Global Strong Solutions To The 3D Inhomogeneous Incompressible Navier-Stokes Equations

In this article,we study the global regularity to the strong solutions of the 3D inhomogeneous incompressible Navier-Stokes equations?N-S equations?with density-temperature-dependent viscosity.This article is mainly composed by three parts:In the first chapter,we make a brief introduction to the initial boundary value problem.First of all,we propose the problem to be studied and list the ways to overcome it;secondly,we introduce the existing results to the N-S equations;finally,we give some basic definitions,common inequalities and formulas.In the second chapter,we study the existence of global strong solutions for the N-S equations with density-temperature-dependent viscosity.The first conclusion is obtained when the initial density is away from vacuum.By using some time-weighted a priori estimates,we establish the global existence of strong solutions to the initial boundary value problems provided the initial energy is small.As for the initial density contains vacuum,we show that strong solutions globally exist under the assumption that||?u₀||_L2is small.It is worth mentioning that we did not make any request for the initial temperature,which undoubtedly made the problem more complicated and interesting.In the third chapter,we concern the regularity of the global strong solutions to the incompressible N-S equations without heat conduction.According to the time-weighted a priori estimates,we prove that there exists a global strong solution to the N-S equations provided initial energy is small.We also obtain the decay estimates for the temperature.