Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations

Linghai Zhang

doi:10.3934/dcdss.2016091

Article Contents

2016, Volume 9, Issue 6: 2181-2200. Doi: 10.3934/dcdss.2016091

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Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations

Linghai Zhang^1,

1.
Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, Pennsylvania 18015

Received: June 2015

Revised: September 2016

Published: December 2016

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Abstract

Consider the Cauchy problems for the $n$-dimensional incompressible Navier-Stokes equations \begin{eqnarray*} \frac{\partial{\bf u}}{\partial t}-\alpha\triangle{\bf u}+({\bf u}\cdot\nabla){\bf u}+\nabla p={\bf f}({\bf x},t),\qquad {\bf u}({\bf x},0)={\bf u}_0({\bf x}). \end{eqnarray*} In this system, the dimension $n\geq 3$, ${\bf u}({\bf x},t)=(u_1({\bf x},t),u_2({\bf x},t),\cdots,u_n({\bf x},t))$ and ${\bf f}({\bf x},t)=(f_1({\bf x},t),f_2({\bf x},t),\cdots,f_n({\bf x},t))$ are real vector valued functions of ${\bf x}=(x_1,x_2,\cdots,x_n)$ and $t$. Additionally, $\alpha>0$ is a positive constant. Suppose that the initial function and the external force satisfy appropriate conditions.
The main purpose of this paper is to make complete use of the uniform energy estimates of the global smooth solutions and couple together a well known Gronwall's inequality to improve the Fourier splitting method to accomplish the decay estimates with sharp rates. The decay estimates with sharp rates of the global smooth solutions of the Cauchy problems for the $n$-dimensional magnetohydrodynamics equations may be established very similarly.

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Mathematics Subject Classification: Primary: 35Q20; Secondary: 35Q10.

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