# Remark on the well-posedness of weakly dispersive equations

Jiao He, Youcef Mammeri

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## To cite this version:

Jiao He, Youcef Mammeri. Remark on the well-posedness of weakly dispersive equations. ESAIM: Proceedings and Surveys, 2018, 64, pp.111-120. 10.1051/proc/201864111. hal-03636446

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Submitted on 3 Jul 2025

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# REMARK ON THE WELL-POSEDNESS OF WEAKLY DISPERSIVE EQUATIONS

JIAO HE<fnref n="1" /> AND YOUCF MAMMERI<fnref n="2" />,<fnref n="3" />

**Abstract.** We improve the results about the well-posedness of the regularized fractional dispersive equation $(1 + D_x^\alpha)u_t + u_x + uu_x = 0$ when $0 < \alpha \le 1$. When $\alpha < 1$, the existence and uniqueness of vanishing viscosity solution is proved.

## INTRODUCTION

Fractional Benjamin-Bona-Mahony (fBBM) equation

$$u_t + u_x + uu_x + D_x^\alpha u_t = 0$$

was introduced by Linares, Pilod and Saut [16] to investigate the role of weak dispersion ($0 < \alpha < 1$) on the solution of the Burgers equation

$$u_t + u_x + uu_x = 0.$$

They showed the local in time well-posedness in Sobolev spaces using energy estimates. Nevertheless, this method does not provide the uniqueness. Indeed, the difference $w = u - v$ between two solutions satisfies (see inequality (4.31) of [16])

$$\frac{d}{dt} \|w\|_{H^{s+\frac{\alpha}{2}}}^2 \le \int_{\mathbb{R}} |(-\Delta)^{s/2}(u+v)| w_x(-\Delta)^{s/2} w|dx + \frac{1}{2} \int_{\mathbb{R}} |(u+v)_x(-\Delta)^{s/2} w|^2 dx \\ + \int_{\mathbb{R}} |R(-\Delta)^{s/2} w| dx + \int_{\mathbb{R}} |(-\Delta)^{s/2}(u+v)_x w(-\Delta)^{s/2} w| dx,$$

where $\int_{\mathbb{R}} |R(-\Delta)^{s/2} w| dx \le \|u+v\|_{H^{s+\frac{\alpha}{2}}} \|w\|_{H^{s+\frac{\alpha}{2}}}^2$. But the last term can not be uniformly controlled by the $H^{s+\frac{\alpha}{2}}$-norm if $0 < \alpha < 1$.

When $0 < \alpha < 1$, it seems difficult to obtain the global well-posedness. It has been shown by Bona and Saut [3] that the linearization around 0 has blow-up solution and Klein, Saut numerically observe blow-up when $0 < \alpha < \frac{1}{3}$ [15]. While $\alpha = 1$, global well-posedness can be obtained thanks to Brezis-Gallouët estimates [5, 18].

The paper is organized as follows. In Section 2, we improve the regularity of the global well-posedness when $\alpha = 1$. In Section 3, we deal with the uniqueness of vanishing viscosity solution for $0 < \alpha < 1$.

<footnote n="1">¹ Institut Camille Jordan, CNRS UMR 5208, Université Claude Bernard Lyon 1, 69622 Villeurbanne, France;
e-mail: jiao.he@math.univ-lyon1.fr</footnote>

<footnote n="2">² Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80069 Amiens, France</footnote>

<footnote n="3">³ Institut de Génétique, Environnement et Protection des Plantes, INRA UMR 1349, 35650 Le Rheu, France;
e-mail: youcef.mammeri@u-picardie.fr</footnote>
# 1. THE REGULARIZED BENJAMIN-ONO EQUATION

When $\alpha = 1$, the equation can be rewritten as the Benjamin-Ono equation under the form

$$u_t + H(u_x)_t + u_x + uu_x = 0,$$

where $H$ is the Hilbert defined by its Fourier symbol $\widehat{H}(u)(\xi) = -i \operatorname{sgn}(\xi) \hat{u}(\xi)$.

**Theorem 1.1.** Let $\alpha = 1$ and $u_0 \in H^{\frac{1}{2}}(\mathbb{R})$. There exists a unique solution $u \in C(\mathbb{R}; H^{\frac{1}{2}}(\mathbb{R}))$ of the initial value problem

$$\left\{ \begin{array}{l} u_t + u_x + uu_x + D_x u_t = 0, \\ u(x, 0) = u_0(x). \end{array} \right. \qquad (1)$$

Moreover, for all $t \in \mathbb{R}$

$$\|u(t)\|_{H^{\frac{1}{2}}} = \|u_0\|_{H^{\frac{1}{2}}},$$

and the map $u_0 \in H^{\frac{1}{2}}(\mathbb{R}) \to u \in C(\mathbb{R}; H^{\frac{1}{2}}(\mathbb{R}))$ is continuous.

The proof is done in two steps: first, a compactness argument is used to obtain a weak solution, then the uniqueness of the weak solution provides the strong continuity of the weak solution.

Let $(u_{n,0})_{n \in \mathbb{N}}$ be a sequence of $H^1(\mathbb{R})$ such that $u_{n,0} \to u_0$ in $H^{\frac{1}{2}}(\mathbb{R})$. We denote by $u_n(t) \in C(\mathbb{R}; H^1(\mathbb{R}))$ the solution of the initial value problem associated with the initial datum $u_{n,0}$. Then it is proved that the energy [18]

$$E(t) = \int_{\mathbb{R}} u_n^2 + \|D_x\|_{H^{\frac{1}{2}}}^2 u_n^2 dx$$

is preserved with respect to time, i.e.

$$\|u_n\|_{H^{\frac{1}{2}}}^2 = \|u_{n,0}\|_{H^{\frac{1}{2}}}^2 \le C,$$

and the sequence $(u_n)_{n \in \mathbb{N}}$ is bounded in $C(\mathbb{R}; H^{\frac{1}{2}}(\mathbb{R}))$.

On the other hand, multiplying (1) by $\partial_t u_n$ and integrating over space gives

$$\int_{\mathbb{R}} |\partial_t u_n|^2 dx = - \int_{\mathbb{R}} \partial_t u_n \partial_x (1 + D_x)^{-1} (u_n + \frac{u_n^2}{2}) dx,$$

and Young's inequality provides, for $\varepsilon > 0$.

$$\begin{aligned} \|\partial_t u_n\|_{L^2}^2 &= \int_{\mathbb{R}} |\partial_t u_n|^2 dx \le \left| \int_{\mathbb{R}} \partial_t u_n \partial_x (1+D_x)^{-1} u_n dx + \int_{\mathbb{R}} \partial_t u_n \partial_x (1+D_x)^{-1} \frac{u_n^2}{2} dx \right| \\ &\le \varepsilon \left| \int_{\mathbb{R}} \partial_t u_n|^2 dx + \frac{1}{2\varepsilon} \int_{\mathbb{R}} |\partial_x(1+D_x)^{-1} u_n|^2 dx + \frac{1}{2\varepsilon} \int_{\mathbb{R}} |\partial_x(1+D_x)^{-1} \frac{u_n^2}{2}|^2 dx \right| \\ &\le \varepsilon \|\partial_t u_n\|_{L^2}^2 + \frac{1}{2\varepsilon} \|\partial_x(1+D_x)^{-1} u_n\|_{L^2}^2 + \frac{1}{2\varepsilon} \left\| \partial_x(1+D_x)^{-1} \frac{u_n^2}{2} \right\|_{L^2}^2 \\ &\le \varepsilon \|\partial_t u_n\|_{L^2}^2 + \frac{1}{4\varepsilon} (\|u_n\|_{L^2}^2 + \|u_n^2\|_{L^2}^2) \\ &\le \varepsilon \|\partial_t u_n\|_{L^2}^2 + \frac{1}{4\varepsilon} (\|u_n\|_{L^2}^2 + \|u_n\|_{L^4}^4) \\ &\le \varepsilon \|\partial_t u_n\|_{L^2}^2 + \frac{1}{4\varepsilon} (\|u_n\|_{L^2}^2 + \|u_n\|_{H^{\frac{1}{2}}}^4). \end{aligned}$$

Taking $\varepsilon = \frac{1}{2}$, it gets

$$\|\partial_t u_n\|_{L^2} \le \|u_n\|_{H^{\frac{1}{2}}}^2 \le \|u_{n,0}\|_{H^{\frac{1}{2}}}^2 \le C.$$
Thus,

$$ ||u_n(t) - u_n(s)||_{L^2} = \left\| \int_s^t \partial_t u_n(r) dr \right\|_{L^2} \le \int_s^t ||\partial_t u_n(r)||_{L^2} dr \le C|t-s|, $$

and the sequence $(u_n)_{n \in \mathbb{N}}$ is uniformly bounded and equicontinuous.

We deduce according to the Rellich theorem for all $T > 0$, there exists $u \in C_w([0, T]; H^{\frac{1}{2}}(\mathbb{R})) \cap C([0, T]; L^2(\mathbb{R}))$ and a subsequence $(u_{n_k})_{k \in \mathbb{N}}$ such that $\forall t \in [0, T]$

$$ u_{n_k}(t) \to u(t) \text{ in } H^{\frac{1}{2}}(\mathbb{R}) \text{ and } u_{n_k}(t) \to u(t) \text{ in } L^2(\mathbb{R}). $$

Since the subsequence $(u_{n_k})_{k \in \mathbb{N}}$ satisfies for all $v \in C_c^\infty(\mathbb{R})$

$$ \int_0^T \int_{\mathbb{R}} v \partial_t u_{n_k} + v \partial_t D_x u_{n_k} + v \partial_x \left( u_{n_k} + \frac{u_{n_k}^2}{2} \right) dx dt = 0, $$

the limit verifies

$$ \int_0^T \int_{\mathbb{R}} u \partial_t v + u \partial_t D_x v + \left( u + \frac{u^2}{2} \right) \partial_x v dx dt = 0. $$

The function $u \in C_w([0, T]; H^{\frac{1}{2}}(\mathbb{R})) \cap C([0, T]; L^2(\mathbb{R}))$ is a weak solution of the equation (1) and

$$ \|u(t)\|_{H^{\frac{1}{2}}} \le \liminf_{k \to \infty} \|u_{n_k}(t)\|_{H^{\frac{1}{2}}} \le \|u_0\|_{H^{\frac{1}{2}}}. \qquad (2) $$

Suppose now that the weak solution of the Cauchy problem $u \in C_w([0, T]; H^{\frac{1}{2}}(\mathbb{R})) \cap C([0, T]; L^2(\mathbb{R}))$ is unique. By reversing time in the regularized Benjamin-Ono equation (1), we have

$$ \|u(-t)\|_{H^{\frac{1}{2}}} \le \|u(0)\|_{H^{\frac{1}{2}}} \text{ or } \|u(0)\|_{H^{\frac{1}{2}}} \le \|u(t)\|_{H^{\frac{1}{2}}}. $$

From inequality (2) and from the uniqueness of the weak solution, we obtain

$$ \|u(t)\|_{H^{\frac{1}{2}}} = \|u_0\|_{H^{\frac{1}{2}}} $$

and the solution $u$ belongs to $C([0, T]; H^{\frac{1}{2}}(\mathbb{R}))$. Note that we also obtain the continuity with respect to the initial data since we proved that for $u_{n,0} \to u_0$ in $H^{\frac{1}{2}}(\mathbb{R})$, then the respective solution $(u_n)$ verifies $u_n \to u$ in $C([0, T]; H^{\frac{1}{2}}(\mathbb{R}))$.

It remains to prove the uniqueness of the weak solution. We are inspired by the method introduced by Yudovich [21]. We need a Trudinger-type estimates proved by Gérard and Grellier [9] in the torus $\mathbb{T}$.

**Lemma 1.2.** There exists a constant $C > 0$ such that for all $2 < p < \infty$, we have

$$ \|u\|_{L^p(\mathbb{R})} \le C \sqrt{p} \|u\|_{H^{\frac{1}{2}}(\mathbb{R})}. $$

*Proof.* The proof is similar to [9] except that $u$ is split as

$$ u = u_{>\lambda} + u_{<\lambda} = \frac{1}{2\pi} \int_{|\xi| \le \lambda} e^{i\xi x} \hat{u}(\xi) d\xi + \frac{1}{2\pi} \int_{|\xi| \ge \lambda} e^{i\xi x} \hat{u}(\xi) d\xi. $$

□
Let $u$ and $v \in C_w([0, T]; H^{\frac{1}{2}}(\mathbb{R})) \cap C([0, T]; L^2(\mathbb{R}))$ be two weak solutions of (1) starting from the same initial datum. Consider the function $g$ defined as

$$g(t) = \|u(t) - v(t)\|_{L^2}^2.$$

Then, for $w := u - v$ and $P(D_x) := \partial_x(1 + D_x)^{-1}$, we have

$$
\begin{aligned}
g'(t) &= 2 \int_{\mathbb{R}} (u-v)(u_t - v_t)dx = -2 \int_{\mathbb{R}} (u-v)P(D_x) \left(u + \frac{u^2}{2} - v - \frac{v^2}{2}\right) \\
&= -2 \int_{\mathbb{R}} wP(D_x)(w) + wP(D_x)(w(u+v)) dx = -2 \int_{\mathbb{R}} wP(D_x)(w(u+v)) dx \\
&= - \int_{\mathbb{R}} w^2 P(D_x)(u+v)dx - \int_{\mathbb{R}} w(P(D_x)w)(u+v)dx - \int_{\mathbb{R}} wRdx \\
&=: \text{I} + \text{II} + \text{III}
\end{aligned}
$$

where

$$R = P(D_x)(w(u + v)) - (u + v)P(D_x)w - wP(D_x)(u + v).$$

Since $w^2 = w^2(1-\frac{1}{p})w^{\frac{2}{p}}$, the Hölder inequality provides

$$
\begin{aligned}
|I| = \int_{\mathbb{R}} w^2 P(D_x)(u+v)dx &\le \|w^2(1-\frac{1}{p})\|_{L^{\frac{p}{p-1}}} \|w^{\frac{2}{p}} P(D_x)(u+v)\|_{L^p} \\
&\lesssim \|w\|_{L^2}^{2(1-\frac{1}{p})} \left( \|u\|_{L^4}^{\frac{2}{p}} + \|v\|_{L^4}^{\frac{2}{p}} \right) \|P(D_x)(u+v)\|_{L^{2p}}.
\end{aligned}
$$

We note that the operator $P(D_x)$ is bounded in $L^p$, for $1 < p < +\infty$ according to the Mikhlin-Hörmander theorem [17, 19]. The Sobolev and the Trudinger inequalities imply

$$|I| \lesssim \|w\|_{L^2}^{2(1-\frac{1}{p})} \left( \|u\|_{L^4}^{\frac{2}{p}} + \|v\|_{L^4}^{\frac{2}{p}} \right) (\|u\|_{L^{2p}} + \|v\|_{L^{2p}}) \lesssim \sqrt{2p} g(t)^{1-\frac{1}{p}} \left( \|u\|_{L^4}^{\frac{2}{p}} + \|v\|_{L^4}^{\frac{2}{p}} \right) (\|u\|_{H^{\frac{1}{2}}} + \|v\|_{H^{\frac{1}{2}}})$$

and from inequality (2)

$$|I| \lesssim \sqrt{p} g(t)^{1-\frac{1}{p}}.$$

Similarly, we have

$$
\begin{aligned}
|II| = \int_{\mathbb{R}} w(P(D_x)w)(u+v)dx &\le \|w^{1-\frac{2}{p}} P(D_x)w\|_{L^{\frac{p}{p-1}}} \|w^{\frac{2}{p}}(u+v)\|_{L^p} \\
&\lesssim \|w^{1-\frac{2}{p}}\|_{L^{\frac{2p}{p-2}}} \|P(D_x)w\|_{L^2} \left( \|u\|_{L^4}^{\frac{2}{p}} + \|v\|_{L^4}^{\frac{2}{p}} \right) \|u+v\|_{L^{2p}} \\
&\lesssim \|w\|_{L^2}^{1-\frac{2}{p}} \|w\|_{L^2} (\|u\|_{L^{2p}} + \|v\|_{L^{2p}}) \\
&\lesssim g(t)^{1-\frac{1}{p}} (\|u\|_{L^{2p}} + \|v\|_{L^{2p}}) \lesssim \sqrt{p} g(t)^{1-\frac{1}{p}} \|u\|_{H^{\frac{1}{2}}} \lesssim \sqrt{p} g(t)^{1-\frac{1}{p}}.
\end{aligned}
$$

We can write thanks to the fractional Leibniz rule [14].

**Lemma 1.3.** We have for $\frac{1}{p} = \frac{1}{q} + \frac{1}{r}$

$$\|P(D)(uv) - vP(D)(u) - uP(D)(v)\|_{L^p} \lesssim \|u\|_{L^q} \|v\|_{L^r}.$$
We deduce

$$|\text{III}| = \int_{\mathbb{R}} wRdx \lesssim \|w\|_{L^2} \|u + v\|_{L^q} \|w\|_{L^r} \lesssim \|w\|_{L^2} (\|u\|_{L^q} + \|v\|_{L^q}) \|w\|_{L^2}^{1-\frac{2}{p}} \|w\|_{L^p}^{\frac{2}{p}}$$

where

$$\frac{1}{r} = \frac{1}{2} - \frac{1}{q} \quad \text{or} \quad \frac{1}{r} = \frac{1}{2} - \frac{1}{p} \left(1 - \frac{2}{p}\right).$$

Thus

$$|\text{III}| \lesssim \sqrt{q} \|w\|_{L^2}^{2-\frac{2}{p}} \|w\|_{L^p}^{\frac{2}{p}} \|u\|_{H^{\frac{1}{2}}} \lesssim \sqrt{q} g(t)^{1-\frac{1}{p}} (\|u\|_{L^p}^{\frac{2}{p}} + \|v\|_{L^p}^{\frac{2}{p}}) \lesssim p^{\frac{1}{p}} \sqrt{q} g(t)^{1-\frac{1}{p}} \|u\|_{H^{\frac{1}{2}}}^{\frac{2}{p}} \lesssim p^{\frac{1}{p}} \sqrt{q} g(t)^{1-\frac{1}{p}}.$$

Taking $p > 2$ large enough so that

$$|\text{III}| \lesssim p^{\frac{1}{p}} \sqrt{\frac{p}{1-\frac{2}{p}}} g(t)^{1-\frac{1}{p}} \lesssim \sqrt{p} g(t)^{1-\frac{1}{p}}$$

implies

$$|g'(t)| \leq C \sqrt{p} g(t)^{1-\frac{1}{p}}, \quad \forall 2 < p < \infty,$$

with $C = C(\|u_0\|_{H^{\frac{1}{2}}}, T)$ independent of $p$. Then

$$g(t)^{\frac{1}{p}} \leq C \frac{t}{\sqrt{p}},$$

or

$$g(t) \leq C \frac{t^p}{p^{\frac{2}{p}}} \to 0, \quad \text{as } p \to \infty.$$

Finally, $g(t) \equiv 0$, and $u = v$.

## 2. THE WEAK DISPERSIVE EQUATION

Let us come back to the initial value problem, for $0 < \alpha < 1$,

$$\begin{cases} u_t + u_x + uu_x + D_x^\alpha \partial_t u = 0, \\ u(x, 0) = u_0(x). \end{cases} \qquad (3)$$

The existence is proved in [16] using energy estimates.

**Theorem 2.1.** Let $0 < \alpha < 1, r > \max(1, \frac{3}{2} - \alpha)$ and $u_0 \in H^r(\mathbb{R})$. Then the Cauchy problem has at least one solution in $H^r(\mathbb{R})$.

We briefly remind the proof in order to highlight the loss of uniqueness (see [16] for details).

*Proof.* For $r = s + \frac{\alpha}{2}$, we define $J^s = (I - \Delta)^{\frac{\alpha}{2}}$ by its Fourier transform

$$\widehat{J^s u}(\xi) := (I - \Delta)^{\frac{\alpha}{2}} u(\xi) = (1 + \xi^2)^{\frac{\alpha}{2}} \hat{u}(\xi).$$

Applying the operator $J^s$ to (3), multiplying by $J^s u$ and integrating by part over space, we obtain

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}} |J^s u|^2 + |J^s D_x^{\frac{\alpha}{2}} u|^2 dx &= \int_{\mathbb{R}} -J^s u J^s (uu_x) dx \\ &= \int_{\mathbb{R}} |J^s u|^2 u_x + u J^s u J^s \partial_x u + R J^s u dx = \frac{1}{2} \int_{\mathbb{R}} |J^s u|^2 u_x + \int_{\mathbb{R}} R J^s u dx, \end{aligned}$$
where, thanks to Leibniz's rule,

with, for all $0 < \epsilon < s$,

The Hölder inequality provides

$$J^s(uu_x) = uJ^s u_x + u_x J^s u + R,$$

$$\|R\|_{L^{\frac{2}{1+\alpha}}} \lesssim \|J^{s-\epsilon}u\|_{L^{\frac{4}{1+\alpha}}} \|J^{\epsilon}u_x\|_{L^{\frac{4}{1+\alpha}}}.$$

$$\int_{\mathbb{R}} |(J^s u)^2 u_x| dx \le \|u_x\|_{L^{\frac{2}{1+\alpha}}} \|J^s u\|_{L^{\frac{2}{1-\alpha}}}^2$$

and

$$\int_{\mathbb{R}} |RJ^s u| dx \le \|R\|_{L^{\frac{2}{1+\alpha}}} \|J^s u\|_{L^{\frac{2}{1-\alpha}}}.$$

Using Sobolev's embedding

$$H^{\frac{\alpha}{2}}(\mathbb{R}) \hookrightarrow L^{\frac{2}{1-\alpha}}, \quad H^{\frac{\alpha}{2}+\epsilon} \hookrightarrow L^{\frac{4}{1+\alpha}}, \quad H^{s+\frac{\alpha}{2}-\epsilon-1} \hookrightarrow L^{\frac{4}{1+\alpha}},$$

we finally find

$$\frac{d}{dt} \|u\|_{H^{s+\frac{\alpha}{2}}}^2 \lesssim \|u\|_{H^{s+\frac{\alpha}{2}}}^2$$

and we conclude with the Gronwall lemma.

Let us try to prove the uniqueness. Let $u$ and $v$ be two solutions of (3) and denote $w = u - v$. We have

$$w_t + \partial_x (1 + D_x^\alpha)^{-1} \left( w + \frac{1}{2} w(u + v) \right) = 0.$$

Using the fractional Leibniz rule and integrating by parts, it gets

$$\frac{1}{2} \int_{\mathbb{R}} |J^s w|^2 + |J^{s+\frac{\alpha}{2}} w|^2 dx \le \int_{\mathbb{R}} |J^s(u+v)w_x J^s w| dx + \frac{1}{2} \int_{\mathbb{R}} |(u+v)_x ||J^s w|^2 dx \\ + \int_{\mathbb{R}} |RJ^s w| dx + \int_{\mathbb{R}} |J^s(u+v)_x w J^s w| dx.$$

Here, for all $0 < \epsilon < s$ and $\frac{1}{p} + \frac{1}{q} = \frac{1}{2}$,

$$\|R\|_{L^2} \le \|J^{s-\epsilon}(u+v)\|_{L^p} \|J^\epsilon w\|_{L^q}.$$

Even if the three first terms are bounded by $\|u+v\|_{H^{s+\frac{\alpha}{2}}} \|w\|_{H^{s+\frac{\alpha}{2}}}^2$, the last one can not be uniformly controlled by the $H^{s+\frac{\alpha}{2}}$-norm when $0 < \alpha < 1$.

To avoid this difficulty, we propose the following regularization [11]

$$\left\{ \begin{array}{l} u_t^\varepsilon + u_x^\varepsilon + u^\varepsilon u_x^\varepsilon + D_x^\alpha u_t^\varepsilon - \varepsilon u_{xx}^\varepsilon = 0, \\ u^\varepsilon(0, x) = u_0^\varepsilon(x). \end{array} \right. \qquad (4)$$

**Lemma 2.2.** Let $0 < \alpha < 1$ and $r \ge 0$. We have

$$\|\partial_x (1 + D_x^\alpha)^{-1} S_t(uv)\|_{H^r} \le C(\varepsilon, t) \|u\|_{H^r} \|v\|_{H^r},$$

where

$$S_t u = \mathcal{F}^{-1} \left( e^{-\frac{i\epsilon+\varepsilon\xi^2}{1+|\xi|^\alpha} t} \right) * u(x), \quad \text{and} \quad C(\varepsilon, t) = C(\varepsilon t)^{-\frac{7-\alpha}{2-\alpha}}.$$
Proof. The idea of the proof is introduced in [4, 13]. By duality, it is enough to show that for all function $w \in S(\mathbb{R})$,

$$ \int_{\mathbb{R}} \partial_x (1 + D_x^\alpha)^{-1} S_t (u(x)v(x)) \overline{w}(x) dx \leq C(t) \|u\|_{H^r} \|v\|_{H^r} \|w\|_{H^{-r}}. $$

The Plancherel identity provides

$$ \begin{aligned} \int_{\mathbb{R}} \partial_x (1 + D_x^\alpha)^{-1} S_t (u(x)v(x)) \overline{w}(x) dx &= \int_{\mathbb{R}} \frac{i\xi}{1 + |\xi|^\alpha} e^{-\frac{i\xi + \xi\xi^2}{1+|\xi|^\alpha} t} \hat{u}(\xi)v(\xi) \overline{\hat{w}}(\xi) d\xi \\ &= \int_{\mathbb{R}} \frac{i\xi}{1 + |\xi|^\alpha} e^{-\frac{i\xi + \xi\xi^2}{1+|\xi|^\alpha} t} \hat{u} * \hat{v}(\xi) \overline{\hat{w}}(\xi) d\xi \\ &= \int_{\mathbb{R}} \int_{\mathbb{R}} \frac{i\xi}{1 + |\xi|^\alpha} e^{-\frac{i\xi + \xi\xi^2}{1+|\xi|^\alpha} t} \hat{u}(\xi - \eta) \hat{v}(\eta) d\eta \overline{\hat{w}}(\xi) d\xi. \end{aligned} $$

Let us denote $\langle \xi \rangle = (1 + \xi^2)^{\frac{1}{2}}$, $\hat{U} = \langle \xi \rangle^r \hat{u}$, $\hat{V} = \langle \xi \rangle^r \hat{v}$, $\hat{W} = \langle \xi \rangle^{-r} \hat{w}$, it is necessary to prove

$$ \left| \int_{\mathbb{R}} \int_{\mathbb{R}} e^{-\frac{i\xi + \xi\xi^2}{1+|\xi|^\alpha} t} \frac{|\xi|\langle\xi\rangle^r}{\langle\xi\rangle^\alpha \langle\xi - \eta\rangle^r \langle\eta\rangle^r} \hat{U}(\xi - \eta) \hat{V}(\eta) \overline{\hat{W}}(\xi) d\xi d\eta \right| \leq C(\varepsilon, t) \|u\|_{H^r} \|v\|_{H^r} \|w\|_{H^{-r}}. $$

The triangle inequality $\langle \xi \rangle^r \lesssim \langle \xi - \eta \rangle^r \langle \eta \rangle^r$ implies

$$ \begin{aligned} & \left| \int_{\mathbb{R}} \int_{\mathbb{R}} e^{-\frac{i\xi + \xi\xi^2}{1+|\xi|^\alpha} t} \frac{|\xi|\langle\xi\rangle^r}{\langle\xi\rangle^\alpha \langle\xi - \eta\rangle^r \langle\eta\rangle^r} \hat{U}(\xi - \eta) \hat{V}(\eta) \overline{\hat{W}}(\xi) d\xi d\eta \right| \\ & \lesssim \int_{\mathbb{R}} \int_{\mathbb{R}} \langle\xi\rangle^{1-\alpha} e^{-\frac{\varepsilon\xi^2}{1+|\xi|^\alpha} t} |\hat{U}(\xi - \eta) \hat{V}(\eta) \overline{\hat{W}}(\xi)| d\xi d\eta \\ & \lesssim \|W\|_{L_{\xi}^2} \left\| \langle\xi\rangle^{1-\alpha} e^{-\frac{\varepsilon\xi^2}{1+|\xi|^\alpha} t} \int_{\mathbb{R}} \hat{U}(\xi - \eta) \hat{V}(\eta) d\eta \right\|_{L_{\xi}^2} \\ & \lesssim \|W\|_{L_{\xi}^2} \left\| \langle\xi\rangle^{1-\alpha} e^{-\frac{\varepsilon\xi^2}{1+|\xi|^\alpha} t} \right\|_{L_{\xi}^2} \left\| \int_{\mathbb{R}} \hat{U}(\xi - \eta) \hat{V}(\eta) d\eta \right\|_{L_{\xi}^{\infty}} \\ & \lesssim \|W\|_{L_{\xi}^2} \|U\|_{L_{\xi}^2} \|V\|_{L_{\xi}^2} \left\| \langle\xi\rangle^{1-\alpha} e^{-\frac{\varepsilon\xi^2}{1+|\xi|^\alpha} t} \right\|_{L_{\xi}^2} \\ & \lesssim \|w\|_{H^{-r}} \|u\|_{H^r} \|v\|_{H^r} \left\| \langle\xi\rangle^{1-\alpha} e^{-\frac{\varepsilon\xi^2}{1+|\xi|^\alpha} t} \right\|_{L_{\xi}^2}. \end{aligned} $$

Since for $x \in \mathbb{R}$, $\theta \in \mathbb{R}$, we have $e^{-x} \leq \frac{1}{x^\theta}$, we deduce

$$ \left\| \langle \xi \rangle^{1-\alpha} e^{-\frac{\varepsilon\xi^2}{1+|\xi|^\alpha} t} \right\|_{L_{\xi}^2} \leq \left\| \langle \xi \rangle^{1-\alpha} \frac{1}{\left( \frac{\varepsilon\xi^2 t}{1+|\xi|^\alpha} \right)^{\theta}} \right\|_{L_{\xi}^2} \leq \frac{1}{(\varepsilon t)^{\theta}} \left\| \frac{1}{\langle \xi \rangle^{(2-\alpha)\theta - (1-\alpha)}} \right\|_{L_{\xi}^2} \lesssim \frac{1}{(\varepsilon t)^{\theta}}, $$

if $(2 - \alpha)\theta - (1 - \alpha) > 1/2$, i.e. $\theta > \frac{3}{2} - \frac{\alpha}{2}$.

□

**Lemma 2.3.** Let $0 < \alpha < 1$, $r \geq 0$ and $u_0^\epsilon \in H^r(\mathbb{R})$, then there exist a time $T_\epsilon = T(\epsilon) > 0$ and a unique solution $u^\epsilon \in C([0, T_\epsilon]; H^r(\mathbb{R}))$.

*Proof.* Thanks to the Duhamel formula, $u^\epsilon$ is solution of (4) if and only if $u^\epsilon$ is the fixed point of $\Phi^\epsilon$ defined as

$$ \Phi^\epsilon u^\epsilon(t) := S_t u_0^\epsilon - \frac{1}{2} \int_0^t S_{t-\tau} \left( (1 + D_x^\alpha)^{-1} \partial_x u^\epsilon(\tau)^2 \right) d\tau $$
where $S_t u = \mathcal{F}^{-1}\left(e^{-\frac{i\xi+\xi\xi^2}{1+|\xi|^\alpha}t}\right) * u(x)$. Let $\bar{B}_T$ be the closed ball

$$ \bar{B}_T := \{u \in C([0,T]; H^r(\mathbb{R})), \|u\|_{L^\infty([0,T], H^r(\mathbb{R}))} \le 2 \|u_0\|_{H^r}\}. $$

We prove that $\Phi^\varepsilon(\bar{B}_T) \subseteq \bar{B}_T$ and $\Phi^\varepsilon$ is a contraction mapping on $\bar{B}_T$. Let $u^\varepsilon \in \bar{B}_T$, we have

$$ \begin{aligned} \|\Phi^\varepsilon u^\varepsilon(t)\|_{H^r(\mathbb{R})} &= \left\| S_t u_0 - \frac{1}{2} \int_0^t S_{t-\tau} (1 + D_x^\alpha)^{-1} \partial_x u^\varepsilon(\tau)^2 d\tau \right\|_{H^r} \\ &\le \|u_0^\varepsilon\|_{H^r} + \int_0^t \left\| S_{t-\tau} (1 + D_x^\alpha)^{-1} \partial_x u^\varepsilon(\tau)^2 \right\|_{H^r} d\tau. \end{aligned} $$

Lemma 2.2 gives

$$ \|S_{t-\tau}(1 + D_x^\alpha)^{-1} \partial_x u^\varepsilon(\tau)^2\|_{H^r} \le C(\varepsilon(t-\tau))^{-\frac{7}{2}-\alpha} \|u^\varepsilon(\tau)\|_{H^r}^2. $$

Note that $0 < \frac{7}{2}-\alpha < 1$, and

$$ \|\Phi^\varepsilon u^\varepsilon(t)\|_{H^r} \le \|u_0^\varepsilon\|_{H^r} + C_{\varepsilon,r} T \|u^\varepsilon\|_{L^\infty([0,T], H^r)}^2. $$

Thus, choosing $T < \frac{1}{C_{\varepsilon,r} \|u_0^\varepsilon\|_{H^r}}$ implies

$$ \|\Phi^\varepsilon u^\varepsilon(t)\|_{L^\infty([0,T], H^r)} \le 2 \|u_0^\varepsilon\|_{H^r}. $$

Let $u_1^\varepsilon, u_2^\varepsilon \in \bar{B}_T$, one gets from Lemma 2.2

$$ \begin{aligned} \|\Phi^\varepsilon u_1^\varepsilon(t) - \Phi^\varepsilon u_2^\varepsilon(t)\|_{H^r} &\le \int_0^t \|S_{t-\tau}(1 + D_x^\alpha)^{-1} \partial_x ((u_1^\varepsilon)^2 - (u_2^\varepsilon)^2)\|_{H^r} d\tau \\ &\le \int_0^t C(\varepsilon(t-\tau))^{-\frac{7}{2}-\alpha} \|u_1^\varepsilon + u_2^\varepsilon\|_{H^r} \|u_1^\varepsilon - u_2^\varepsilon\|_{H^r} d\tau \\ &\le C_{\varepsilon,r} T \|u_0^\varepsilon\|_{H^r} \|u_1^\varepsilon - u_2^\varepsilon\|_{L^\infty([0,T]; H^r)}, \end{aligned} $$

and $\Phi^\varepsilon$ is a contraction mapping on $\bar{B}_T$ as soon as $T < \frac{1}{C_{\varepsilon,r} \|u_0^\varepsilon\|_{H^r}}$. Finally, there exists a unique fixed point $u^\varepsilon := \Phi^\varepsilon(u^\varepsilon)$ in $\bar{B}_T$ for $T < \frac{1}{C_{\varepsilon,s} \|u_0^\varepsilon\|_{H^r}}$.

To obtain the continuity with respect to the initial data, for $u_0^\varepsilon$ and $v_0^\varepsilon$ in $H^r(\mathbb{R})$ with $\|u_0^\varepsilon\|_{H^r} \le M$, $\|v_0^\varepsilon\|_{H^r} \le M$, we consider $u^\varepsilon, v^\varepsilon$ the respective solution. Let $0 \le t \le T = \frac{1}{C_{\varepsilon,s}M}$, we find similarly

$$ \begin{aligned} \|u^\varepsilon(t) - v^\varepsilon(t)\|_{H^r} &\le \|S_t(u_0^\varepsilon - v_0^\varepsilon)\|_{H^r} + \int_0^t \|S_{t-\tau}(1 + D_x^\alpha)^{-1} \partial_x ((u^\varepsilon)^2 - (v^\varepsilon)^2)\|_{H^r} d\tau \\ &\le \|u_0^\varepsilon - v_0^\varepsilon\|_{H^r} + C_{\varepsilon,r} T M \|u^\varepsilon - v^\varepsilon\|_{L^\infty([0,T]; H^r)}, \end{aligned} $$

or in other words, since $1 - C_{\varepsilon,r}TM > 0$,

$$ \|u^\varepsilon(t) - v^\varepsilon(t)\|_{H^r} \le \frac{1}{1 - C_{\varepsilon,r} T M} \|u_0^\varepsilon - v_0^\varepsilon\|_{H^r}. $$

□

**Lemma 2.4.** Let $0 < \alpha < 1$, $r \ge 2 - \frac{\alpha}{2}$ and $u_0^\varepsilon \in H^r(\mathbb{R})$, then $T_\varepsilon$ can be chosen independent of $\varepsilon$.
Proof. Let $r = s + \frac{\alpha}{2}$. Applying $J^s$ to (4), multiplying by $J^s u^\varepsilon$ and integrating by parts over space, it comes

$$ \frac{d}{dt} \left( \int_{\mathbb{R}} (J^s u^\varepsilon)^2 + (J^s D_x^{\frac{\alpha}{2}} u^\varepsilon)^2 dx \right) + 2\varepsilon \int_{\mathbb{R}} (J^s u_x^\varepsilon)^2 dx = \int_{\mathbb{R}} -J^s u^\varepsilon J^s (u^\varepsilon u_x^\varepsilon) dx. $$

We obtain from Kato-Ponce commutator estimates [12]

$$ \int_{\mathbb{R}} J^s u^\varepsilon J^s (u^\varepsilon u_x^\varepsilon) dx = \frac{1}{2} \int_{\mathbb{R}} u_x^\varepsilon (J^s u^\varepsilon)^2 dx + \int_{\mathbb{R}} RJ^s u^\varepsilon dx \leq C \|u^\varepsilon\|_{H^{s+\frac{\alpha}{2}}}^3. $$

It follows that

$$ \frac{d}{dt} \|u^\varepsilon\|_{H^{s+\frac{\alpha}{2}}}^2 \leq C \|u^\varepsilon\|_{H^{s+\frac{\alpha}{2}}}^3 $$

or in other words $\|u^\varepsilon\|_{H^{s+\frac{\alpha}{2}}}^2 \leq y(t)$ where $y(t)$ satisfies the ordinary differential equation

$$ \begin{cases} y'(t) = Cy(t)^{\frac{3}{2}} \\ y(0) = \|u_0^\varepsilon\|_{H^{s+\frac{\alpha}{2}}(\mathbb{R})}^2, \end{cases} $$

which solution is given by

$$ y(t) = \frac{y(0)}{(1 - Cy(0)^{1/2}t)^2} = \frac{\|u_0^\varepsilon\|_{H^{s+\frac{\alpha}{2}}}^2}{(1 - C \|u_0^\varepsilon\|_{H^{s+\frac{\alpha}{2}}(\mathbb{R})}^2)^2}, $$

and $u^\varepsilon$ can be extended until $T = \frac{1}{2C \|u_0^\varepsilon\|_{H^{s+\frac{\alpha}{2}}(\mathbb{R})}^2}$. $\square$

**Theorem 2.5.** Let $0 < \alpha < 1$, $r \geq 2 - \frac{\alpha}{2}$ and $u_0 \in H^r(\mathbb{R})$. Then there exist a time $T > 0$ and a unique vanishing viscosity solution $u \in C([0, T]; H^r(\mathbb{R}))$ of the initial value problem (3). In other words, the solution $u^\varepsilon$ of the initial value problem (4) converges, when $\varepsilon \to 0$, uniformly to $u$ solution of (3) in $C([0, T]; H^r(\mathbb{R}))$.

Proof. We prove that $(u^\varepsilon)_{\varepsilon \geq 0}$ is a Cauchy sequence in the Sobolev space $H^r(\mathbb{R})$. Let $u^\varepsilon$ and $v^\delta$ be two solutions of (4). The difference $w = u^\varepsilon - v^\delta$ satisfies

$$ w_t + w_x + D_x^\alpha \partial_t w - \delta w_{xx} + (u^\varepsilon w - \frac{w^2}{2})_x = (\varepsilon - \delta) u_{xx}^\varepsilon. \quad (5) $$

One multiplies (5) by $w$ and integrates over space to obtain

$$ \frac{d}{dt} \int_{-\infty}^{+\infty} w^2 + (D_x^{\frac{\alpha}{2}} w)^2 dx = \int_{-\infty}^{+\infty} -2\delta w_x^2 - u_x^\varepsilon w^2 + 2(\varepsilon - \delta)u_{xx}^\varepsilon w dx \leq \int_{-\infty}^{+\infty} -u_x^\varepsilon w^2 + 2(\varepsilon - \delta)u_{xx}^\varepsilon w dx $$

and Sobolev's inequality provide

$$ \frac{d}{dt} \int_{-\infty}^{+\infty} w^2 + (D_x^{\frac{\alpha}{2}} w)^2 dx \leq C \|u\|_{H^{2-\frac{\alpha}{2}}} \|w\|_{H^{\frac{\alpha}{2}}}^2 + C(\varepsilon - \delta) \|u\|_{H^{2-\frac{\alpha}{2}}} \|w\|_{H^{\frac{\alpha}{2}}}. $$

Finally

$$ \frac{d}{dt} \|w\|_{H^{\frac{\alpha}{2}}}^2 \leq C \|w\|_{H^{\frac{\alpha}{2}}}^2 + C(\varepsilon - \delta) \|w\|_{H^{\frac{\alpha}{2}}}. $$

and the Gronwall Lemma offers a limit of $(u^\varepsilon)_{\varepsilon \geq 0}$ in $H^{\frac{\alpha}{2}}(\mathbb{R})$.

From the preceding lemma, the map $t \in [0, T] \to u^\varepsilon(t)$ is continuous and uniformly bounded. In particular, the sequence $(u^\varepsilon(t))_{\varepsilon \geq 0}$ is weakly convergent in $H^s(\mathbb{R})$, $s \geq \frac{\alpha}{2}$ to $u(t)$ a weakly continuous and uniformly bounded function. We deduce

$$ t \in [0, T] \to (1 + D_x^\alpha)^{-1} (\partial_x u + u \partial_x u) \in H^{s+1-\alpha}(\mathbb{R}) $$
is weakly continuous and

$$u(t) = u_0 - \int_0^t (1 + D_x^\alpha)^{-1} (\partial_x u + u \partial_x u) \, \mathrm{d}x \quad \text{in } H^{s+1-\alpha}(\mathbb{R})$$

is unique. Indeed, let $v \in H^{s+1-\alpha}(\mathbb{R})$ defined by

$$v(t) = v_0 - \int_0^t (1 + D_x^\alpha)^{-1} (\partial_x v + v \partial_x v) \, \mathrm{d}x,$$

we obtain with similar computations

$$\frac{d}{dt} \|u - v\|_{H^{\frac{\alpha}{2}}}^2 \leq C \|u - v\|_{H^{\frac{\alpha}{2}}}^2$$

and the Gronwall lemma allows to conclude.

## REFERENCES

[1] J.L. BONA, AND R. SMITH, *The initial-value problem for the Korteweg-de Vries equation*, Philos. Trans. Roy. Soc. London Ser. A 278, 1287 (1975) 555-601.

[2] T.B. BENJAMIN, J.L. BONA, AND J.J MAHONY, *Model equations for long waves in nonlinear dispersive systems*, Philos. Trans. Roy. Soc. London Ser. A 272, 1220 (1972) 47-78.

[3] J.L. BONA, AND J. C. SAUT, *Dispersive blow-up II. Schrödinger-type equations, optical and oceanic rogue waves*, Chinese Ann. Math. Ser. B 31, 6 (2010) 793-818.

[4] J.L. BONA, AND N. TZVETKOV, *Sharp well-posedness results for the BBM equation*, Discrete Contin. Dyn. Syst 23, 7 (2009) 1241-1252.

[5] H. BRÉZIS, AND T. GALLOUËT, *Nonlinear Schrödinger evolution equations*, Nonlinear Anal. 4 (1980) 677-681.

[6] N. BURQ, AND F. PLANCHON, *On well-posedness for the Benjamin-Ono equation*, Mathematische Annalen 340, 3 (2008) 497-542.

[7] J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO, *Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$*, Journal American Math. Soc. 16, 3 (2003) 705-749.

[8] A. FRIEDMAN, *Partial differential equations of parabolic type*, Courier Corporation, 2013.

[9] P. GÉRARD, AND S. GRELLIER, *The cubic Szegö equation*, Annales École Normale Sup. 43, 5 (2010) 761-810.

[10] A. IONESCU, AND C. KENIG, *Global well-posedness of the Benjamin-Ono equation in low-regularity spaces*, Journal American Math. Soc. 20,3 (2007) 753-798.

[11] R.J. IÓRIO JR., *KdV, BO and friends in weighted Sobolev spaces*, Functional-Analytic Methods for Partial Differential equations, Lecture Notes in Math. 1450, Springer-Verlag, Berlin, (1990) 104-121.

[12] T. KATO, AND G. PONCE, *Commutator estimates and the Euler and Navier-Stokes equations*, Commun. Pure Appl. Math. 41,7 (1988) 891-907.

[13] C.E. KENIG, G. PONCE, AND L. VEGA, *A bilinear estimate with applications to the KdV equation*, J. American Math. Soc. 9,2 (1996) 573-603.

[14] C.E. KENIG, G. PONCE, AND L. VEGA, *Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle*, Commun. Pure Appl. Math. 46, 4 (1993) 527-620.

[15] C. KLEIN, AND J.C. SAUT, *A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation*, Physica D 295 (2015) 46-65.

[16] F. LINARES, D. PILOD, AND J.C. SAUT, *Dispersive perturbations of Burgers and hyperbolic equations I: local theory*, SIAM Journal Math. Ana. 46, 2 (2014) 1505-1537.

[17] Y. MAMMERI, *On the decay in time of solutions of some generalized regularized long waves equations*, CPAA Comm. Pure Appl. Ana. 7, 3, (2008), 513-532.

[18] Y. MAMMERI, *Long time bounds for the periodic Benjamin-Ono-BBM equation*, Nonlinear Analysis TMA 71, 10 (2009) 5010-5021.

[19] E. STEIN, *Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals*, Princeton University Press, (1993).

[20] T. TAO, *Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbb{R})$, Journal Hyperbolic Diff. Eq. 1 (2004) 27-49.

[21] V.I. YUDOVICH, *Non-stationary flows of an ideal incompressible fluid*, Zhurnal Vychislitel'noi Matematikii Matematicheskoi Fiziki, 3, 6 (1963) 1032-1066.