On the 3D Euler equations with Coriolis force in borderline Besov spaces

TL;DR

This paper proves long-time existence and uniqueness of solutions for the 3D Euler equations with Coriolis force in borderline Besov spaces, especially under high rotation speeds, extending previous regularity results.

Contribution

It introduces a broader class of initial data and establishes uniform estimates and a blow-up criterion for the equations in borderline Besov spaces.

Findings

Long-time solvability for high rotation speeds

Uniform estimates independent of rotation speed

A blow-up criterion of BKM type

Abstract

We consider the 3D Euler equations with Coriolis force (EC) in the whole space. We show long-time solvability in Besov spaces for high speed of rotation $\Omega $ and arbitrary initial data. For that, we obtain $\Omega$-uniform estimates and a blow-up criterion of BKM type in our framework. Our initial data class is larger than previous ones considered for (EC) and covers borderline cases of the regularity. The uniqueness of solutions is also discussed.

Equations

$$\{\begin{array}{rl}& \frac{\partial u}{\partial t}+\mathbb{P}\Omega e_3\times u+\mathbb{P}\left(u\cdot \nabla \right)u=0\text{ in }{\mathbb{R}}^3\times (0,\infty )\\ & \nabla \cdot u=0\text{ in }{\mathbb{R}}^3\times (0,\infty )\\ & u(x,0)=u_0(x)\text{ in }{\mathbb{R}}^3\end{array},$$

$$\sum _{j\in \mathbb{Z}}{\hat{\varphi }}_j(\xi )=1\text{mbox}forall\xi \in {\mathbb{R}}^3\backslash \{0\},$$

$${\hat{S}}_k(\xi )=1-\sum _{j\ge k+1}{\hat{\varphi }}_j(\xi )$$

$$\Vert f{\Vert }_{{\dot{B}}_{p,q}^s}:=\Vert \{2^{sj}\Vert {\Delta }_{j}f{\Vert }_{L^p}{\}}_{j\in \mathbb{Z}}{\Vert }_{l^q(\mathbb{Z})}<\infty .$$

$$\Vert f{\Vert }_{B_{p,q}^s}:=\Vert \{2^{sj}\Vert {\Delta }_{j}f{\Vert }_{L^p}{\}}_{j\in \mathbb{N}}{\Vert }_{l^q(\mathbb{N})}+\Vert \psi \ast f{\Vert }_{L^p}.$$

$$\Vert f{\Vert }_{B_{p,q}^s}\sim \Vert f{\Vert }_{{\dot{B}}_{p,q}^s}+\Vert f{\Vert }_{L^p}.$$

$$C^{-1}{2}^{jk}\Vert f{\Vert }_{L^p}\le \Vert D^{k}f{\Vert }_{L^p}\le C{2}^{jk}\Vert f{\Vert }_{L^p}.$$

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

$$\Vert f{\Vert }_{L^{\infty }}\le C\Vert f{\Vert }_{B_{p,q}^s},$$

$$\Vert \nabla f{\Vert }_{L^{\infty }}\le \Vert \nabla f{\Vert }_{B_{p,q}^{s-1}}\le \Vert f{\Vert }_{B_{p,q}^s}.$$

$$\Vert fg{\Vert }_{{\dot{B}}_{p,q}^s}$$

$$\Vert fg{\Vert }_{B_{p,q}^s}$$

$$[v\cdot \nabla ,{\Delta }_j]u=v\cdot \nabla ({\Delta }_{j}u)-{\Delta }_j(v\cdot \nabla u).$$

$${\left(\sum _{j\in \mathbb{Z}}{2}^{sjq}\Vert [v\cdot \nabla ,{\Delta }_j]\theta {\Vert }_{L^p}^q\right)}^{1/q}\le C\left(\Vert \nabla v{\Vert }_{L^{\infty }}\Vert \theta {\Vert }_{{\dot{B}}_{p,q}^s}+\Vert \nabla \theta {\Vert }_{L^{\infty }}\Vert v{\Vert }_{{\dot{B}}_{p,q}^s}\right).$$

$${\left(\sum _{j\in \mathbb{Z}}{2}^{sjq}\Vert [v\cdot \nabla ,{\Delta }_j]\theta {\Vert }_{L^p}^q\right)}^{1/q}\le C\left(\Vert \nabla v{\Vert }_{L^{\infty }}\Vert \theta {\Vert }_{{\dot{B}}_{p,q}^s}+\Vert \theta {\Vert }_{L^{\infty }}\Vert v{\Vert }_{{\dot{B}}_{p,q}^{s+1}}\right).$$

$${\left(\sum _{j\in \mathbb{Z}}{2}^{jqs}\Vert (S_{j-2}u\cdot \nabla ){\Delta }_{j}u-{\Delta }_j(u\cdot \nabla )u{\Vert }_{L^p}^q\right)}^{1/q}\le C\Vert \nabla u{\Vert }_{L^{\infty }}\Vert u{\Vert }_{B_{p,q}^s},$$

$$P_{\pm }v:=\frac{1}{2}\left(\mathbb{P}v\pm i\frac{D}{|D|}\times v\right),$$

$$\{\begin{array}{rl}& \frac{\partial u^{\delta }}{\partial t}-\delta \Delta u^{\delta }+\mathbb{P}\Omega e_3\times u^{\delta }+\mathbb{P}\left(u^{\delta }\cdot \nabla \right)u^{\delta }=0\text{ in }{\mathbb{R}}^3\times (0,\infty ),\\ & \nabla \cdot u^{\delta }=0\text{ in }{\mathbb{R}}^3\times (0,\infty ),\\ & u^{\delta }(x,0)=u_0(x)\text{ in }{\mathbb{R}}^3.\end{array}$$

$$\Vert e^{t\Delta }f{\Vert }_{B_{p,q}^{s_1}}\le C(1+t^{-\frac{1}{2}(s_1-s_0)})\Vert f{\Vert }_{B_{p,q}^{s_0}},$$

$$\underset{0<t<T}{\sup }{\Vert {\int }_0^t{e}^{\delta (t-\tau )\Delta }\mathbb{P}\left(u(\tau )\cdot \nabla \right)v(\tau )d\tau \Vert }_{B_{p,1}^{3/p}}\le C\underset{0<t<T}{\sup }\Vert u(t){\Vert }_{B_{p,1}^{3/p}}\Vert v{\Vert }_{L^1(0,T;B_{p,1}^{3/p+1})},$$

$${\Vert {\int }_0^t{e}^{\delta (t-\tau )\Delta }\mathbb{P}\left(u(\tau )\cdot \nabla \right)v(\tau )d\tau \Vert }_{L^1(0,T;B_{p,1}^{3/p+k})}\le C\left(T+T^{\frac{1}{2}}{\delta }^{-\frac{1}{2}}\right)\underset{0<t<T}{\sup }\Vert u(t){\Vert }_{B_{p,1}^{3/p+k-1}}\Vert v{\Vert }_{L^1(0,T;B_{p,1}^{3/p+k})},$$

$${\Vert {\int }_0^t{e}^{\delta (t-\tau )\Delta }\mathbb{P}\left(u(\tau )\cdot \nabla \right)v(\tau )d\tau \Vert }_{B_{p,1}^{3/p}}\le C{\int }_0^t\Vert \left(u(\tau )\cdot \nabla \right)v(\tau ){\Vert }_{B_{p,1}^{3/p}}d\tau .$$

$$\Vert \left(u(\tau )\cdot \nabla \right)v(\tau ){\Vert }_{B_{p,1}^{3/p}}\le C\Vert u(\tau ){\Vert }_{B_{p,1}^{3/p}}\Vert v(\tau ){\Vert }_{B_{p,1}^{3/p+1}}$$

$$\begin{array}{rl}{\Vert {\int }_0^t{e}^{\delta (t-\tau )\Delta }\mathbb{P}\left(u(\tau )\cdot \nabla \right)v(\tau )d\tau \Vert }_{B_{p,1}^{3/p}}& \le C{\int }_0^t\Vert u(\tau ){\Vert }_{B_{p,1}^{3/p}}\Vert v(\tau ){\Vert }_{B_{p,1}^{3/p+1}}d\tau \\ & \le C\underset{0<t<T}{\sup }\Vert u(t){\Vert }_{B_{p,1}^{3/p}}\Vert v{\Vert }_{L^1(0,T;B_{p,1}^{3/p+1})},\end{array}$$

$$\begin{array}{rl}\Vert {\int }_0^t& e^{\delta (t-\tau )\Delta }\mathbb{P}\left(u(\tau )\cdot \nabla \right)v(\tau )d\tau {\Vert }_{B_{p,1}^{3/p+k}}\\ & \le C{\int }_0^t\{1+{\delta }^{-\frac{1}{2}}(t-\tau {)}^{-\frac{1}{2}}\}\left(\Vert u(\tau ){\Vert }_{B_{p,1}^{3/p}}\Vert v(\tau ){\Vert }_{B_{p,1}^{3/p+k}}+\Vert u(\tau ){\Vert }_{B_{p,1}^{3/p+k-1}}\Vert v(\tau ){\Vert }_{B_{p,1}^{3/p+1}}\right)d\tau \\ & \le C{\int }_0^t\{1+{\delta }^{-\frac{1}{2}}(t-\tau {)}^{-\frac{1}{2}}\}\Vert u(\tau ){\Vert }_{B_{p,1}^{3/p+k-1}}\Vert v(\tau ){\Vert }_{B_{p,1}^{3/p+k}}d\tau \\ & \le C\underset{0<t<T}{\sup }\Vert u(t){\Vert }_{B_{p,1}^{3/p+k-1}}\left\{\Vert v{\Vert }_{L^1(0,T;B_{p,1}^{3/p+k})}+{\delta }^{-\frac{1}{2}}{\int }_0^t(t-\tau {)}^{-\frac{1}{2}}\Vert v(\tau ){\Vert }_{B_{p,1}^{3/p+k}}d\tau \right\},\end{array}$$

$$\begin{array}{rl}\Vert {\int }_0^t& e^{\delta (t-\tau )\Delta }\mathbb{P}\left(u(\tau )\cdot \nabla \right)v(\tau )d\tau {\Vert }_{L^1(0,T;B_{p,1}^{3/p+k})}\\ & \le C\underset{0<t<T}{\sup }\Vert u(t){\Vert }_{B_{p,1}^{3/p+k-1}}\left\{T\Vert v{\Vert }_{L^1(0,T;B_{p,1}^{3/p+k})}+{\delta }^{-\frac{1}{2}}{\int }_0^T\Vert v(\tau ){\Vert }_{B_{p,1}^{3/p+k}}{\int }_{\tau }^T(t-\tau {)}^{-\frac{1}{2}}dtd\tau \right\}\\ & \le C\left(T+T^{\frac{1}{2}}{\delta }^{-\frac{1}{2}}\right)\underset{0<t<T}{\sup }\Vert u(t){\Vert }_{B_{p,1}^{3/p+k-1}}\Vert v{\Vert }_{L^1(0,T;B_{p,1}^{3/p+k})}.\end{array}$$

$$u^{\delta }\in C([0,T_{\delta ,\Omega }];B_{p,1}^{3/p+1}({\mathbb{R}}^3))\cap AC([0,T_{\delta ,\Omega }];B_{p,1}^{3/p}({\mathbb{R}}^3))\cap L^1(0,T_{\delta ,\Omega };B_{p,1}^{3/p+2}({\mathbb{R}}^3))$$

$$u^{\delta }(t)=e^{\delta t\Delta }{u}_0-{\int }_0^t{e}^{\delta (t-\tau )\Delta }\mathbb{P}\Omega e_3\times u^{\delta }(\tau )d\tau -{\int }_0^t{e}^{\delta (t-\tau )\Delta }\mathbb{P}\left(u^{\delta }(\tau )\cdot \nabla \right)u^{\delta }(\tau )d\tau$$

$$\Vert e^{\delta t\Delta }f{\Vert }_{L^1(0,T;B_{p,1}^{3/p+2})}\le C(T+T^{\frac{1}{2}}{\delta }^{-\frac{1}{2}})\Vert f{\Vert }_{B_{p,1}^{3/p+1}},$$

$$\underset{0\le t\le T}{\sup }\Vert e^{\delta t\Delta }{u}_0{\Vert }_{B_{p,1}^{3/p+1}}+L_{\delta ,T}^{-1}\Vert e^{\delta t\Delta }{u}_0{\Vert }_{L^1(0,T;B_{p,1}^{3/p+2})}\le C_0\Vert u_0{\Vert }_{B_{p,1}^{3/p+1}},$$

$$\mathcal{B}(u^{\delta })(t)=e^{\delta t\Delta }{u}_0-{\int }_0^t{e}^{\delta (t-\tau )\Delta }\mathbb{P}\Omega e_3\times u^{\delta }(\tau )d\tau -{\int }_0^t{e}^{\delta (t-\tau )\Delta }\mathbb{P}\left(u^{\delta }(\tau )\cdot \nabla \right)u^{\delta }(\tau )d\tau$$

Full text

AMS MSC: 35Q31, 76U05, 76B03, 35A07, 42B35

Key: Euler equations; Coriolis force; Long-time solvability; Blow up; Besov-spaces

1 Introduction

We consider the free incompressible Euler equations with Coriolis force

[TABLE]

where $u(x,t)=\left(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t)\right)$ stands for the velocity field, $\mathbb{P}=(\delta_{jk}+R_{j}R_{k})_{1\leq j,k\leq 3}$ is the Leray-Helmholtz projection and $R_{j}$ denotes the $j$-th Riesz transform. The Coriolis parameter $\Omega\in\mathbb{R}$ corresponds to twice the speed of rotation around the vertical unit vector $e_{3}=(0,0,1).$ The initial velocity is denoted by $u_{0}=u_{0}(x)=\left(u_{0,1}(x),u_{0,2}(x),u_{0,3}(x)\right)$ and satisfies the compatibility condition $\nabla\cdot u_{0}=0$. The reader is referred to the book [12] for more details about the physical model. Throughout the paper, we denote spaces of scalar and vector functions abusively in the same way; for example, we write $u_{0}\in H^{s}(\mathbb{R}^{3})$ instead of $u_{0}\in(H^{s}(\mathbb{R}^{3}))^{3}.$

The system (1.1) has been studied by several authors in the case $\Omega=0$ that corresponds to the classical Euler equations (E). In what follows we give a brief review of some of these results. In the framework of Sobolev spaces, Kato [20] showed that (E) has a unique local-in-time solution $u\in C\left(\left[0,T\right];H^{s}\left(\mathbb{R}^{3}\right)\right)\cap C^{1}(\left[0,T\right];H^{s-1}\left(\mathbb{R}^{3}\right))$ for $u_{0}\in H^{s}\left(\mathbb{R}^{3}\right)$ with an integer $s\geq 3$ where $T=T(\left\|u_{0}\right\|_{H^{s}(\mathbb{R}^{3})})$. In [21], Kato and Ponce proved that if $s>2/p+1$, $1<p<\infty$ and $u_{0}\in H_{p}^{s}\left(\mathbb{R}^{2}\right),$ then there exists a unique 2D global solution $u\in C\left(\left[0,\infty\right);H_{p}^{s}\left(\mathbb{R}^{2}\right)\right)$. Later, in [22] they considered $n\geq 2$ and proved that for $s>n/p+1,$ $1<p<\infty$ and $u_{0}\in H_{p}^{s}\left(\mathbb{R}^{n}\right),$ there exist $T>0$ and a unique solution $u\in C\left(\left[0,T\right];H_{p}^{s}\left(\mathbb{R}^{n}\right)\right)\cap C^{1}\left(\left[0,T\right];H_{p}^{s-1}\left(\mathbb{R}^{n}\right)\right).$ Temam [29] extended the results of Kato [20] to $H^{m}$ and $W^{m,p}$ in bounded domains (see also Ebin-Marsden [15] and Bourguignon-Brezis [6]). For existence and uniqueness results in Holder $C^{k,\gamma}$ and Triebel-Lizorkin $F_{p,q}^{s}$ spaces, the reader is referred to [11] and [7, 8], respectively.

In the context of Besov spaces, Chae [9] and Zhou [31] proved that (E) has a unique solution $u\in C([0,T];\linebreak{B_{p,1}^{n/p+1}\left(\mathbb{R}^{n}\right)})$ for $1<p<\infty$ and $n\geq 3$ (see also [30] for $n=2$). After, the borderline cases $p=\infty$ [25] and $p=1$ [26] was considered by Pak and Park. Takada [28] showed existence-uniqueness in Besov type spaces based on weak-$L^{p}$ with $1<p<\infty$ and $n\geq 3.$ The exponent $s=\frac{n}{p}+1$ is critical for (E) in $H_{p}^{s}$ and $B_{p,q}^{s}$-spaces. In fact, Bourgain and Li [5] showed that (E) is ill-posed in $H_{p}^{s}$ and $B_{p,q}^{n/p+1}$ for $1\leq p<\infty$, $1<q\leq\infty$ and $n=2,3$. So, it is natural to consider $q=1$ when $s=\frac{n}{p}+1$. The critical case is also of special interest because the regularity index $s-1=\frac{n}{p}$ of the vorticity $\nabla\times u$ corresponds to a critical case of Sobolev type embeddings. Motivated by the symbol $>,$ the case $s>\frac{n}{p}+1$ has been named in the literature as supercritical.

For $\Omega\neq 0$, Dutrifoy [14] showed long-time existence of solutions for (1.1) with lower bound on the existence-time $T_{\Omega}\gtrsim\log{\log{|\Omega|}}$ provided that $|\Omega|$ is large enough and $u_{0}$ belongs to a certain Sobolev type class. Also, Dutrifoy [13] and Charve [10] obtained analogous results for quasigeostrophic systems. Recently, for $s>s_{0}=\frac{3}{2}+1$ and $u_{0}\in H^{s}(\mathbb{R}^{3}),$ Koh, Lee and Takada [23] proved that there exists a unique local in time solution $u$ for (1.1) in the class $C([0,T];H^{s}(\mathbb{R}^{3}))\cap C^{1}([0,T];H^{s-1}(\mathbb{R}^{3})).$ Moreover, assuming that $s>s_{1}=\frac{5}{2}+1,$ they showed that their solutions can be extended to long-time intervals $[0,T_{\Omega}]$ provided that the speed of rotation is large enough. For the viscous case, we refer the reader to the works [1, 2, 12, 18] for global well-posedness in Sobolev spaces with $\left|\Omega\right|$ large enough and to the papers [16, 19] (and their references) for results about global well-posedness with $\Omega$-uniform smallness condition on initial data in different types of critical spaces (e.g., in Fourier Besov spaces).

In view of the previous results for (1.1) and (E), it is natural to wonder about the borderline cases $s_{0}$ and $s_{1}.$ In this paper we extend the results of [23] by treating these two cases in the framework of Besov spaces. To be more precise, we consider the critical regularity $s_{0}$ and show local-in-time existence and uniqueness of solutions for initial data in the critical Besov space $B_{2,1}^{s_{0}}$ with smallness condition on the existence-time uniformly in $\Omega\in\mathbb{R}.$ After, for large Coriolis parameter $\left|\Omega\right|,$ we obtain long-time solvability of (1.1) in $B_{2,1}^{s}$ in the borderline case $s=s_{1}$.  It is worth to observe that $H^{s}\subset B_{2,1}^{5/2}$ and  $H^{s}\subset B_{2,1}^{7/2}$ for $s>\frac{3}{2}+1$ and $s>\frac{5}{2}+1,$ respectively, and so our result provides a larger class for both local and long time solvability of (1.1).

Our main result reads as follows.

Theorem 1.1**.**

1. $(i)$

Let $u_{0}\in B_{2,1}^{5/2}(\mathbb{R}^{3})$ satisfy $\nabla\cdot u_{0}=0$. There exists $T=T(\|u_{0}\|_{B_{2,1}^{5/2}})>0$ such that (1.1) has a unique solution $u\in C([0,T];B_{2,1}^{5/2}(\mathbb{R}^{3}))\cap C^{1}([0,T];B_{2,1}^{3/2}(\mathbb{R}^{3}))$, for all $\Omega\in\mathbb{R}$. 2. $(ii)$

Let $0<T<\infty$ and $u_{0}\in B_{2,1}^{7/2}(\mathbb{R}^{3})$ be such that $\nabla\cdot u_{0}=0$. There exists $\Omega_{0}=\Omega_{0}(T,\|u_{0}\|_{B_{2,1}^{7/2}})>0$ such that (1.1) has a unique solution $u\in C([0,T];B_{2,1}^{7/2}(\mathbb{R}^{3}))\cap C^{1}([0,T];B_{2,1}^{5/2}(\mathbb{R}^{3}))$ provided that $|\Omega|\geq\Omega_{0}.$

Considering $\Omega=0,$ item $(i)$ recovers the local existence result by Chae [9] and Zhou [31] for Euler equations in $B_{p,1}^{n/p+1}(\mathbb{R}^{n})$ in the case $p=2$ and $n=3$. Assuming further regularity on the initial data, item $(ii)$ shows that local solutions can be extended to arbitrary large time $T>0$ provided that $\left|\Omega\right|$ is large enough and so it resembles results for the 2D Euler equations (see [30, 9]). In fact, we recall that existence of smooth solutions for the 3D Euler equations is an outstanding open problem. Long-time solvability type results for (1.1) with arbitrary data show a smoothing effect connected to the speed of rotation $\Omega$ (see [12]).

Finally, we comment on some technical points in our results. The general strategy of this paper consists in three basic steps: approximation scheme; a priori $\Omega$-uniform estimates and passing to the limit for obtaining local-in time solutions; blow-up criterion and long-time solvability. This is the same one employed by [23] in $H^{s}$-spaces however here we need to carry out the necessary estimates in the borderline Besov spaces $B_{2,1}^{s_{0}}$ and $B_{2,1}^{s_{1}}$. In order to pass the limit in the approximation scheme $\{u^{\delta}\}_{\delta>0}$, the authors of [23] relied on the Hilbert structure of $H^{s}$-spaces. Since our setting has not such property, we need to control $u^{\delta}$ by means of estimates involving localization and $B_{2,1}^{s}$-norms (see, e.g., Lemma 3.3, Proposition 3.4 and proof of Theorem 1.1). In order to cover the endpoints $s_{0}$ and $s_{1}$ of the ranges in [23], we are inspired by previous results for the Euler equations (E) [9, 25, 31] and consider $B_{2,1}^{s}$-spaces (and the embedding $B_{2,1}^{3/2}\hookrightarrow L^{\infty}$ in $\mathbb{R}^{3}$) that allow us to have $\nabla u\in L^{\infty}$ for $s=s_{0}$ (which is not true in $H^{s}$) and control globally in time $U(t)=\int_{0}^{t}\|\nabla u(\tau)\|_{L^{\infty}}\ d\tau$ for large $\Omega$ when $s=s_{1}$. The quantity $U(t)$ is used to derive a blow-up criterion and obtain long-time solutions. Also, we show Lemma 4.1 that deals with the time-continuity of weak solutions for (1.1) and is useful to prove time-regularity of solutions obtained as limit of the approximation scheme. For that matter, we extend [26, Lemma 2.1] (that considered solutions of (E) in $B_{1,1}^{d+1}(\mathbb{R}^{d})$) to the Euler Coriolis equations in $B_{p,1}^{3/p+1}(\mathbb{R}^{3})$ with $1\leq p<\infty.$

The plan of this paper is as follows. The next section is devoted to some preliminaries about product and commutator estimates in Besov spaces and projection operators linked to the Coriolis term. In Section 3, we deal with the approximation scheme $\{u^{\delta}\}_{\delta>0}$ and show local existence on $[0,T]$ with $T>0$ independent of $\delta$ and $\Omega.$ The proof of Theorem 1.1 is given in Section 4 through three subsections: item $(i)$ in subsection 4.1, blow-up criterion in subsection 4.2, and item $(ii)$ in subsection 4.3.

2 Function spaces and projection operators

This section is devoted to some preliminaries about Besov spaces. We refer the reader to [4] for more details on these spaces and their properties. Also, we recall two projection operators that will be useful for our purposes.

Let $\mathcal{S}(\mathbb{R}^{3})$ and $\mathcal{S}^{\prime}(\mathbb{R}^{3})$ stand for the Schwartz class and the space of tempered distributions, respectively. Let $\widehat{f}$ denote the Fourier transform of $f\in\mathcal{S}^{\prime}$. Consider a nonnegative radial function $\phi_{0}\in\mathcal{S}(\mathbb{R}^{3})$ satisfying $0\leq\widehat{\phi}_{0}(\xi)\leq 1$ for all $\xi\in\mathbb{R}^{3}$, $\mbox{supp}\ \widehat{\phi}_{0}\subset\{\xi\in\mathbb{R}^{3}:\frac{1}{2}\leq|\xi|\leq 2\}$ and

[TABLE]

where $\phi_{j}(x):=2^{3j}\phi_{0}(2^{j}x).$ For $k\in\mathbb{Z}$, we define the function $S_{k}\in\mathcal{S}$ as

[TABLE]

and denote $\psi=S_{0}$. For $f\in\mathcal{S}^{\prime}(\mathbb{R}^{3}),$ the Littlewood-Paley operator $\Delta_{j}$ is defined by $\Delta_{j}f:=\phi_{j}\ast f.$

Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$ and let $\mathcal{P}$ denote the set of polynomials with $3$ variables. The homogeneous Besov space ${\dot{B}}_{p,q}^{s}(\mathbb{R}^{3})$ is the set of all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{3})/\mathcal{P}$ such that

[TABLE]

The inhomogeneous version of ${\dot{B}}_{p,q}^{s}$, denoted by $B_{p,q}^{s}(\mathbb{R}^{3})$, is defined as the set of all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{3})$ such that

[TABLE]

The pairs $({\dot{B}}_{p,q}^{s},\|\cdot\|_{{\dot{B}}_{p,q}^{s}})$ and $(B_{p,q}^{s},\|\cdot\|_{B_{p,q}^{s}})$ are Banach spaces. For $s>0$, we have the equivalence

[TABLE]

Lemma 2.1** (Bernstein inequality).**

Assume that $f\in L^{p}$, $1\leq p\leq\infty$, and $\mbox{supp}\ \widehat{f}\subset\{\xi\in\mathbb{R}^{3}:2^{j-2}\leq|\xi|<2^{j}\}$. Then there exists a constant $C=C(k)>0$ such that

[TABLE]

Remark 2.2**.**

As a consequence of the above lemma we have the following equivalence

[TABLE]

We also recall the estimate (see, e.g., [28])

[TABLE]

where $s>n/p$ with $1\leq p,q\leq\infty$, or $s=n/p$ with $1\leq p\leq\infty$ and $q=1$. Thus, for $s>n/p+1$ with $1\leq p,q\leq\infty$ or $s=n/p+1$ with $1\leq p\leq\infty$ and $q=1$, we have the estimates

[TABLE]

The following lemma contains product estimates in the framework of Besov spaces (see [9]).

Lemma 2.3**.**

Let $s>0$, $1\leq p,q\leq\infty$, $1\leq p_{1},p_{2}\leq\infty$ and $1\leq r_{1},r_{2}\leq\infty$ satisfy $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{r_{1}}+\frac{1}{r_{2}}.$ Then there exists a universal constant $C>0$ such that

[TABLE]

In the next two lemmas we recall estimates in ${\dot{B}}_{p,q}^{s}$ and $B_{p,q}^{s}$ for the commutator (see [9, 28])

[TABLE]

Lemma 2.4**.**

Let $1<p<\infty$ and $1\leq q\leq\infty$.

1. $(i)$

Let $s>0$, $v\in{\dot{B}}_{p,q}^{s}(\mathbb{R}^{n})$ with $\nabla v\in L^{\infty}(\mathbb{R}^{n})$ and $\nabla\cdot v=0,$ and $\theta\in{\dot{B}}_{p,q}^{s}(\mathbb{R}^{n})$ with $\nabla\theta\in L^{\infty}(\mathbb{R}^{n})$. Then, there exists a universal constant $C>0$ such that

[TABLE] 2. $(ii)$

Let $s>-1$, $v\in{\dot{B}}_{p,q}^{s+1}(\mathbb{R}^{n})$ with $\nabla v\in L^{\infty}(\mathbb{R}^{n})$ and $\nabla\cdot v=0,$ and $\theta\in{\dot{B}}_{p,q}^{s}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$. Then, there exists a universal constant $C>0$ such that

[TABLE]

Lemma 2.5**.**

Let $1<p<\infty$ and let $s>3/p+1$ with $1\leq q\leq\infty$ or $s=3/p+1$ with $q=1$. Then, there exists a constant $C>0$ such that

[TABLE]

for all $u\in B_{p,q}^{s}(\mathbb{R}^{3})$ with $\nabla\cdot u=0$.

In order to handle the Coriolis term, we will need the following projection operators $P_{\pm}:L^{2}(\mathbb{R}^{3})^{3}\longrightarrow L^{2}(\mathbb{R}^{3})^{3}$ given by

[TABLE]

where $\frac{D}{|D|}\times$ is defined by means of the Fourier transform as $(\frac{D}{|D|}\times v)^{\widehat{}}(\xi):=\frac{\xi}{|\xi|}\times\widehat{v}(\xi)$.

The next lemma contains basic properties of $P_{\pm}$ and can be found in [14, 23].

Lemma 2.6**.**

The projections $P_{\pm}$ satisfy $P_{\pm}\mathbb{P}=P_{\pm}$. Moreover, if $\ \nabla\cdot v=0$ we have that $v=P_{+}v+P_{-}v$, $\mathbb{P}\left(e_{3}\times v\right)=-i\frac{D_{3}}{|D|}(P_{+}v-P_{-}v)$, $P_{\pm}P_{\pm}=P_{\pm}$, and $P_{\pm}P_{\mp}=0$.

3 Approximation scheme

Let $u_{0}$ be the initial velocity in (1.1). For $0<\delta<1$, we consider the approximate parabolic problem

[TABLE]

We are going to show that the above problem has a solution for each $\delta>0$ in a suitable class involving Besov spaces. For that matter, first we recall some estimates for the heat semigroup $\{e^{t\Delta}\}_{t\geq 0}$ in $B_{p,q}^{s}$ (see, e.g., [24]).

Lemma 3.1**.**

Let $s_{0}\leq s_{1}$ and $1\leq p,q\leq\infty$. Then there exists a constant $C>0$ (independent of $p,q$ and $t>0$) such that

[TABLE]

for all $f\in{B_{p,q}^{s_{0}}}(\mathbb{R}^{3})$.

We start by showing estimates for the bilinear term of the mild formulation for (3.1).

Lemma 3.2**.**

Let $0<\delta<1$ and $1<p<\infty$.

1. $(i)$

There exists $C>0$ such that

[TABLE]

for all $u\in C([0,T];{B_{p,1}^{3/p}}(\mathbb{R}^{3}))$ and $v\in L^{1}(0,T;{B_{p,1}^{3/p+1}}(\mathbb{R}^{3})).$ 2. $(ii)$

Let $k=1,2.$ There exists $C>0$ such that

[TABLE]

for all $u\in C([0,T];{B_{p,1}^{3/p}}(\mathbb{R}^{3}))$ with $\nabla\cdot u=0$ and $v\in L^{1}(0,T;{B_{p,1}^{3/p+k}}(\mathbb{R}^{3})).$

Proof. For $1<p<\infty,$ we have that $\|e^{\delta t\Delta}\|_{B_{p,1}^{3/p}\rightarrow B_{p,1}^{3/p}}\leq 1$ and $\mathbb{P}$ is bounded in $B_{p,1}^{3/p}.$ So, we can estimate

[TABLE]

From Lemmas 2.3 and 2.1, it follows that

[TABLE]

and then

[TABLE]

for all $0<t<T$, which gives (3.2).

By Minkowski inequality and Lemmas 3.1, 2.3 and 2.1, we have that

[TABLE]

for all $0<t<T$ and $k=1,2$. We can now compute the norm $\left\|\cdot\right\|_{L^{1}(0,T)}$ in (3.3) to obtain

[TABLE]

$\diamond$

Before proceeding, we recall that $AC([0,T];X)$ denotes the set of all $X$-valued absolutely continuous functions on $[0,T]$. The next lemma ensures the existence of strong solution for (3.1). The proof follows essentially the same steps of [23, Lemma 3.1.] but using estimates in Besov spaces instead of Sobolev spaces.

Lemma 3.3**.**

Let $1<p<\infty$, $\delta\in(0,1)$ and $\Omega\in\mathbb{R}$. Assume that $u_{0}\in B_{p,1}^{3/p+1}(\mathbb{R}^{3})$ and $\nabla\cdot u_{0}=0$. Then there exists a positive time $T_{\delta,\Omega}=T(\delta,|\Omega|,\|u_{0}\|_{B_{p,1}^{3/p+1}})$ such that (3.1) has a unique strong solution $u^{\delta}$ satisfying

[TABLE]

Proof.  Firstly, we consider the mild formulation for (3.1)

[TABLE]

and show the existence of a local in time solution. Lemma 3.1 yields the estimate

[TABLE]

for all $f\in{B_{p,1}^{3/p+1}}$. Thus, for all $0<T<\infty$ we have

[TABLE]

where $C_{0}>0$ is a constant and $L_{\delta,T}=(T+T^{\frac{1}{2}}\delta^{-\frac{1}{2}}).$

Consider the map

[TABLE]

and the complete metric space

[TABLE]

whose norm is given by

[TABLE]

We claim that the map $\mathcal{B}$ is a contraction map on $Z_{T}$ for small $T>0$.

In fact, using that $\|e^{\delta t\Delta}\|_{B_{p,1}^{3/p+1}\rightarrow B_{p,1}^{3/p+1}}\leq 1$ and $\mathbb{P}$ is bounded in $B_{p,1}^{3/p+1}$ for $1<p<\infty$, we have that

[TABLE]

Taking the supremum over $t\in[0,T],$ we get a constant $C>0$ such that

[TABLE]

for all $u\in C([0,T];{B_{p,1}^{3/p+1}}(\mathbb{R}^{3}))$. Similarly,

[TABLE]

for all $t\in[0,T]$. An integration of (3.8) over $[0,T]$ yields the estimate

[TABLE]

for all $u\in L^{1}(0,T;{B_{p,1}^{3/p+2}}(\mathbb{R}^{3})).$

Next we can apply (3.7), (3.9) and Lemma 3.2 in order to estimate

[TABLE]

for all $u^{\delta},v^{\delta}\in Z_{T}$. Moreover, using (3.6) and (3.10) with $v^{\delta}=0,$ we obtain

[TABLE]

for all $u^{\delta}\in Z_{T}$. Next we choose $T=T_{\delta,\Omega}=T(\delta,|\Omega|,\|u_{0}\|_{B_{p,1}^{3/p+1}})>0$ such that

[TABLE]

Inserting (3.12) into (3.11) and (3.10), we get that $\mathcal{B}(Z_{T_{\delta,\Omega}})\subset Z_{T_{\delta,\Omega}}$ and

[TABLE]

which gives the claim. By the Banach Fixed Point Theorem, there exists a unique solution $u^{\delta}\in Z_{T_{\delta,\Omega}}$ for (3.5).

We claim that $u^{\delta}\in Z_{T_{\delta,\Omega}}$ is a strong solution for (3.1) in the class (3.4). By the above estimates and using that $u^{\delta}\in C([0,T];B_{p,1}^{3/p+1}(\mathbb{R}^{3}))\cap L^{1}(0,T;B_{p,1}^{3/p+2}(\mathbb{R}^{3}))$, it is not difficult to see that

[TABLE]

and $\delta\Delta v^{\delta}\in L^{1}(0,T_{\delta,\Omega};B_{p,1}^{3/p}(\mathbb{R}^{3}))$ where

[TABLE]

Thus, $\partial_{t}v^{\delta}\in L^{1}(0,T_{\delta,\Omega};B_{p,1}^{3/p}(\mathbb{R}^{3}))$ and then $v^{\delta}\in AC([0,T_{\delta,\Omega}];B_{p,1}^{3/p}(\mathbb{R}^{3})).$ Moreover, $e^{\delta t\Delta}u_{0}\in\linebreak{\ AC([0,T_{\delta,\Omega}];B_{p,1}^{3/p}(\mathbb{R}^{3}))}$. By standard arguments (see Kato [20] and Pazy [27]), we obtain the desired claim. For more details see [23]. The uniqueness follows from the fact that $u^{\delta}$ is the unique solution for (3.5) in the class $Z_{T_{\delta,\Omega}}$. $\diamond$

In what follows, we prove that there exists $T>0$ independent of $\delta\in(0,1)$ and $\Omega\in\mathbb{R}$ such that the solution $u^{\delta}$ exists on $[0,T]$. For that, we need some a priori uniform estimates for $u^{\delta}$ in the space $B_{2,1}^{5/2}.$

Proposition 3.4**.**

Assume that $u_{0}\in B_{2,1}^{5/2}(\mathbb{R}^{3})$ and $\nabla\cdot u_{0}=0$. There exists $T=T(\|u_{0}\|_{B_{2,1}^{5/2}})>0$ such that (3.1) has a unique strong solution

[TABLE]

for all $0<\delta<1$ and $\Omega\in\mathbb{R}$. Furthermore, $\{u^{\delta}\}_{\delta\in(0,1)}$ is bounded in $C([0,T];B_{2,1}^{5/2}(\mathbb{R}^{3}))$.

Proof. Applying the Littlewood-Paley operator $\Delta_{j}$ to the equation in (3.1), taking the $L^{2}$-norm product with $\Delta_{j}u^{\delta}(t)$, and using $\nabla\cdot\Delta_{j}u^{\delta}=0$ and the skew-symmetric of $e_{3}\times$, we have that

[TABLE]

Notice that the second term in the right hand side of (3.13) is non-negative. So, using that

[TABLE]

and recalling the definition of the commutator $[u^{\delta}(t)\cdot\nabla,\Delta_{j}]$, we get

[TABLE]

By the Cauchy-Schwarz inequality, it follows that

[TABLE]

Multiplying by $2^{5/2j}$, applying the $l^{1}(\mathbb{Z})$-norm and Lemma 2.4, we can estimate

[TABLE]

By Remark 2.2, it follows that

[TABLE]

On the other hand, taking the $L^{2}$-norm product with $u^{\delta}(t)$ in (3.1), we arrive at

[TABLE]

Above, we have used the skew-symmetric of $e_{3}\times$ and $\langle(u^{\delta}(t)\cdot\nabla)u^{\delta}(t),u^{\delta}(t)\rangle_{L^{2}}=0$ because $\nabla\cdot u^{\delta}=0$. Then,

[TABLE]

Denote by $\|\cdot\|_{B_{2,1}^{s}}^{\ast}$ the equivalent norm $\|\cdot\|_{L^{2}}+\|\cdot\|_{\dot{B}_{2,1}^{s}}$ in $B_{2,1}^{s}$ (see (2.1)). By (3.14) and (3.15), we have that

[TABLE]

Using (3.16) and that $K_{1}\|\cdot\|_{B_{2,1}^{s}}\leq\|\cdot\|_{B_{2,1}^{s}}^{\ast}\leq K_{2}\|\cdot\|_{B_{2,1}^{s}}$ for some $K_{1},K_{2}>0$, it follows that

[TABLE]

for $0\leq t<(CK_{2}\|u_{0}\|_{B_{2,1}^{5/2}})^{-1}$. Taking $T=T(\|u_{0}\|_{B_{2,1}^{5/2}})=(2CK_{2}\|u_{0}\|_{B_{2,1}^{5/2}})^{-1}$ and $L=2K_{2}/K_{1}$, we obtain

[TABLE]

Notice that $T>0$ is independent of $\delta\in(0,1)$ and $\Omega\in\mathbb{R}$. If $T_{\delta,\Omega}<T$, by (3.12) and (3.17) we can take $T_{\delta,\Omega}^{\prime}=T_{\delta,\Omega}^{\prime}(\|u_{0}\|_{B_{2,1}^{5/2}})>0$ small enough and solve (3.1) on $[T_{\delta,\Omega},T_{\delta,\Omega}+T_{\delta,\Omega}^{\prime}]$ with the initial value $u^{\delta}(T_{\delta,\Omega})\in B_{2,1}^{5/2}(\mathbb{R}^{3})$. It follows that the solution $u^{\delta}$ can be extended to the interval $[0,T_{\delta,\Omega}+T_{\delta,\Omega}^{\prime}]$. Invoking again the same procedure, we can extend $u^{\delta}$ (if necessary) to $[0,T_{\delta,\Omega}+2T_{\delta,\Omega}^{\prime}]$, $[0,T_{\delta,\Omega}+3T_{\delta,\Omega}^{\prime}]$ and so on, and obtain a solution $u^{\delta}$ for (3.1) on $[0,T]$ satisfying (3.17). $\diamond$

4 Proof of Theorem 1.1

In this section we prove Theorem 1.1 through three subsections.

4.1 Proof of item $(i)$

For $0<\delta_{1}<\delta_{2}<1$, we can write

[TABLE]

We will show that there exists a limit $u\in C([0,T];B_{2,1}^{3/2}(\mathbb{R}^{3}))$ such that

[TABLE]

We start by obtaining estimates in ${B}_{2,1}^{3/2}$ for the difference $u^{\delta_{1}}-u^{\delta_{2}}$ uniformly in $[0,T]$. Computing the $L^{2}$-inner product of (4.1) with $u^{\delta_{1}}-u^{\delta_{2}}$, and afterwards using the skew-symmetry of $\left(e_{3}\times\cdot\right)$, $\nabla\cdot(u^{\delta_{1}}-u^{\delta_{2}})=0$, Holder inequality, and Remark 2.2, we obtain

[TABLE]

Integrating over $(0,t)$ and using (3.17), we arrive at the estimate

[TABLE]

By Gronwall inequality and (4.3), we have that there exists $C>0$ such that

[TABLE]

and, consequently, as $\delta_{2}\rightarrow 0^{+}$ we have

[TABLE]

Let $0<\theta<1$ and $s_{1},s_{2},s_{3}\geq 0$ be such $s_{3}=(1-\theta)s_{1}+\theta s_{2}$. By Gagliardo-Nirenberg type inequality in Besov spaces (see [17]), we can estimate

[TABLE]

Considering $s_{1}=0,$ $s_{2}=5/2$ and $s_{3}=\theta s_{2}$ in (4.5), and using (3.17), $B_{2,2}^{0}=L^{2}$ and (4.4) , we obtain

[TABLE]

for each fixed $\theta\in(0,1).$ Hence, by completeness and uniqueness of the limit in the distributional sense, $u^{\delta}\rightarrow u$ in $L^{\infty}(0,T;B_{2,1}^{\tilde{s}})$ for all $0<\tilde{s}<5/2$. In particular, taking $\tilde{s}=3/2$ and recalling that $u^{\delta}\in C([0,T];B_{2,1}^{3/2}(\mathbb{R}^{3})),$ we obtain (4.2).

Also, in view of (3.17), it follows that $(u^{\delta})_{\delta\in(0,1)}$ is bounded in $L^{\infty}(0,T;B_{2,1}^{5/2}(\mathbb{R}^{3}))$. Then, we can extract a subsequence $(u^{\delta^{(j)}})_{j=1}^{\infty}$ that converges to $u$ weakly-$\star$ in $L^{\infty}(0,T;B_{2,1}^{5/2}(\mathbb{R}^{3}))$. Thus we have that

[TABLE]

and

[TABLE]

Next we claim that $u$ is a solution for (1.1). For the nonlinear term, by using integration by parts, Lemma 2.3, Remark 2.2, (3.17) and (4.7), we can estimate

[TABLE]

which implies

[TABLE]

Also, we have that

[TABLE]

and

[TABLE]

Then

[TABLE]

Therefore, since $u^{\delta}$ satisfies (3.1), we obtain from (4.8), (4.9) and the continuous inclusion $B_{2,1}^{3/2}\subset B_{2,1}^{1/2}$ that

[TABLE]

In view of the above estimates and (4.6), we can see that both sides of (4.10) belong to $C([0,T];B_{2,1}^{3/2}(\mathbb{R}^{3}).$ Thus, equality (4.10) holds in $B_{2,1}^{3/2}(\mathbb{R}^{3})$ and $u\in AC([0,T];B_{2,1}^{3/2}(\mathbb{R}^{3}))\cap L^{\infty}(0,T;B_{2,1}^{5/2}(\mathbb{R}^{3}))$ is a solution for (1.1), as claimed.

The next lemma deals with the time-continuity of solutions for (1.1). In particular, for $p=2$ it implies the time-continuity of the solution $u$ in $B_{2,1}^{5/2}(\mathbb{R}^{3})$ obtained as limit of the approximation scheme. Notice that in fact it holds for $1\leq p<\infty.$

Lemma 4.1**.**

Let $0<T<\infty$ and $1\leq p<\infty$. If $u$ is a solution for (1.1) in $L^{\infty}(0,T;B_{p,1}^{3/p+1}(\mathbb{R}^{3}))$ with initial velocity $u_{0}\in B_{p,1}^{3/p+1}(\mathbb{R}^{3})$ satisfying $\nabla\cdot u_{0}=0$, then $u\in C([0,T];B_{p,1}^{3/p+1}(\mathbb{R}^{3}))$.

Proof. Firstly, by Lemma 2.3, we have that $\partial_{t}u\in L^{\infty}(0,T;B_{p,1}^{3/p}(\mathbb{R}^{3}))$. Thus

[TABLE]

For every $k\in\mathbb{N}$, we denote $w_{k}:=S_{k}u$. We are going to prove that the sequence $\{w_{k}\}_{k\in\mathbb{N}}$ converges to $u$ in $L^{\infty}(0,T;B_{p,1}^{3/p}(\mathbb{R}^{3}))$. Applying the Littlewood-Paley operator in (1.1), for each $j\in\mathbb{N}$ we obtain

[TABLE]

Since $\Delta_{j}u$ is absolutely continuous on $[0,T]$ with values in $L^{p}(\mathbb{R}^{3})$ and $\nabla\cdot S_{j-2}u=0$, we can estimate

[TABLE]

It follows that

[TABLE]

The first term in the right-hand side converges to zero as $k\rightarrow\infty$ because $u_{0}\in B_{p,1}^{3/p+1}(\mathbb{R}^{3})$. By Lemma 2.3, Lemma 2.5 and the fact that $u(t)\in B_{p,1}^{3/p+1}(\mathbb{R}^{3})$, we have that the second and third terms in the right-hand side also converge to zero as $k\rightarrow\infty$. Therefore, the sequence $\{w_{k}\}_{k\in\mathbb{N}}$ converges to $u$ in $L^{\infty}(0,T;B_{p,1}^{3/p+1}(\mathbb{R}^{3}))$. Moreover, we get

[TABLE]

Estimate (4.11) and the fact that $u\in C([0,T];B_{p,1}^{3/p}(\mathbb{R}^{3}))$ imply that each $w_{k}\in C([0,T];B_{p,1}^{3/p+1}(\mathbb{R}^{3}))$. Therefore, the limit $u$ also belongs to $C([0,T];B_{p,1}^{3/p+1}(\mathbb{R}^{3}))$. $\diamond$

Now, taking $p=2$ in Lemma 4.1, since $u\in L^{\infty}(0,T;B_{2,1}^{5/2}(\mathbb{R}^{3}))$ and $u_{0}\in B_{2,1}^{5/2}(\mathbb{R}^{3})$ we have that $u\in C([0,T];B_{2,1}^{5/2}(\mathbb{R}^{3}))$, and then $u$ satisfies

[TABLE]

This shows that $u\in C^{1}([0,T];B_{2,1}^{3/2}(\mathbb{R}^{3}))$, and therefore $u$ is a strong solution for (1.1) in the class

[TABLE]

**Uniqueness. **Let $u$ and $v$ be strong solutions for (1.1) in the class (4.13) with the same initial data $u_{0}(x)$. Subtracting the corresponding equations satisfied by $u$ and $v$, we get

[TABLE]

Computing the $L^{2}$-inner product of (4.14) with $u-v$, we obtain

[TABLE]

and then

[TABLE]

Since $\|u\|_{L^{\infty}(0,T;B_{2,1}^{5/2})}<\infty,$ we can use Gronwall inequality to obtain $\|u(t)-v(t)\|_{L^{2}}=0$ for all $t\in[0,T]$, and then $u\equiv v.$ $\diamond$

4.2 Blow-up criterion

In this part, we prove a blow-up criterion of BKM type (see [3]). We will use it to prove item (ii) of Theorem 1.1.

Proposition 4.2**.**

Let $u_{0}\in B_{2,1}^{5/2}(\mathbb{R}^{3})$ with $\nabla\cdot u_{0}=0$. Assume that

[TABLE]

is a solution for (1.1). For some $T^{\prime}>T,$ $u$ can be extended to $[0,T^{\prime})$ with $u\in C([0,T^{\prime});B_{2,1}^{5/2}(\mathbb{R}^{3}))\cap\linebreak{C^{1}([0,T^{\prime});B_{2,1}^{3/2}(\mathbb{R}^{3}))}$ provided that $\int_{0}^{T}\|\nabla u(t)\|_{L^{\infty}}\ dt<\infty$.

Proof. Item (i) of Theorem 1.1 assures that the existence-time $T>0$ depends only on the initial data norm $\|u_{0}\|_{B_{2,1}^{5/2}}$. Computing the $L^{2}$-inner product of (1.1) with $u$, using the symmetry of $e_{3}\times u$ and $\nabla\cdot u=0$, one can deduce

[TABLE]

Moreover, we can apply the operator $\Delta_{j}$ in (1.1), multiply the result by $\Delta_{j}u$ and after use $\langle(u\cdot\nabla)\Delta_{j}u,\Delta_{j}u\rangle_{L^{2}}=0$ to get the identity

[TABLE]

Using the Schwartz inequality and integrating (4.18) over $(0,t)$, we obtain

[TABLE]

Now we multiply (4.19) by $2^{(5/2)j}$ and afterwards take the $l^{1}(\mathbb{Z})$-norm to deduce

[TABLE]

By Lemma 2.4 (i), there exists $C>0$ such that

[TABLE]

Putting together (4.17) and (4.20), we have that

[TABLE]

where $C_{3}$ and $C_{4}$ are positive constants. By Gronwall inequality, we get

[TABLE]

Therefore, by standard arguments, if $\int_{0}^{T}\|\nabla u(t)\|_{L^{\infty}}\ dt<\infty$ then $u$ can be continued to $[0,T]$ and so to $[0,T^{\prime})$ for some $T^{\prime}>T$ (by item (i) of Theorem 1.1). $\diamond$

The contrapositive assertion of Proposition 4.2 gives the following remark.

Remark 4.3**.**

Let $u_{0}\in B_{2,1}^{5/2}(\mathbb{R}^{3})$ with $\nabla\cdot u_{0}=0$. Assume that $u$ is a solution for (1.1) in the class (4.16). If $T=T^{\ast}<\infty$ is the maximal existence-time, then

[TABLE]

4.3 Proof of item $(ii)$

Let $u_{0}\in B_{2,1}^{7/2}(\mathbb{R}^{3})$ with $\nabla\cdot u_{0}=0$ and let $u\in C([0,T_{\ast});B_{2,1}^{7/2}(\mathbb{R}^{3}))\cap C^{1}([0,T_{\ast});B_{2,1}^{5/2}(\mathbb{R}^{3}))$ be the solution of (1.1) with maximal existence-time $T_{\ast}>0$. Applying the projection operators $P_{\pm}$ in (1.1), we get

[TABLE]

Denoting $A_{\pm}:=\pm i\Omega\frac{D_{3}}{|D|}$ and using Duhamel principle, we have that

[TABLE]

Before proceeding, we recall the Strichartz estimates of [23] which states that if $2\leq r,\theta\leq\infty$ with $(r,\theta)\neq(2,\infty)$ and $\frac{1}{r}+\frac{1}{\theta}\leq\frac{1}{2}$ then

[TABLE]

Let $2<r<\infty.$ A scaling argument in (4.23) leads us to

[TABLE]

for all $j\in\mathbb{Z}$ and $\Omega\in\mathbb{R}\setminus\{0\}$, where $C=C(r)$ is a constant.

In what follows, we derive an estimate in $B_{\infty,1}^{1}$ for the solution $u$. Using $u=P_{+}u+P_{-}u$ (see Lemma 2.6), we only need to show the estimate for $P_{+}u$ and $P_{-}u$. First notice that

[TABLE]

Moreover, by (4.23), we have that

[TABLE]

For $2<r<\infty$ and $\Omega\in\mathbb{R}\setminus\{0\}$, the last two inequalities yield

[TABLE]

For the nonlinear term, using similar arguments we obtain

[TABLE]

Therefore

[TABLE]

Estimates (4.25) and (4.26) imply that

[TABLE]

for all $0<T<T_{\ast}$. Next, we define

[TABLE]

Using the embedding $B_{\infty,1}^{1}(\mathbb{R}^{3})\hookrightarrow W^{1,\infty}(\mathbb{R}^{3})$, (4.21) and estimate (4.27), we obtain

[TABLE]

Then, there exist positive constants $C_{5}$ and $C_{6}$ (independent of $\Omega$) such that

[TABLE]

For $0<T<\infty$, we consider

[TABLE]

We will show that $\widehat{T}_{\ast}=\min\{T,T_{\ast}\}$. For that, suppose that $\widehat{T}_{\ast}<\min\{T,T_{\ast}\}$ by contradiction. Then there exists $\widehat{T}$ such that $\widehat{T}_{\ast}<\widehat{T}<\min\{T,T_{\ast}\}$. In view of $u\in C([0,\widehat{T}];B_{2,1}^{7/2}(\mathbb{R}^{3}))$, we have that $U(t)$ is uniformly continuous on $[0,\widehat{T}]$ and

[TABLE]

Taking a sufficiently large $\Omega\in\mathbb{R}\setminus\{0\}$ in such a way that

[TABLE]

and using (4.28), (4.29) and (4.30), it follows that

[TABLE]

Thus, there exists $L$ such that $\widehat{T}_{\ast}<L<\widehat{T}$ and $U(L)\leq C_{5}T^{1-\frac{1}{r}}\|u_{0}\|_{B_{2,1}^{7/2}}$, contradicting the definition of $\widehat{T}_{\ast}$. Therefore, if (4.30) holds true we have that $\widehat{T}_{\ast}=\min\{T,T_{\ast}\}.$ If $T_{\ast}<T$, it follows that $T_{\ast}=\widehat{T}_{\ast}=\sup H_{T}$ and then

[TABLE]

for all $0\leq t<T_{\ast}$, and so $U(T_{\ast})<\infty$. In view of the blow-up criterion (see Remark 4.3), we are done. $\diamond$