Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data

Keyan Wang; Yao Xiao

doi:10.3934/dcds.2020158

Article Contents

2020, Volume 40, Issue 5: 2987-3011. Doi: 10.3934/dcds.2020158

This issue Previous Article General initial data for a class of parabolic equations including the curve shortening problem Next Article Dynamical systems with a prescribed globally bp-attracting set and applications to conservative dynamics

Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data

Keyan Wang^1, , and
Yao Xiao^2,

1.
School of Mathematics,, Shanghai University of Finance and Economics, Shanghai 200433, China
2.
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

^* Corresponding author: Keyan Wang
^* Corresponding author: Keyan Wang

Published: March 2020

The first author is supported by NSFC grant No. 11971290.

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Abstract

In this paper, we prove that for the ill-prepared initial data of the form

$ \begin{equation} \nonumber u_0^\epsilon(x) = (v_0^h(x_\epsilon), \epsilon^{-1}v_0^3(x_\epsilon))^T,\quad x_\epsilon = (x_h, \epsilon x_3)^T, \end{equation} $

the Cauchy problem of the incompressible Navier-Stokes equations on $ \mathbb{R}^3 $ is locally well-posed for all $ \epsilon > 0 $, provided that the initial velocity profile $ v_0 $ is analytic in $ x_3 $ but independent of $ \epsilon $.

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Mathematics Subject Classification: Primary: 76D03, 76D05; Secondary: 35Q30.

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