Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations of Korteweg Type

Qi Zhang

doi:10.4236/apm.2021.114022

Abstract

In this paper, we study the Cauchy problem of the density-dependent Boussinesq equations of Korteweg type on the whole space with a vacuum. It is proved that there exists a unique strong solution for the two-dimensional Cauchy problem established that the initial density and the initial temperature decay not extremely slow. Particularly, it is allowed to be arbitrarily large for the initial data and vacuum states for the initial density, even including the compact support. Moreover, when the density depends on the Korteweg term with the viscosity coefficient and capillary coefficient, we obtain a consistent priority estimate by the energy method, and extend the local strong solutions to the global strong solutions. Finally, when the pressure and external force are not affected, we deform the fluid models of Korteweg type, we can obtain the large time decay rates of the gradients of velocity, temperature and pressure.

Highlights

IntroductionFor small initial data, [8] [9] provided the existence problems of the global strong solutions for Korteweg system in Besov space
We study the Cauchy problem of the density-dependent Boussinesq equations of Korteweg type on the whole space with a vacuum
For Equation (1.1) we propose the relationship between velocity field, fluid temperature and pressure so as to solve the difficulties caused by vacuum

Summary

Introduction

For small initial data, [8] [9] provided the existence problems of the global strong solutions for Korteweg system in Besov space. Germain-LeFloch [13] studied the existence, convergence and compactness of the compressible Navier-Stokes-Korteweg model Both the vacuum and nonvacuum weak solutions were obtained. Assuming that the influence of temperature is not considered, the Equation (1.1) can be simplified into a general incompressible Korteweg model, liu-wang-zheng [16] studied the strong solution of Cauchy problem in this model. If there is no influence of fluid temperature, i.e., θ = 0 , (1.1) reduces to the fluid of Korteweg type, Theorem 1.2 extends the results of Liu and Wang [27] to the Cauchy problem of global solutions in two-dimensional space.

Preliminaries

A Priori Estimates of the Solution Lower Order Estimates

L2 ρ u

L2 Lq

H1 xaρ

Conclusions