Persistence of the solution to the Euler equations in the end-point critical Triebel-Lizorkin space $F^{d+1}_{1, \infty}(\mathbb{R}^d)$

TL;DR

This paper investigates the persistence of solutions to the Euler equations for incompressible fluids within the critical Triebel-Lizorkin space at the endpoint, clarifying conditions for solution stability.

Contribution

It provides new insights into the local persistence of solutions in the endpoint Triebel-Lizorkin space $F^{d+1}_{1, ext{infinity}}$, a critical function space for Euler equations.

Findings

Solutions persist locally in the endpoint Triebel-Lizorkin space

Clarifies conditions for solution stability at the critical endpoint

Advances understanding of Euler equations in advanced function spaces

Abstract

Local stay of the solutions to the Euler equations for an ideal incompressible fluid in the end-point Triebel-Lizorkin spaces $F^s_{1, \infty}(\mathbb{R}^d)$ with $s \geq d + 1$ is clarified.

Equations

$$\frac{\partial }{\partial t}u+(u,\nabla )u$$

$$\text{mbox}divu$$

$$s>\frac{d}{p}+1,1<p,q<\infty \text{mbox}ors\ge d+1,p=1,1<q<\infty .$$

$$s=\frac{d}{p}+1,1<p\le \infty ,q=1\text{mbox}ands\ge d+1,p=1,q=\infty$$

$$\hat{f}(\xi )=\mathcal{F}(f)(\xi )={\int }_{{\mathbb{R}}^n}f(x)e^{-ix\cdot \xi }dx$$

$${\Vert {\left\{f_j\right\}}_{j\in \mathbb{Z}}\Vert }_{l^p}(x):={\left(\sum _{j\in \mathbb{Z}}{\left|f_j(x)\right|}^p\right)}^{1/p}$$

$${\Delta }_{j}f=\{\begin{array}{ll}\widehat{{\phi }_j}(D)f={\phi }_j\ast f& \text{for }j\ge 0\\ \hat{\Phi }(D)f=\Phi \ast f& \text{for }j=-1\\ 0& \text{for }j\le -2\end{array}$$

$${\dot{\Delta }}_{j}f=\{\begin{array}{ll}{\Delta }_{j}f& \text{for }j\ge 0\\ \widehat{{\phi }_j}(D)f={\phi }_j\ast f& \text{for }j\le -1.\end{array}$$

$${\Delta }_{-1}+\sum _{j=0}^{\infty }{\Delta }_j=I\text{mbox}and\sum _{j=-\infty }^{\infty }{\dot{\Delta }}_j=I,$$

$${\Vert f\Vert }_{{\dot{F}}_{1,\infty }^s}:={\int }_{{\mathbb{R}}^d}\underset{j\in \mathbb{Z}}{\sup }\left|2^{js}{\dot{\Delta }}_{j}f\right|(x)dx<\infty ,$$

$$\Vert f{\Vert }_{F_{1,\infty }^s}:={\int }_{{\mathbb{R}}^d}\underset{j\in \mathbb{Z}}{\sup }\left|2^{js}{\Delta }_{j}f\right|(x)dx<\infty .$$

$${\Vert f\Vert }_{L^1}+{\Vert f\Vert }_{{\dot{F}}_{1,\infty }^s}.$$

$${\Vert D^{k}f\Vert }_{{\dot{F}}_{p,q}^s}\lesssim {\Vert f\Vert }_{{\dot{F}}_{p,q}^{s+k}}\lesssim {\Vert D^{k}f\Vert }_{{\dot{F}}_{p,q}^s}.$$

$$|u(z)|\lesssim (1+|z|{)}^{\frac{d}{r}},$$

$$\underset{z\in {\mathbb{R}}^d}{\sup }\frac{1}{t}\frac{|\nabla u(x-z)|}{(1+t|z|{)}^{\frac{d}{r}}}\lesssim \underset{z\in {\mathbb{R}}^d}{\sup }\frac{|u(x-z)|}{(1+t|z|{)}^{\frac{d}{r}}}\lesssim M(|u{|}^r)(x{)}^{\frac{1}{r}},$$

$$|\psi (z)|{\left(1+|z|\right)}^{d/r}\le U(z),z\in {\mathbb{R}}^d$$

$$\left|[\psi {]}_s\ast \left(gf\right)\right|(x)\lesssim {\left(\frac{t}{s}\right)}^{\frac{d}{r}}M(g)(x){\left[M(|f{|}^{r_1})(x)\right]}^{\frac{\gamma }{r_1}}{\left[M(|f{|}^{r_2})(x)\right]}^{\frac{\delta }{r_2}}$$

$$\left|[\psi {]}_s\ast \left(gf\right)\right|(x)$$

$$\le \left[{\int }_{{\mathbb{R}}^d}\left|\psi (z)\right|\left|g(x-s^{-1}z)\right|{\left(1+\frac{t}{s}|z|\right)}^{d/r}dz\right]$$

$$\times \underset{z\in {\mathbb{R}}^d}{\sup }\frac{\left|f\left(x-s^{-1}z\right)\right|}{{\left(1+t/s\left|z\right|\right)}^{d/r}}.$$

$$\underset{z\in {\mathbb{R}}^d}{\sup }\frac{\left|f\left(x-s^{-1}z\right)\right|}{{\left(1+t/s\left|z\right|\right)}^{d/r}}$$

$$\lesssim {\left[M(|f{|}^{r_1})(x)\right]}^{\frac{\gamma }{r_1}}{\left[M(|f{|}^{r_2})(x)\right]}^{\frac{\delta }{r_2}}.$$

$${\int }_{{\mathbb{R}}^d}\left|\psi (z)\right|$$

$$\le \max \{{\ell }^{-\frac{d}{r}},1\}{\int }_{{\mathbb{R}}^d}\left|\psi (z)\right|{\left(\frac{t}{s}+\frac{t}{s}\left|z\right|\right)}^{d/r}\times |g(x-s^{-1}z)|dz$$

$$\lesssim {\left(\frac{t}{s}\right)}^{\frac{d}{r}}\left([\varphi {]}_s\ast |g|\right)(x)\lesssim {\left(\frac{t}{s}\right)}^{\frac{d}{r}}M(g)(x),$$

$$\left|{\Delta }_k\left(gf\right)\right|(x)\lesssim 2^{(j-k)\frac{d}{r}}M(g)(x){\left[M(|f{|}^{r_1})(x)\right]}^{\frac{\gamma }{r_1}}{\left[M(|f{|}^{r_2})(x)\right]}^{\frac{\delta }{r_2}}$$

$${\Vert {\Vert {\left\{M(g_k)\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^q}\Vert }_{L^p}\lesssim {\Vert {\Vert {\left\{g_k\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^q}\Vert }_{L^p}.$$

$$\left|\psi (z)\right|{\left(1+|z|\right)}^{ds}\le U(z),z\in {\mathbb{R}}^d$$

$${\Vert {\Vert {\left\{[\psi {]}_{2^{k+\ell }}\ast {\dot{\Delta }}_k{f}_k)\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^q}\Vert }_{L^1}\lesssim {\Vert {\Vert {\left\{{\dot{\Delta }}_k{f}_k\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^q}\Vert }_{L^1}.$$

$${\Vert {\Vert {\left\{[\psi {]}_{2^k}\ast {\dot{\Delta }}_k{f}_k)\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^q}\Vert }_{L^1}$$

$$\lesssim {\Vert {\Vert {\left\{M(|{\dot{\Delta }}_k{f}_k{|}^{r_1}{)}^{\frac{\gamma }{r_1}}M(|{\dot{\Delta }}_k{f}_k{|}^{r_2}{)}^{\frac{\delta }{r_2}}\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^q}\Vert }_{L^1}$$

$$\lesssim {\Vert {\Vert {\left\{M(|{\dot{\Delta }}_k{f}_k{|}^{r_1}{)}^{\frac{\gamma }{r_1}}\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^{\frac{q}{\gamma }}}\Vert }_{L^{\frac{1}{\gamma }}}{\Vert {\Vert {\left\{M(|{\dot{\Delta }}_k{f}_k{|}^{r_2}{)}^{\frac{\delta }{r_2}}\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^{\frac{q}{\delta }}}\Vert }_{L^{\frac{1}{\delta }}}$$

$$={\Vert {\Vert {\left\{{\dot{\Delta }}_k{f}_k\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^q}^{\gamma }\Vert }_{L^{\frac{1}{\gamma }}}{\Vert {\Vert {\left\{{\dot{\Delta }}_k{f}_k\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^q}^{\delta }\Vert }_{L^{\frac{1}{\delta }}}$$

$$={\Vert {\Vert {\left\{{\dot{\Delta }}_k{f}_k\right\}}_{k\in \mathbb{Z}}\Vert }_{{\ell }^q}\Vert }_{L^1},$$

$${\Vert fg\Vert }_{F_{1,\infty }^s}\lesssim {\Vert f\Vert }_{L^{\infty }}{\Vert g\Vert }_{F_{1,\infty }^s}+{\Vert g\Vert }_{L^{\infty }}{\Vert f\Vert }_{F_{1,\infty }^s}.$$

Full text

Correspondence: Hee Chul Pak, hpak@dankook.ac.kr

1. Introduction

The perfect incompressible inviscid fluid is governed by the Euler equations:

[TABLE]

Here $u(x,t)=(u_{1},u_{2},\cdots,u_{d})$ is the velocity of a fluid flow and $p(x,t)$ is the scalar pressure.

Existence and uniqueness theories of solutions of the 2 or 3 dimensional Euler equations have been worked on by many mathematicians and physicists. L. Lichtenstein, N. Gunther and Wolibner started the subject on Hölder classes. D. Ebin and J. Marsden, J. Bourguignon and H. Brezis, R. Temam, T. Kato and G. Ponce studied this subject on Sobolev spaces. Some researches on the Euler equations in Besov spaces $B_{p,q}^{s}(\mathbb{R}^{d})$ have been done by M. Vishik [20], D. Chae [6] and Pak-Park [17]. For a detailed survey of this issue, we refer [1, 16, 7, 9, 3, 4] and references therein. Bourgain and Li proved strong ill-posedness results for the Euler equations associated with initial data in (borderline) Besov spaces, Sobolev spaces or the space $C^{m}$. For the survey of the ill-posedness issue we refer [3, 4].

The existence and uniqueness of the solutions of the Euler equations in general Triebel-Lizorkin spaces $F_{p,q}^{s}(\mathbb{R}^{d})$ were first investigated by D. Chae in [5], and his research covers the cases of

[TABLE]

The unique existence of solution in the critical space $F_{1,1}^{s}(\mathbb{R}^{d})=B_{1,1}^{s}(\mathbb{R}^{d})$ ($s\geq d+1$) is proved by Pak-Park in [18], and very recently, the continuity of the solution map in this space is reported in [13]. The remaining critical and sub-critical cases

[TABLE]

of well- or ill-posedness issue are not clearly reported, yet. In this paper, we clarify the persistence of the solution for the critical end-point case $p=1$ and $q=\infty$ with $s\geq d+1$. We now state our results as follows.

Theorem 1.1**.**

*Let $s$ be a real number greater than or equal to $d+1$.

1. (Persistence and uniqueness) For any divergence free vector field $u_{0}\in F_{1,\infty}^{s}(\mathbb{R}^{d})$, there exists a positive time $T>0$ such that the initial value problem for the Euler equations (1.1) with initial velocity $u_{0}$ has a unique solution $u$ with $u(t)\in F_{1,\infty}^{s}(\mathbb{R}^{d})$ for all $t\in[0,T]$.

Moreover, this solution $u$ also satisfies

2. (2-D global existence of the solution) When $d=2$, the solution $u$ does not blow-up within finite time. That is, $u(t)\in F_{1,\infty}^{s}(\mathbb{R}^{2})$ for all $t\in[0,\infty)$.*

One of the main analytic points to deal with the end-point critical spaces is that the usual techniques may not be utilized in those critical spaces. For example, the (vector-valued) Hardy-Littlewood maximal operator is not $L^{1}$-bounded and even the Calderon-Zygmund theory does not work. Furthermore, the smooth approximation via mollifier can not be applicable due to the fact that the Schwartz class is not dense in the $\ell^{\infty}$-hierarchy spaces, which causes uncertainty of the existence result in the space $F_{1,\infty}^{s}(\mathbb{R}^{d})$. However, we could detour and organize some known estimates and techniques to derive our conclusion without too many analytic difficulties.

We pursue self-contained and detailed descriptions of the arguments to avoid possible gaps that sometimes happen on this subject[8, 13].

Notations: Throughout this paper,

1. (1)

for $x\in\mathbb{R}^{d}$, $x_{i}$ is the $i$-th component of $x$ 2. (2)

$B(x,r):=\{y\in\mathbb{R}^{d}:|x-y|<r\}$ 3. (3)

$\displaystyle\frac{\partial f}{\partial x_{k}}=\partial_{x_{k}}f$ or simply $\partial_{k}f$ 4. (4)

for $k>0$ and a function $\phi$ on $\mathbb{R}^{d}$, $[\phi]_{k}(x):=k^{d}\phi(kx)$ for $x\in\mathbb{R}^{d}$ 5. (5)

for $f\in\mathcal{S}({\mathbb{R}}^{d})$, the Fourier transform $\hat{f}={\mathcal{F}}(f)$ of $f$ on ${\mathbb{R}}^{d}$ is defined by

[TABLE] 6. (6)

$D^{s}u=\sqrt{-\Delta}^{\;s}u:={\mathcal{F}}^{-1}(|\cdot|^{s}{\mathcal{F}}(u))$ 7. (7)

for $1\leq p<\infty$ and a sequence of functions $\left\{f_{j}\right\}\!\!_{j\in\mathbb{Z}}$,

[TABLE] 8. (8)

the notation $X\lesssim Y$ means that $X\leq CY$, where $C$ is a fixed but unspecified constant. Unless explicitly stated otherwise, $C$ may depend on the dimension $d$ and various other parameters (such as exponents), but not on the functions or variables $(u,v,f,g,x_{i},\cdots)$ involved.

2. A-priori estimate

Let $\mathcal{S}(\mathbb{R}^{d})$ denote the Schwartz class. We consider a nonnegative radial function $\chi\in\mathcal{S}(\mathbb{R}^{d})$ satisfying $\mbox{supp}\>\chi\subset\{\xi\in{\mathbb{R}}^{d}:|\xi|\leq 1\}$, and $\chi=1$ for $|\xi|\leq\frac{3}{4}$. Set $h_{j}(\xi):=\chi(2^{-j-1}\xi)-\chi(2^{-j}\xi)$ and let $\varphi_{j}$ and $\Phi$ be defined by $\varphi_{j}:=\mathcal{F}^{-1}(h_{j})$, $j\geq 0$ and $\Phi:=\mathcal{F}^{-1}(\chi)$. For any $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$, we define the operator $\Delta_{j}$ by

[TABLE]

and the operator $\dot{\Delta}_{j}$ by

[TABLE]

Then we have an analog of a partition of unity:

[TABLE]

where $I$ represents the identity operator. The partial sum operators $S_{k}f$ and $\dot{S}_{k}f$ are $\displaystyle{S_{k}f=\sum_{j=-\infty}^{k}\Delta_{j}f}$ and $\displaystyle{\dot{S}_{k}f=\sum_{j=-\infty}^{k}\dot{\Delta}_{j}f}$, respectively.

For $s\in\mathbb{R}$, the homogenous Triebel-Lizorkin space $\dot{F}_{1,\infty}^{s}(\mathbb{R}^{d})$ is the collection of all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ modulo polynomials such that

[TABLE]

and the nonhomogeneous Triebel-Lizorkin space $F_{1,\infty}^{s}(\mathbb{R}^{d})$ is the space of all tempered distributions $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ obeying

[TABLE]

We observe that for $s>0$, the Triebel-Lizorkin norm $\left\lVert f\right\rVert_{F_{1,\infty}^{s}}$ is equivalent to the nonhomogeneous norms

[TABLE]

Remark 2.1*.*

Let $1\leq p<\infty$, $1\leq q\leq\infty$ and $s\in\mathbb{R}$. For any $k\in\mathbb{N}$, we have Bernstein type inequality:

[TABLE]

2.1. Basic estimates

We begin with Lemma 2.2 which is introduced by Peetre. Lemma 2.2 has been developed for the proof of the independent choice of the mother (bump) function $\chi$ employed at page 2. The version given in this paper is a refinement introduced by Triebel [19]. For the proof, we refer the page 71 in [12].

Lemma 2.2**.**

Let $0<r<\infty$. Then for all $t>0$ and for any $C^{1}$-function $u$ on $\mathbb{R}^{d}$ whose Fourier transform is supported in the ball $|\xi|\lesssim t$ and that satisfies

[TABLE]

we have

[TABLE]

where $M$ is the Hardy-Littlewood maximal operator.

Paley-Wiener-Schwartz theorem says that every distribution $u$ on $\mathbb{R}^{d}$ whose Fourier transform is supported in the ball $|\xi|\leq c_{0}t$ is an entire function of $d$-complex variables. The following proposition is a generalizations of one presented by Guo and Li [13].

Proposition 2.3**.**

Let $r,r_{1},r_{2}>0$, $s,t,\ell>0$ with $\frac{s}{t}<\ell$ and $\gamma,\delta>0$ with $\gamma+\delta=1$ and $\frac{\gamma}{r_{1}}+\frac{\delta}{r_{2}}=\frac{1}{r}$. Let $\psi$ be a measurable function such that

[TABLE]

for some nonnegative radial decreasing integrable function $U$. Then we have that for $x\in\mathbb{R}^{d}$,

[TABLE]

for all $g\in L^{1}_{loc}(\mathbb{R}^{d})$ and any tempered distribution $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ whose frequency support $\mbox{supp}\>\widehat{f}$ is contained in the ball $|\xi|\lesssim t$ together with the condition $|f(x)|\lesssim(1+|x|)^{d\theta}$ ($\theta=\min\{\frac{\gamma}{r_{1}},\frac{\delta}{r_{2}}\}$) for all $x\in\mathbb{R}^{d}$.

Proof. We take $g\in L^{1}_{loc}(\mathbb{R}^{d})$ and a distribution $f$ satisfying $\mbox{supp}\>\widehat{f}\,\subseteq B(0,c_{0}t)$ for some $c_{0}>0$ and $|f(x)|\lesssim(1+|x|)^{\frac{d}{r}}$ for all $x\in\mathbb{R}^{d}$. Then we have that for $x\in\mathbb{R}^{d}$,

[TABLE]

Lemma 2.2 yields

[TABLE]

By the assumption that $\frac{s}{t}<\ell$, we also get

[TABLE]

where $\phi(z):=|\psi(z)|\left(\frac{t}{s}+\frac{t}{s}\left|z\right|\right)^{d/r}$ and $[\phi]_{s}(s):=s^{d}\phi\left(sx\right)$. $\Box$

As an application, we have the following statement.

Corollary 2.4**.**

Let $\ell_{0},r,r_{1},r_{2}>0$ and $\gamma,\delta>0$ with $\gamma+\delta=1$ and $\frac{\gamma}{r_{1}}+\frac{\delta}{r_{2}}=\frac{1}{r}$. Then for $j,\,k\in\mathbb{Z}$ with $j>k-\ell_{0}$, we have that

[TABLE]

for any $g\in L^{1}_{loc}(\mathbb{R}^{d})$ and any measurable function $f$ whose frequency support $\mbox{supp}\>\widehat{f}$ is contained in the ball $|\xi|\lesssim 2^{j}$ together with the condition $|f(x)|\lesssim(1+|x|)^{d\theta}$ ($\theta=\min\{\frac{\gamma}{r_{1}},\frac{\delta}{r_{2}}\}$) for all $x\in\mathbb{R}^{d}$.

One of main tools for the estimates of Triebel-Lizorkin spaces is the continuity of the Hardy-Littlewood maximal operator $M$ for the vector valued functions. In fact, for any sequence $\left\{g_{k}\right\}_{k\in\mathbb{Z}}$ in $L^{p}(\mathbb{R}^{d})$, one has

[TABLE]

This holds for the case of $1<p<\infty$ and $1<q\leq\infty$, and unfortunately not for $p=1$. We present a useful alternative of the estimate (2.7) for $L^{1}$-hierarchy spaces:

Corollary 2.5**.**

Let $1<q\leq\infty$ and let $\psi$ be a measurable function such that $\hat{\psi}$ has a compact support and

[TABLE]

for some $s>0$ and some nonnegative radial decreasing integrable function $U$. Then for a fixed integer $\ell$,

[TABLE]

Proof. We apply Proposition 2.3 to have

[TABLE]

where $\gamma+\delta=1$, $\frac{\gamma}{r_{1}}+\frac{\delta}{r_{2}}=\frac{1}{r}$.111In the following, $\gamma,\delta,r_{1},r_{2}$ and $r$ are always positive real numbers. This implies the estimate. $\Box$

Equipped with Proposition 2.3 and its corollaries, we can verify a Moser type inequality in $F_{1,\,\infty}^{s}(\mathbb{R}^{d})$ and a commutator estimate in $\dot{F}_{1,\infty}^{s}(\mathbb{R}^{d})$. The ways of proofs have been well-developed. The proofs are placed at Appendices.

Proposition 2.6** (Moser type inequality in $F_{1,\infty}^{s}(\mathbb{R}^{d})$).**

Let $s>0$. For scalar functions $f$ and $g$, we have

[TABLE]

The bracket operator $[\cdot,\cdot]$ below is defined by

[TABLE]

Proposition 2.7** (Commutator estimate in $\dot{F}_{1,\infty}^{s}(\mathbb{R}^{d})$).**

Let $s>0$. For a scalar function $f$ and a divergence-free vector field $u$, we have

[TABLE]

2.2. Boundedness of Leray projection operator in $\dot{F}_{1,\infty}^{s}(\mathbb{R}^{d})$

We treat the continuity of the Leray projection $\mathbb{P}$ in $\dot{F}_{1,\infty}^{s}(\mathbb{R}^{d})$, which is defined by

[TABLE]

for an appropriate vector field $u$ in $\mathbb{R}^{d}$. We present the following lemma:

Lemma 2.8**.**

For any $k=1,2,\cdots,d$, we have

[TABLE]

Proof. The corresponding symbol $m_{k}$ with respect to $\partial_{k}\Delta^{-1}$ is $m_{k}(\xi)=\frac{\xi_{k}}{|\xi|^{2}}$ with $\xi=(\xi_{1},\xi_{2},\cdots,\xi_{d})\in\mathbb{R}^{d}$, that is,

[TABLE]

We choose a function $\tilde{{\bf 1}}\in C_{0}^{\infty}(\mathbb{R}^{d})$ with $\tilde{{\bf 1}}=1$ on $\mathrm{supp}\>{h}_{0}=\{\xi:1/2\leq|\xi|\leq 2\}$ and $\tilde{{\bf 1}}(0)=0$. From the fact that $m_{k}$ is a symbol of homogeneous of order $-1$, we have

[TABLE]

where $\mathrm{K}:=\mathcal{F}^{-1}(\tilde{{\bf 1}}(\xi)\,m_{k}(\xi)).$ For any positive integer $N_{0}$, we observe that

[TABLE]

where $\alpha,\,\beta$ are multi-indices in $\mathbb{R}^{d}$ and $c_{\alpha,\,\beta}$ are integers whose exact values do not matter. For $r>0$, we may take an integer $N_{0}$ so large that $N_{0}>\frac{d}{2}+\frac{d}{r}$ to have

[TABLE]

where $N_{1}$ is a suitable integer greater than $\frac{d}{2}$. Then the function $\displaystyle\frac{1}{(1+|x|^{2})^{N_{1}}}$ is a nonnegative radial decreasing integrable function, and hence by Corollary 2.5 we derive that

[TABLE]

From (2.10), the arguments are completed. $\Box$

Lemma 2.8 implies:

Corollary 2.9**.**

The Leray projection $\mathbb{P}$ is continuous on $\dot{F}_{1,\infty}^{s}(\mathbb{R}^{d})$, that is,

[TABLE]

3. Persistence of the solution in $F_{1,\infty}^{s}(\mathbb{R}^{d})$

In this section we will prove the first and second properties of the main theorem stated at page 1.1. For $s\geq d+1$, a divergence free vector field

[TABLE]

is given. According to the well-known result in [17] together with the fact that $u_{0}\in F_{1,\infty}^{s}(\mathbb{R}^{d})\subset B_{\infty,1}^{1}(\mathbb{R}^{d})$([14] and Bernstein’s lemma), there exists a time $T>0$ and a solution $u$ to the Euler equation (1.1) in $C([0,T);B_{\infty,1}^{1}(\mathbb{R}^{d}))$.222 The temporal continuity is due to the denseness of smooth functions in $B_{\infty,1}^{1}(\mathbb{R}^{d})$. See the arguments in [17] or [18]. Thanks to this, it suffices to prove that the solution $u(t)$ stays in $F_{1,\infty}^{s}(\mathbb{R}^{d})$ for a while.

First, we consider the trajectory map $X(x,\,t)$ defined by

[TABLE]

Then from the Euler equation (1.1), we have

[TABLE]

and so

[TABLE]

In order to estimate the pressure term, we recall that

[TABLE]

Lemma 3.1**.**

For $s>0$, we have

[TABLE]

Proof. Note that

[TABLE]

Successive applications of Lemma 2.8, Proposition 2.6 and Remark 2.1 imply

[TABLE]

To estimate the term $\left\lVert\Delta_{-1}\nabla p\right\rVert_{{\bf L}^{1}}$, we observe that

[TABLE]

Applying the fundamental solution of the Laplace’s equation, we get that for $d\geq 3$,

[TABLE]

where $\tilde{{\bf 1}}\in C_{0}^{\infty}(\mathbb{R}^{d})$ with $\tilde{{\bf 1}}(x)=1$ for $|x|\leq 1$. We have the same result for the two dimensional case $d=2$. Plugging the estimates (3.4) and (3.5) in the inequality (3.2), we get the result. $\Box$

Lemma 3.1 leads to

[TABLE]

By applying $\dot{\Delta}_{j}$ and adding the term $(u,\,\nabla)\dot{\Delta}_{j}u$ on both sides of the Euler equation (1.1) at $(X(x,\,t),\,t)$, we get

[TABLE]

Integrate with respect to time $t$ over $[0,t]$, and then take $F^{s}_{1,\infty}$-norm on both sides of (3.7) to obtain

[TABLE]

Owing to the estimate (3.4) and Proposition 2.7, we can derive that

[TABLE]

Hence, combining (3.9) and (3.6), we find that

[TABLE]

By virtue of Gronwall’s inequality, (3.11) leads to

[TABLE]

for some positive constant $C_{0}$. Let

[TABLE]

Then from (3.12), we note that

[TABLE]

Solving the separable ordinary differential inequality (3.13), we see that

[TABLE]

if $0\leq t<\frac{1}{C^{2}_{0}\|u_{0}\|_{F_{1,\infty}^{s}}}:=T_{0}$. Hence it is shown that

[TABLE]

This illustrates that the solution $u(t)$ stays in $F_{1,\infty}^{s}$, at least, for $t\in[0,T_{0})$. This completes the proof of the local existence and uniqueness of the solution.

For the two dimensional case $d=2$, the estimate (3.10) can be rewritten as

[TABLE]

for some positive constant $C$. Then from the well-known facts that $\|u\|_{{W}^{1,\infty}}\lesssim\|u\|_{{B}^{1}_{\infty,1}}$ and $\|u(t)\|_{B^{1}_{\infty,1}}<\infty$ for all $t\in[0,\infty)$, we establish that $u(t)$ stays inside the space ${F}^{s}_{1,\infty}(\mathbb{R}^{d})$ for all time $t$. $\Box$

Appendices

We present the proofs of Proposition 2.6 and Proposition 2.7. The key point is to exhibit a detour way to avoid the discontinuity of the Hardy-Littlewood maximal operator $M:L^{1}(\mathbb{R}^{d})\to L^{1}(\mathbb{R}^{d})$ with respect to $L^{1}$-norm. The arguments are borrowed from [13].

Bony’s paraproduct formula decomposes the product $fg$ into three parts as follows:

[TABLE]

where the para-product $T_{f}g$ and the remainder $R\left(f,\,g\right)$ defined by

[TABLE]

respectively [2]. Then we note that

[TABLE]

because $\Delta_{k}\left(S_{j-4}f\Delta_{j}g\right)=0$ if $\left|j-k\right|\geq 3$. We also see that

[TABLE]

since $\mathrm{supp}\>\mathcal{F}\left(\Delta_{k}\left(\Delta_{j}f\Delta_{j+l}g\right)\right)=\varnothing$ if $j\leq k-6$.

Similarly, the homogeneous version of the Bony’s paraproduct formula is of the form:

[TABLE]

where the (homogeneous) Bony’s para-product $T_{f}g$ and (homogeneous) remainder defined by

[TABLE]

Proof of Proposition 2.6. Bony’s paraproduct decomposition can be read as

[TABLE]

By the formula (3.17) and Corollary 2.4, we have that for any $r\in(0,\,\infty)$,

[TABLE]

Then, for fixed $0<r<1$, the continuity of vector-valued Hardy-Littlewood maximal operator (the inequality (2.7)) can be employed to get

[TABLE]

Similarly, we can get

[TABLE]

For the remainder term, the formula (3.18) and Corollary 2.4 lead to

[TABLE]

where $\gamma+\delta=1$ with $\dfrac{\gamma}{r_{1}}+\dfrac{\delta}{r_{2}}=\dfrac{1}{r}$. We choose $r>0$ so large that we can have $s>\dfrac{d}{r}$. The integrand of (3.22) is equal to

[TABLE]

where the sequences $a:=\{a_{j}\}_{j\in\mathbb{Z}}$ and $b:=\{b_{j}\}_{j\in\mathbb{Z}}$ are defined by

[TABLE]

and for $j\in\mathbb{Z}$,

[TABLE]

Then, Young’s inequality for $l^{q}$-sequences implies the estimate

[TABLE]

Take $0<r_{1},\,r_{2}<1$ and then Höler’s inequality and the inequality (2.7) yield

[TABLE]

Combining the estimates (3.20), (3.21) and (3.23), we obtain the result (2.8). $\Box$

Proof of Proposition 2.7. Let $u^{l}$ denote the $l$-th component of $u$ for $1\leq l\leq d$. By the homogeneous Bony’s paraproduct decomposition, we have

[TABLE]

Estimate of I : By the fact that $\dot{S}_{k-4}\dot{\Delta}_{j}f=0$ if $k\leq j+2$ and Young’s inequality for $\ell^{q}$-series with arguments used at page 3.22, the first term (I) can be estimated as

[TABLE]

Estimate of II : We note that $\mathrm{supp}\>\mathcal{F}(\dot{\Delta}_{k}(\dot{\Delta}_{j}f))\neq\varnothing$ if $k\leq j-2$ or $k\geq j+2$ to have

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Noting that $\mathrm{supp}\>\mathcal{F}((\dot{\Delta}_{k+m}u^{l})(\dot{\Delta}_{k}(\dot{\Delta}_{j}\partial_{l}f)))\subseteq\{\xi:|\xi|\leq 2^{k+5}\},$ we can choose $\hat{\tilde{\mathbf{1}}}\in C_{0}^{\infty}\left(\mathbb{R}^{d}\right)$ with $\hat{\tilde{\mathbf{1}}}=1$ on $\{\xi:|\xi|\leq 2^{5}\}$ to get

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because $u$ is divergence free. Therefore, by Corollary 2.4, Bernstein lemma, Young’s inequality and Remark 2.1, we have that for $0<r<1$,

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Estimate of III : From the identity (3.17) and the fact that

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if $k\geq j+2$ or $k\leq j-2$, we have

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Since the vector field $u$ is divergence free, we can derive that

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The mean value theorem provides

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for some $\xi_{x}(y)$ located on the line segment between $x$ and $y$. Applying Proposition 2.3, we have

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where $0<r<1$ and ${\bf\theta}(z):=z\partial_{l}\varphi(z)\,\in{\mathcal{S}}(\mathbb{R}^{d})$. Hence we conclude that

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Estimate of IV : The argument used at (3.20) says that for $0<r<1$,

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Estimate of V : The support of $\mathcal{F}(\dot{\Delta}_{j}(\dot{\Delta}_{m}f\dot{\Delta}_{m+k}g))$ is null set if $m<j-5$ as we pointed out at page 3.18. So, by the divergence-free condition of $\bf u$, we have that

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Applying Remark 2.1 and Corollary 2.4, we obtain that

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where $\gamma+\delta=1$ and $\frac{\gamma}{r_{1}}+\frac{\delta}{r_{2}}=\frac{1}{r}$. We choose $r>0$ so large that we can have $s>\frac{d}{r}$, and we also choose $0<r_{1},r_{2}<1$. Then by Young’s inequality for $\ell^{q}$-series as the same argument at page 3.22 together with the Bernstein lemma, we get

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Collecting the estimates of the terms (I) $\sim$ (V) altogether, we obtain the inequality (2.9). $\Box$