The Improved Fourier Splitting Method and Decay Estimates of the Global Solutions of the Cauchy Problems for Nonlinear Systems of Fluid Dynamics Equations
Year: 2016
Author: Linghai Zhang
Annals of Applied Mathematics, Vol. 32 (2016), Iss. 4 : pp. 396–417
Abstract
Consider the Cauchy problems for an $n$-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall’s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980’s to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for $n$-dimensional incompressible Navier-Stokes equations, for the $n$-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2016-AAM-20651
Annals of Applied Mathematics, Vol. 32 (2016), Iss. 4 : pp. 396–417
Published online: 2016-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: nonlinear systems of fluid dynamics equations global weak solutions decay estimates uniform energy estimates Fourier transformation Plancherel’s identity Gronwall’s inequality improved Fourier splitting method.