数理解析研究所講究録
第 1985 巻 2016 年 54-59

54

WEAK NORM INFLATION IN A FAMILY OF SOBOLEV SPACESFOR THE $2D$ EULER EQUATIONSGERARD MISIOLEKDEPARTMENT OF MATHEMATICS,
UNIVERSITY OF NOTRE DAMEANDTSUYOSHI YONEDA
DEPARTMENT OF MATHEMATICS,
TOKYO INSTITUTE OF TECHNOLOGY

1. INTRODUCTION

Consider the incompressible Euler equations1$D_{t}u:=u_{t}+u\cdot\nabla u=-\nabla\pi, t\geq 0, x\in \mathbb{R}^{2}$(1.1)$divu=0$$u(0)=u_{0}$where $u=u(t, x)$ is the velocity of the fluid and $\pi=\pi(t, x)$ is the pressure function. The
study of the Cauchy problem (1.1) has a long history going back to the work of Gyunter[5], Lichtenstein [8] and Wolibner [12] in the $1920s$ and $1930s$ . Modern expositions of
those as well as more refined results that have been obtained can be found for example
in the recent textbooks of Chemin [4], Majda and Bertozzi [9] or Bahouri, Chemin andDanchin [1].
Recently, in a series of papers Bourgain and Li [2, 3] introduced a method based on large
Lagrangian deformations to prove strong local ill-posedness results in borderline Sobolev
for any $1\leq p<\infty$
spaces $W^{n/p+1,p}$ for any $1\leq p<\infty$ and in Besov spaces
and $1<q\leq\infty$ and $n=2$ or 3. Roughly speaking, using multi-scale vortices theauthors construct a background Lagrangian flow with symmetries which they use together
with careful estimates of singular integral operators to show that the flow has a large
deformation eﬀect. They then define a suitable high frequency perturbation in such a
way so that its support intersects a neighbourhood of the point where the flow has thelargest gradient which leads to norm inflation. In this paper we describe a diﬀerent thoughrelated mechanism based on a limiting procedure that involves a scale of Banach spaces.
On the one hand the norm inflation result we prove is weaker than that obtained by
Bourgain and Li. On the other hand our method seems to be applicable in the borderline
function spaces that were left out of the analysis in [2, 3]. Details for the case of the Besov
spacewill appear elsewhere. Our objective here is to introduce this technique in aspaces. More precisely, we will prove the following
simpler setting of the Sobolev
result.
$B_{p,q}^{n/p+1}$$B_{2,1}^{2}$$W^{s_{\}}p}$lThis note may be regarded as complementary to the paper [10].

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Theorem 1.1. Letof smooth solutions
sequencesuch thatbe a sequence of positive numbers. There is a sequenceand aof (1.1) with rapidly decaying initial data

$M_{j}\nearrow\infty$$\{u_{j}\}_{j=1}^{\infty}$$\{u_{0,j}\}_{j=1}^{\infty}$$p_{j}\searrow 2$$\Vert u_{0,j}\Vert_{W^{2,p_{J\sim}}}<1$

for some

and$\Vert u_{j}(t)\Vert_{W^{2,p_{J}}}\geq M_{j}$$0<t\leq M_{j}^{-3}.$

In what follows we will need the fact that the Euler equations (1.1) are locally wellposedin the sense of Hadamard in the little H\"older space. A proof of this result is given in [11].

2. LARGE LAGRANGIAN DEFORMATION
The vorticity formulation of the Cauchy problem for the Euler equations in two dimensions has the form$\omega_{t}+u\cdot\nabla\omega=0, t\geq 0, x\in \mathbb{R}^{2}$(2.2)$u=K*\omega$$\omega(0)=\omega_{0}$where$\omega=\partial_{1}u_{2}-\partial_{2}u_{1}$

(2.3)

and

$K(x)= \frac{1}{2\pi}(-\frac{x_{2}}{|x|^{2}}, \frac{x_{1}}{|x|^{2}})$and$\nabla^{\perp}=(-\frac{\partial}{\partial x_{2}}, \frac{\partial}{\partial x_{1}})$denote the Biot-Savart kernel and the symplectic gradient, respectively. Recall that theLagrangian flow of is obtained by solving
$u$

(2.4)

$\frac{d}{dt}\eta(t, x)=u(t, \eta(t, x))$$\eta(0, x)=x$

and defines a curve in the group of diﬀeomorphisms ofIn order to construct a flow with a large deformation gradient we need to choose asuitable initial vorticity. Given a smooth radial bump function $0\leq\psi\leq 1$ with support
in the ball $B(O, 1/4)$ let$\mathbb{R}^{2}.$(2.5)$\psi_{0}(x_{1}, x_{2})=\sum_{\epsilon 1,\epsilon 2^{=\pm 1}}\epsilon_{1}\epsilon_{2}\psi(x_{1}-\epsilon_{1}, x_{2}-\epsilon_{2})$Next, fix a positive integer$N\geq 1$ define(2.6)$N_{0}$.

and for any numbers $p>0$ and $M\gg 1$ and any integer$\omega_{0}(x)=M^{-2}N^{-\frac{1}{p}}\sum_{N_{0}\leq k\leq N_{0}+N}\psi_{k}(x)$

,

where
$\psi_{k}(x)=2^{(-1+\frac{2}{p})k}\psi_{0}(2^{k}x)$whose supports are disjoint for any $k\geq 1$ and contained in the sets

supp

(2.7)

It follows thatfurthermore$\omega_{0}$$\psi_{k}\subset\bigcup_{1_{\rangle}\epsilon 2}B((\epsilon_{1}2^{-k}, \epsilon_{2}2^{-k}), 2^{-(k+2)})$.

is smooth, compactly supported and odd in both variables$x_{1},$ $x_{2}$

and

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Lemma 2.1. If $2<p<\infty$ then for any integer $N\geq 1$ we have
(2.8)$\Vert\omega_{0}\Vert_{W^{1,p_{\sim}}}<M^{-2}.$

Proof. A straightforward calculation is omitted. I
Let $u=K*\omega$ be the associated velocity field and consider its Lagrangian flow
is smooth and preserves the coordinate
given by (2.4). It is not diﬃcult to verify that
as well as the symmetries of the initial vorticity . In fact, it is hyperbolic
axes
near the origin (which is a stagnation point) and we have the following estimate$\eta(t)$$\eta(t)$$x_{1},$$x_{2}$$\omega_{0}$Proposition 2.2. Let $M\gg 1$ . For any suﬃciently large integer $N\geq 1$ and any $2<p<$
suﬃciently close to 2 we have$\infty$(2.9)$\sup_{0\leq t\leq M^{-3}}\Vert D\eta(t)\Vert_{\infty}\geq M.$

Proof. Cf. [10]; Prop. 6. I
andis a closed subspace of the H\"older space
Recall that the little H\"older spaceconsists of those functions that in addition satisfy the vanishing condition $\lim_{harrow 0}\sup_{|x-y|<h}|x-$$y|^{-\alpha}|Df(x)-Df(y)|=0$ . It is a Banach space and is equipped with the same norm
as the standard H\"older space. We will also need the following wellposedness result$C^{1,\alpha}$$c^{1,\alpha}$$\Vert\cdot\Vert_{1,\alpha}$

Proposition 2.3. The data-to-solution map of the Euler equations (1.1) is continuous infor any $0<\alpha<1.$$c^{1,\alpha}\cap L^{2}$

Proof. Cf. [11]. Note that the restriction to$L^{2}$

is needed to ensure appropriate decay at

infinity. 1

3. PROOF OF THEOREM 1.1.
and $2<p_{j}<\infty$ for which the estimate (2.9)
and choose sequencesLetof Proposition 2.2 holds.2 Proceeding as in [2] we will perturb the initial vorticity (2.6)
and work with the associated Lagrangian flows. Theorem 1.1 will be a direct consequenceof the following result.
$N_{j}\geq 1$$M_{j}\nearrow\infty$

Theorem 3.1. There is a sequenceof smooth, compactly supported functions withthe following properties, and
1. there is a constant $C>0$ such that
2. there is $0<t^{*}\leq M_{j}^{-3}$ such thatfor all suﬃciently large positive integers$\omega_{0,j}^{n}$$\Vert\omega_{0,j}^{n}\Vert_{W^{1,p_{J}}}\leq C$$\Vert\omega_{j}^{n}(t^{*})\Vert_{W^{1,p_{J}}}\geq M_{j}^{1/3}$$n.$for some
observe that if$0<t\leq M_{j}^{-3}$ then there is nothing to prove and therefore we may assume that

Proof of Theorem 3.1. First, given(3.10)$\Vert\omega_{j}(t)\Vert_{W^{1,p_{J}}}\geq M_{j}^{1/3}$$M_{j}\gg 1$$\Vert\omega_{j}(t)\Vert_{W^{1,p_{J}}}\leq M_{j}^{1/3}$

for all $0\leq t\leq M_{j}^{-3}.$

isand therefore so is its flow
Since $p_{j}>2$ the associated velocity fieldand. Using (2.9) we can then pickis at
for which the absolute value of one of the entries in the Jacobi matrix$u_{j}=\nabla^{\perp}\triangle^{-1}\omega_{j}$$\eta_{j}(t)=(\eta_{j}^{1}(t), \eta_{j}^{2}(t))$$C^{1}$$0\leq t^{*}\leq M_{j}^{-3}$$x^{*}=(x_{1}^{*}, x_{2}^{*})\in \mathbb{R}^{2}$$D\eta_{j}(t^{*}, x^{*})$$2For$

example, it is suﬃcient to take $N_{j}=10M_{j}^{10}$ and$p_{j}=2^{\frac{N_{0}+N}{N_{0}+N_{J}-1}}.$57
least as large asof we haveand by continuity deduce that in a suﬃciently small -neighbourhood$\delta$$M_{j}$$x^{*}$(3.11)$| \frac{\partial\eta_{j}^{2}}{\partial x_{2}}(t^{*}, x)|\geq M_{j}$for all$|x-x^{*}|<\delta.$

To construct a sequence of high-frequency perturbations of the initial vorticity inin the Fourier space with support in theconsider a smooth bump function. Let $\xi_{0}=(2,0)$ and define
andunit ball and such that$W^{1,p_{J}}$$\hat{\chi}\in C_{c}^{\infty}(\mathbb{R}^{2})$$0\leq\hat{\chi}\leq 1$$\int_{\mathbb{R}^{2}}\hat{\chi}(\xi)d\xi=1$$\hat{\rho}(\xi)=\hat{\chi}(\xi-\xi_{0})+\hat{\chi}(\xi+\xi_{0}) , \xi\in \mathbb{R}^{2}.$Observe thatsatisfies$\hat{\rho}$

is supported in the union of the unit balls with centers at(3.12)$\xi_{0}$and$-\xi_{0}$and$\rho(0)=\int_{\mathbb{R}^{2}}\hat{\rho}(\xi)d\xi=2$

and for any$k\in \mathbb{Z}_{+}$(3.13)

and$\lambda>0$

define$\beta_{j}^{k,\lambda}(x)=\frac{\lambda^{-1+\frac{2}{p_{J}}}}{\sqrt{k}}\sum_{2\epsilon 1,\epsilon=\pm 1}\epsilon_{1}\epsilon_{2}\rho(\lambda(x-x_{\epsilon}^{*}))\sin kx_{1}$.
whereTo proceed we need the following estimates.
$x_{\epsilon}^{*}=(\epsilon_{1}x_{1}^{*}, \epsilon_{2}x_{2}^{*})$

Lemma 3.2. Let $2<p_{j}<\infty,$ $2\leq p\leq\infty$ and $\sigma>0$ . For any suﬃciently large positive
we haveinteger and$\lambda>0$

$k$

1.

$\Vert\beta_{j}^{k,\lambda}\Vert_{W^{1,p_{J\sim}}}<\lambda^{-1\sqrt{k}}$2. $\Vert-2\sim$

3.

$\Vert\partial_{lL^{p_{\sim}}}\triangle^{-1}\beta_{j}^{k,\lambda}\Vert<k^{-\frac{1}{2}}\lambda^{-2+\frac{2}{p_{J}}-\frac{2}{p}}$where $l=1$ , 2.

Proof. The first estimate is routine and therefore we focus on the remaining two. It will
be convenient to use the Fourier transform(3.14)$\hat{\beta}_{j}^{k,\lambda}(\xi)=\frac{1}{2i}k^{-\frac{1}{2}}\lambda^{-3+\frac{2}{p_{J}}}\sum_{\epsilon\epsilon_{1,2}=\pm 1}\sum_{m=1}^{2}(-1)^{m+1}\epsilon_{1}\epsilon_{2}\hat{\rho}(\lambda^{-1}\xi_{m}^{k})e^{-2\pi i\langle x_{\epsilon}^{*},\xi_{m}^{k}\rangle}$. Let be the conjugate Lebesgue exponent of . Using thewhere
Hausdorﬀ-Young inequality we have$\xi_{m}^{k}=(\xi_{1}+\frac{(-1)^{m}}{2\pi}k, \xi_{2})$

$p’$

$p$

$\Vert\triangle^{\underline{1}\pm z}2\partial_{l}\triangle^{-1}\beta_{j}^{k,\lambda}\Vert_{Lp}\sim<\Vert|\cdot|^{\sigma}\hat{\beta}_{j}^{k,\lambda}\Vert_{Lp’}$$\sim<k^{-\frac{1}{2}}\lambda^{-3+\frac{2}{p_{j}}}\sum_{m=1}^{2}(\int_{R^{2}}|\xi|^{\sigma p’}|\hat{\rho}(\lambda^{-1}\xi_{m}^{k})|^{p’}d\xi)^{1/p’}$58
Changing the variables and estimating further we obtain$\sim<k^{-\frac{1}{2}}\lambda^{-3+\frac{2}{p_{J}}+\frac{2}{p}}\sum_{m=1}^{2}(\int_{R^{2}}((\xi_{1}-\frac{(-1)^{m}}{2\pi}k)^{2}+\xi_{2}^{2})^{\sigma p’/2}|\hat{\rho}(\lambda^{-1}\xi)|^{p’}d\xi)^{1/p’}$$\sim<k^{-\frac{1}{2}}\lambda^{-1+\frac{2}{r_{J}}-\frac{2}{r}}\sum_{m=1}^{2}(\int_{\mathbb{R}^{2}}((\lambda\xi_{1}-\frac{(-1)^{m}}{2\pi}k)^{2}+(\lambda\xi_{2})^{2})^{\sigma p’/2}|\hat{\rho}(\zeta)|^{p’}d\xi)^{1/p’}$$\sim<k^{-\frac{1}{2}}\lambda^{-1+\frac{2}{p_{J}}-\frac{2}{p}}(\lambda^{\sigma}+k^{\sigma})$.

By the same calculation as above we also have$\Vert\partial_{l}\triangle^{-1}\beta_{j}^{k,\lambda}\Vert_{Lp\sim}<\Vert|\cdot|^{-1}\hat{\beta}_{j}^{k,\lambda}\Vert_{Lp’}$$\sim<k^{-\frac{1}{2}}\lambda^{-1+\frac{2}{p_{J}}-\frac{2}{p}}\sum_{m=1}^{2}\int_{\sup p\{\hat{\rho}\}}((\lambda\xi_{1}-\frac{(-1)^{m}}{2\pi}k)^{2}+(\lambda\xi_{2})^{2})^{-p’/2}|\hat{\rho}(\xi)|^{p’}d\xi$$\sim<k^{-\frac{1}{2}}\lambda^{-2+\frac{2}{p_{j}}-\frac{2}{p}}$for any suﬃciently large
Next, setting(3.12) we have$k$$\beta_{j}^{n}=\beta_{j}^{k,\lambda}$

and$\lambda.$

where

$I$

$k=\lambda^{2}$

and $\lambda=3n$ with $n\gg 1$ and using (3.11) andLemma 3.3. Let $2<p_{j}<\infty,$ $0<t^{*}\leq M_{j}^{-3}$ and $n\gg 1$ be as above. Then

1.
2.
where

$\Vert\partial_{2}\beta_{j}^{n}\eta_{j}^{2}(t^{*})\Vert_{L^{p_{J_{\sim}}}}$$\Vert\partial_{1}\beta_{j}^{n}\partial_{2}\eta_{j}^{2}(t^{*})\Vert_{L^{p_{J}}}\sim>M(1+\mathcal{O}(n^{-\frac{1}{2}}))-Cn^{-1}$$C$

depends on$\Vert\hat{\rho}\Vert_{L^{p_{J}’}}$

and$\sup_{0\leq t\leq 1}\Vert u_{j}(t)\Vert_{C^{1}}$and where$\frac{1}{p_{J}}+\frac{1}{p_{J}}=1.$llLemll. 1
Proof. Cf. [10]; Lem.

Consider the sequence of initial vorticities
with $n\geq 1$ . By
Lemma 2.1 and Lemma 3.2 (part 1) it is clearly infor any $2<p_{j}<\infty$ and so letbe the corresponding solution of the vorticity equations (2.2).
Observe that choosing the parameterensures that both right hand sides ofthe expressions in parts 2 and 3 of Lemma 3.2 converge to zero as. Furthermore,choosing $p>2/\sigma$ and $0<\alpha<\sigma-2/p$ and using continuity of the solution map in little
H\"older spaces in Lemma 2.3 we obtain$\omega_{0,j}^{n}(x)=\omega_{0,j}(x)+\beta_{j}^{n}(x)$$W^{1,p_{J}}$$\omega_{j}^{n}(t)\in C([0,1], W^{1,p_{j}}(\mathbb{R}^{2}))$$0< \sigma<\frac{1}{2}$$narrow\infty$(3.15)$\sup_{0\leq t\leq 1}\Vert\nabla^{\perp}\triangle^{-1}(\omega_{j\sim}^{n}(t)-\omega_{j}(t))\Vert<arrow 0$

asand therefore standard application of Gronwall’s inequality to the flow equation$narrow\infty$(2.4) gives(3.16)$\theta_{n}=\sup_{0\leq t\leq 1}\Vert\eta_{j}^{n}(t)-\eta_{j}(t)\Vert_{C^{1}}arrow 0$

where
vorticity$\eta_{j}^{n}(t)$is the flow of the velocity field$\omega_{0,j}^{n}.$a$s$$narrow\infty$$u_{j}^{n}=\nabla^{\perp}\triangle^{-1}\omega_{j}^{n}$corresponding to the initial

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Combining (3.16) with the fact that the Lagrangian flows are volume-preserving andusing conservation of vorticity we now have$\Vert\omega_{j}^{n}(t^{*})\Vert_{W^{1,p_{J\sim}}}>\Vert d\omega_{0,j}^{n}(\nabla^{\perp}\eta_{j}^{n,2}(t^{*}))\Vert_{L^{p_{J}}\sim}>\Vert d\omega_{0,j}^{n}(\nabla^{\perp}\eta_{j}^{2}(t^{*}))\Vert_{L^{p_{J}}}-\theta_{n}\Vert d\omega_{0,j}^{n}\Vert_{L^{p_{\mathcal{J}}}}$(3.17)$\sim>\Vert d\beta_{j}^{n}(\nabla^{\perp}\eta_{j}^{2}(t^{*}))\Vert_{L^{p_{J}}}-\Vert d\omega_{0,j}(\nabla^{\perp}\eta_{j}^{2}(t^{*}))\Vert_{L^{p_{J}}}-\theta_{n}\Vert d\omega_{0,j}^{n}\Vert_{L^{p_{J}}}.$

Finally, it suﬃces to observe that by the assumption (3.10) the middle term on the rightof (3.17) is bounded by$\Vert d\omega_{0,j}(\nabla^{\perp}\eta_{j}^{2}(t^{*}))\Vert_{L^{p_{J}}\sim}<\Vert\omega_{j}(t^{*})\Vert_{W^{1,p_{J\sim}}}<M_{j}^{1/3}$

while by Lemma 3.3 we have$\Vert d\beta_{j}^{n}(\nabla^{\perp}\eta_{j}^{2}(t^{*}))\Vert_{L^{p_{J}}}\sim>\Vert\partial_{1}\beta_{j}^{n}\partial_{2}\eta_{j}^{2}(t^{*})\Vert_{L^{p_{J}}}-\Vert\partial_{2}\beta_{j}^{n}\eta_{j}^{2}(t^{*})\Vert_{L^{p_{J}}\sim}M_{j}$for any suficiently large $n\gg 1$ . The proof of Theorem 3.1 is completed. I
Acknowledgments. Part of this work was done while GM was the Ulam Chair Visiting
Professor at the University of Colorado, Boulder. TY was partially supported by JSPS
KAKENHI Grant Number 25870004.

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$C^{m}$$L_{s}^{p}(R^{2})$$L_{s}^{p}(\mathbb{R}^{2})$$C^{1}$$B_{\infty,1}^{1},$

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME, IN 46556, USA$E$-mail address: gmisiole@nd.edu

DEPARTMENT OF MATHEMATICS, TOKYO INSTITUTE OF TECHNOLOGY, MEGURO-KU, TOKYO 1528551, JAPAN$E_{-ma}nt$ address: yoneda@math.titech.ac.jp