Volume 32, Issue 4
The Improved Fourier Splitting Method and Decay Estimates of the Global Solutions of the Cauchy Problems for Nonlinear Systems of Fluid Dynamics Equations

Linghai Zhang

Ann. Appl. Math., 32 (2016), pp. 396-417.

Published online: 2022-06

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  • 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.

  • AMS Subject Headings

35Q20

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@Article{AAM-32-396, author = {Zhang , Linghai}, title = {The Improved Fourier Splitting Method and Decay Estimates of the Global Solutions of the Cauchy Problems for Nonlinear Systems of Fluid Dynamics Equations}, journal = {Annals of Applied Mathematics}, year = {2022}, volume = {32}, number = {4}, pages = {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.

}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aam/20651.html} }
TY - JOUR T1 - The Improved Fourier Splitting Method and Decay Estimates of the Global Solutions of the Cauchy Problems for Nonlinear Systems of Fluid Dynamics Equations AU - Zhang , Linghai JO - Annals of Applied Mathematics VL - 4 SP - 396 EP - 417 PY - 2022 DA - 2022/06 SN - 32 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/aam/20651.html KW - 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. AB -

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.

Linghai Zhang. (2022). The Improved Fourier Splitting Method and Decay Estimates of the Global Solutions of the Cauchy Problems for Nonlinear Systems of Fluid Dynamics Equations. Annals of Applied Mathematics. 32 (4). 396-417. doi:
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