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| 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 |
| (Dept. of Math., Lehigh University, 14 East Packer Avenue,Bethlehem, Pennsylvania 18015 USA) |
| DOI： |
| 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. |
| Key words: 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 |