主講人：黎野平 南通大學教授
時間：2024年5月20日8：30
地點：三號樓332室
舉辦單位：數(shù)理學院
主講人介紹：黎野平，南通大學數(shù)學與統(tǒng)計學院教授、博士研究生導師、湖北“楚天學者”特聘教授。先后在湖北大學、武漢大學和香港中文大學獲教育學學士學位、理學碩士學位和博士學位。主要致力于非線性偏微分方程的研究，尤其是來自物理、材料、生物和醫(yī)學等自然科學中的各類非線性偏微分方程和非線性耦合方程組。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》等國際、國內的重要學術期刊雜志上發(fā)表論文100余篇，其中SCI90余篇。同時，主持完成國家自然科學基金3項和教育部博士點博導專項、上海市教委創(chuàng)新項目以及江蘇省自然科學基金等省部級科研項目10余項；現(xiàn)在正主持國家自然科學基金面上項目1項和參加國家自然科學基金重點項目1項。
內容介紹：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.