Navier–Stokes equations: local existence, uniqueness and blow-up of solutions in Sobolev–Gevrey spaces

ABSTRACT

This work establishes local existence and uniqueness as well as blow-up criteria for solutions $u(x,t)$ of the Navier–Stokes equations in Sobolev–Gevrey spaces $H_{a,\sigma }^s\left({\mathbb{R}}^3\right)$. More precisely, if it is assumed that the initial data $u_0$ belongs to $H_{a,\sigma }^{s_0}\left({\mathbb{R}}^3\right)$, with $s_0\in (\frac{1}{2},\frac{3}{2})$, we prove that there is a time T>0 such that $u\in C\left(\right[0,T];H_{a,\sigma }^s({\mathbb{R}}^3\left)\right)$ for $a>0,\sigma \ge 1$ and $s\le s_0$. If the maximal time interval of existence of solutions is finite, $0\le t<T^{\ast }$, then, we prove, for example, that the blow-up inequality $\begin{array}{rl}{C}_1\exp \left\{C_2\right(T^{\ast }-t{)}^p\left\}\right(T^{\ast }-t{)}^{-q}& \le \parallel u\left(t\right){\parallel }_{H_{a,\sigma }^s\left({\mathbb{R}}^3\right)},\\ q=\frac{2(s\sigma +{\sigma }_0)+1}{6\sigma },p& =-\frac{1}{3\sigma },\end{array}$ holds for $0\le t<T^{\ast },s\in (\frac{1}{2},s_0]$, a>0, $\sigma >1$ ($2{\sigma }_0$ is the integer part of $2\sigma$).

Keywords:

• Navier–Stokes equations
• local existence and uniqueness of solution
• blow-up criteria
• Sobolev–Gevrey spaces

AMS Mathematics Subject Classifications:

• 35B44
• 35Q30
• 76D03
• 76D05
• 76W05

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Wilberclay G. Melo  http://orcid.org/0000-0003-4388-3491

Notes

1. $H_{a,\sigma }^s\left({\mathbb{R}}^3\right):=\{f\in S^{\prime }({\mathbb{R}}^3):\parallel f{\parallel }_{H_{a,\sigma }^s\left({\mathbb{R}}^3\right)}^2:={\int }_{{\mathbb{R}}^3}(1+|\xi {|}^2{)}^s{\mathrm{e}}^{2a|\xi {|}^{\frac{1}{\sigma }}}|\hat{f}\left(\xi \right){|}^2\mathrm{d}\xi <\mathrm{\infty }\}$ is the Sobolev–Gevrey space endowed with the inner product $⟨f,g{⟩}_{H_{a,\sigma }^s\left({\mathbb{R}}^3\right)}:={\int }_{{\mathbb{R}}^3}(1+|\xi {|}^2{)}^s{\mathrm{e}}^{2a|\xi {|}^{\frac{1}{\sigma }}}\hat{f}(\xi )\cdot \hat{g}(\xi )\mathrm{d}\xi$ ($S^{\prime }\left({\mathbb{R}}^3\right)$ is the set of tempered distributions). Here $\mathcal{F}\left(f\right)\left(\xi \right)=\hat{f}\left(\xi \right):={\int }_{{\mathbb{R}}^3}{\mathrm{e}}^{-i\xi \cdot x}f\left(x\right)\mathrm{d}x$ and ${\mathcal{F}}^{-1}\left(f\right)\left(\xi \right):=\left(2\pi {)}^{-3}{\int }_{{\mathbb{R}}^3}{\mathrm{e}}^{i\xi \cdot x}f\right(x)\mathrm{d}x,\mathrm{\forall }\xi \in {\mathbb{R}}^3.$

2. ${\dot{H}}^s\left({\mathbb{R}}^3\right)=\{f\in S^{\prime }({\mathbb{R}}^3):\parallel f{\parallel }_{{\dot{H}}^s\left({\mathbb{R}}^3\right)}^2:={\int }_{{\mathbb{R}}^3}|\xi {|}^{2s}|\hat{f}(\xi ){|}^2\mathrm{d}\xi <\mathrm{\infty }\}$ is the homogenous Sobolev space.

3. The Sobolev–Gevrey space ${\dot{H}}_{a,\sigma }^s\left({\mathbb{R}}^3\right):=\{f\in S^{\prime }({\mathbb{R}}^3):\parallel f{\parallel }_{{\dot{H}}_{a,\sigma }^s\left({\mathbb{R}}^3\right)}^2:={\int }_{{\mathbb{R}}^3}|\xi {|}^{2s}{\mathrm{e}}^{2a|\xi {|}^{\frac{1}{\sigma }}}|\hat{f}(\xi ){|}^2\mathrm{d}\xi <\mathrm{\infty }\}$ is endowed with the inner product $⟨f,g{⟩}_{{\dot{H}}_{a,\sigma }^s\left({\mathbb{R}}^3\right)}:={\int }_{{\mathbb{R}}^3}|\xi {|}^{2s}{\mathrm{e}}^{2a|\xi {|}^{\frac{1}{\sigma }}}\hat{f}\left(\xi \right)\cdot \hat{g}\left(\xi \right)\mathrm{d}\xi$.

4. In the Navier–Stokes equations (1(1) $\begin{array}{rl}& u_t+u\cdot \mathrm{\nabla }u+\mathrm{\nabla }p=\mu \mathrm{\Delta }u,x\in {\mathbb{R}}^3,t\in [0,T^{\ast }),\\ & \text{div}u=0,x\in {\mathbb{R}}^3,t\in [0,T^{\ast }),\\ & u(x,0)=u_0\left(x\right),x\in {\mathbb{R}}^3,\end{array}$(1) ), we have that $f\cdot \mathrm{\nabla }g=\sum _{i=1}^3{f}_i{D}_{i}g$ where $f=(f_1,f_2,f_3)$ and $g=(g_1,g_2,g_3)$ and $D_i=\mathrm{\partial }/\mathrm{\partial }{x}_i$ (i = 1, 2, 3).

5. The tensor product is given by $f\otimes g:=(g_{1}f,g_{2}f,g_{3}f),$ where $f,g:{\mathbb{R}}^3\to {\mathbb{R}}^3$.

Additional information

Funding

Natã Firmino Rocha was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) grant 1579575, Ezequiel Barbosa was partially supported by CNPq.