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 - <p style="text-align: justify;">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.</p>