報 告 人：黎野平 教授 

報告題目：Asymptotic behavior of the solutions for the 1D compressible NSK equations in the half line    

報告時間：2024年06月05日（周三）下午16：00-17：00

報告地點：靜遠樓1508學術報告廳

主辦單位：數學與統計學院、數學研究院、科學技術研究院 

報告人簡介：

       黎野平，南通大學數學與統計學院教授、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。

報告摘要：

       In this talk, I am going to present the time-asymptotic behavior of strong solutions to the initial-boundary value problem of the 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.