Pointwise wave behavior of the Navier-Stokes equations in half space

Linglong Du; Haitao Wang

doi:10.3934/dcds.2018055

Article Contents

2018, Volume 38, Issue 3: 1349-1363. Doi: 10.3934/dcds.2018055

This issue Previous Article Long-time behaviour of a radially symmetric fluid-shell interaction system Next Article Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian

Pointwise wave behavior of the Navier-Stokes equations in half space

Linglong Du^1, and
Haitao Wang^2, ,

1.
Department of Applied Mathematics, Donghua University, Shanghai 201620, China
2.
Institute of Natural Sciences and School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Haitao Wang, haitaowang.math@gmail.com.
* Corresponding author: Haitao Wang, haitaowang.math@gmail.com.

Received: March 31, 2017

Revised: August 31, 2017

Published: June 2018

Du is supported by NSFC(Grant No. 11526049 and 11671075) and the Fundamental Research Funds for the Central Universities (No. 2232016D-22)

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Abstract

In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays with algebraic rate.

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Mathematics Subject Classification: 76N10, 35B40, 35A08.

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