \Lambda_2 \otimes \Lambda_2) : \nabla \varphi dx dt \\ & + \int_{\mathbb{R}_+ \times \mathbb{R}^2} P(\rho) \mathrm{div}(\varphi) dx dt - \frac{\kappa}{2} \int_{\mathbb{R}_+ \times \mathbb{R}^2} \nabla \rho \cdot \nabla(\mathrm{div}(\varphi)) dx dt \\ & - 2\kappa \int_{\mathbb{R}_+ \times \mathbb{R}^2} (\nabla \sqrt{\rho} \otimes \nabla \sqrt{\rho}) : \nabla \varphi dx dt = 0. \end{aligned} \quad (2.131) $$

Moreover we have using (1.16)

$$ \begin{aligned} & \int_{\mathbb{R}^2} \left( \frac{1}{2} \rho_n^0 |u_n^0|^2(x) + \kappa |\nabla \sqrt{\rho_n^0}(x)|^2 + j_\gamma(\rho_n^0(x)) \right) dx \\ & \xrightarrow{n \to \infty} \int_{\mathbb{R}^2} \left( \frac{1}{2} \rho^0 |u^0|^2(x) + \kappa |\nabla \sqrt{\rho^0}(x)|^2 + j_\gamma(\rho^0(x)) \right) dx. \end{aligned} $$

Then passing to the limit inf in (2.11) and using Fatou's lemma we have for any $t \ge 0$.

$$ E(t) \le E(0). \quad (2.132) $$
Furthermore since $\text{div}(\rho_n \mathbb{P}u_n) = 0$ for any $n \in \mathbb{N}$ we deduce from Proposition (2.6) that $\text{div}(\rho \Lambda_2) = 0$ in the sense of distribution.

**Acknowledgement** BH has been partially funded by the ANR project INFAMIE ANR-15-CE40-0011. We thank Corentin Audiard for some useful comments.

**Funding** Open access funding provided by COUPERIN.

**Declarations**

**Competing Interests** The authors declare no competing interests.

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**References**

1. Antonelli, P., Marcati, P.: On the finite energy weak solutions to a system in quantum fluid dynamics. Commun. Math. Phys. **287**(2), 657–686 (2009)

2. Antonelli, P., Marcati, P.: The quantum hydrodynamics system in two space dimensions. Arch. Ration. Mech. Anal. **203**(2), 499–527 (2012)

3. Antonelli, P., Marcati, P., Zheng, H.: Genuine hydrodynamics analysis to 1-D QHD system: existence, dispersion and stability. Commun. Math. Phys. **383**, 2113–2161 (2021)

4. Antonelli, P., Marcati, P., Zheng, H.: An intrinsically hydrodynamic approach to multidimensional QHD systems. Arch. Ration. Mech. Anal. **247**, 24 (2023)

5. Audiard, C.: Small energy traveling waves for the Euler-Korteweg system. Nonlinearity **30**(9), 3362–3400 (2017)

6. Audiard, C.: Existence of multi-traveling waves in capillary fluids. Proc. R. Soc. Edinb., Sect. A, Math. **150**(6), 2905–2936 (2020)

7. Audiard, C., Haspot, B.: Global well-posedness of the Euler–Korteweg system for small irrotational data. Commun. Math. Phys. **351**(1), 201–247 (2017)

8. Audiard, C., Haspot, B.: From Gross-Pitaevskii equation to Euler Korteweg system, existence of global strong solutions with small irrotational initial data. Ann. Mat. Pura Appl. **197**(3), 721–760 (2018)

9. Benzoni-Gavage, S., Mazet, L., Descombes, S., Jamet, D.: Structure of Korteweg models and stability of diffuse interfaces. Interfaces Free Bound. **7**(4), 371–414 (2005)

10. Benzoni-Gavage, S., Danchin, R., Descombes, S.: On the well-posedness for the Euler-Korteweg model in several space dimensions. Indiana Univ. Math. J. **56**, 1499–1579 (2007)

11. Bresch, D., Gisclon, M., Lacroix-Violet, I.: On Navier-Stokes-Korteweg and Euler-Korteweg systems: application to quantum fluid models. Arch. Ration. Mech. Anal. **233**(3), 975–1025 (2019)

12. Carles, R., Danchin, R., Madelung, J.-C.S.: Gross-Pitaevskii and Korteweg. Nonlinearity **25**(10), 2843–2873 (2012)

13. Christ, M., Kiselev, A.: Maximal operators associated to filtrations. J. Funct. Anal. **179**, 409–425 (2001)

14. Coste, C.: Nonlinear Schrödinger equation and superfluid hydrodynamics. Eur. Phys. J. B, Condens. Matter Complex Syst. **1**(2), 245–253 (1998)

15. Ferry, D., Zhou, J.-R.: Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling. Phys. Rev. B **48**, 7944–7950 (1993)

16. Fujita, B.H.: Kato solution for compressible Navier-Stokes equation with axisymmetric initial data and zero Mach number limit. Commun. Contemp. Math. **22**(05), 1950041 (2020)

17. Giesselman, J., Tzavaras, A.: Stability properties of the Euler-Korteweg system with non-monotone pressure. Appl. Anal. **96**(9), 1528–1546 (2017)

18. Grant, J.: Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations. J. Phys. A, Math. Nucl. Gen. **6**, 151–153 (1973)
19. Hientzsch, L.E.: Nonlinear Schrödinger equations and quantum fluids non vanishing at infinity: Incom-pressible limit and quantum vortices. PhD Thesis, Gran Sasso Science Institute (2019)

20. Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14. Am. Math. Soc., Providence (2001)

21. Linares, F., Ponce, G.: Introduction to Nonlinear Dispersive Equations, 2nd edn. Springer, Berlin, 316 p. (2015)

22. Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Vol 2, Compressible Models. Oxford University Press, London (1996)

23. Loffredo, M., Morato, L.: On the creation of quantum vortex lines in rotating He II. Nuovo Cimento **108B**, 205–215 (1993). Conditions at infinity

24. Serre, D.: Invariants et dégénérescence symplectique de l'équation d'Euler des fluides parfaits incompresibles. C. R. Acad. Sci. Paris **298**, 349–352 (1984)

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