Asymptotic behavior of the solutions for the 1D compressible NSK equations in the half line


主讲人:黎野平 南通大学教授




主讲人介绍:黎野平,南通大学数学与统计学院教授、博士研究生导师、湖北“楚天学者”特聘教授。先后在湖北大学、武汉大学和香港中文大学获教育学学士学位、理学硕士学位和博士学位。主要致力于非线性偏微分方程的研究,尤其是来自物理、材料、生物和医学等自然科学中的各类非线性偏微分方程和非线性耦合方程组。在《Mathematical Models and Methods in Applied Sciences》,《SIAM Journal of Mathematical Analysis》,《Calculus of Variations and Partial Differential Equations》,《Journal of Differential Equations》和《Communications in Mathematical Sciences》等国际、国内的重要学术期刊杂志上发表论文100余篇,其中SCI90余篇。同时,主持完成国家自然科学基金3项和教育部博士点博导专项、上海市教委创新项目以及江苏省自然科学基金等省部级科研项目10余项;现在正主持国家自然科学基金面上项目1项和参加国家自然科学基金重点项目1项。

内容介绍:In this talk, I am going to present the time-asymptotic behavior of strong solutions to the initial-boundary value problem of the compressible fluid models of Korteweg type with density-dependent viscosity and capillarity on the half-line $R^+$. The case when the pressure $p(v)=v^{-\gamma}$, the viscosity $\mu(v)=\tilde{\mu} v^{-\alpha}$ and the capillarity $\kappa(v)=\tilde{\kappa} v^{-\beta}$ for the specific volume $v(t,x)>0$ is considered, where $\alpha,\beta, \gamma\in\mathbb{R}$ are parameters, and $\tilde{\mu},\tilde{\kappa}$ are given positive constants. I focus on the impermeable wall problem where the velocity $u(t,x)$ on the boundary $x=0$ is zero. If $\alpha,\beta$ and $\gamma$ satisfy some conditions and the initial data have the constant states $(v_+, u_+)$ at infinity with $v_+, u_+>0$, and have no vacuum and mass concentrations, we prove that the one-dimensional compressible Navier-Stokes-Korteweg system admits a unique global strong solution without vacuum, which tends to the 2-rarefction wave as time goes to infinity. Here both the initial perturbation and the strength of the rarefaction wave can be arbitrarily large. As a special case of the parameters $\alpha,\beta$ and the constants $\tilde{\mu},\tilde{\kappa}$, the large-time behavior of large solutions to the compressible quantum Navier-Stokes system is also obtained for the first time. Our analysis is based on a new approach to deduce the uniform-in-time positive lower and upper bounds on the specific volume and a subtle large-time stability analysis.This is a joint work with Prof. Chen Zhengzheng.