Analyticity for the (generalized) Navier-Stokes equations with rough initial data

Abstract

We study the Cauchy problem for the (generalized) incompressible Navier-Stokes equations \begin{align} u_t+(-Δ)^αu+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0. \nonumber \end{align} We show the analyticity of the local solutions of the Navier-Stokes equation ($\alpha =1$) with any initial data in critical Besov spaces ${\dot{B}}_{p,q}^{n/p-1}({\mathbb{R}}^n)$ with $1<p<\infty ,1\le q\le \infty$ and the solution is global if $u_0$ is sufficiently small in ${\dot{B}}_{p,q}^{n/p-1}({\mathbb{R}}^n)$. In the case $p=\infty$, the analyticity for the local solutions of the Navier-Stokes equation ($\alpha =1$) with any initial data in modulation space $M_{\infty ,1}^{-1}({\mathbb{R}}^n)$ is obtained. We prove the global well-posedness for a fractional Navier-stokes equation ($\alpha =1/2$) with small data in critical Besov spaces ${\dot{B}}_{p,1}^{n/p}({\mathbb{R}}^n)(1\le p\le \infty )$ and show the analyticity of solutions with small initial data either in ${\dot{B}}_{p,1}^{n/p}({\mathbb{R}}^n)(1\le p<\infty )$ or in ${\dot{B}}_{\infty ,1}^0({\mathbb{R}}^n)\cap M_{\infty ,1}^0({\mathbb{R}}^n)$. Similar results also hold for all $\alpha \in (1/2,1)$.

31 pages

Cited by in corpus (2)

• Ill-posedness for the Navier-Stokes equations in critical Besov spaces ${\dot{B}}_{\infty ,q}^{-1}$
• Dynamical Behavior for the Solutions of the Navier-Stokes Equation