学术报告

报告题目:Asymptotic Behavior of Solutions to the IBVP of the Compressible Navier-Stokes-Korteweg Equations

报告时间：2021年10月19日16:00-17:00

报告地点：25教学楼14楼学术报告厅

报告摘要:In this talk, I am going to present the time-asymptotic behavior of strong solutions to the initial-boundary value problem of the isothermal compressible fluid models of Korteweg type with density-dependent viscosity and capillarity on the half-line $\mathbb{R}^+$. 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.

报告人简介： 黎野平，南通大学理学院教授、博士研究生导师、湖北“楚天学者”特聘教授。先后在武汉大学和香港中文大学获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》、《Journal of Differential Equations》等数学主流学术期刊上发表论文90余篇，其中SCI收录70余篇。主持完成国家自然科学基金项目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科研项目多项；现主持国家自然科学基金面上项目1项并参加国家自然科学基金面上项目2项。