报告题目：Asymptotic behavior of the solutions for the 1D compressible NSK equations in the half line

时间：2025年6月24日（周二）

地点：数学与统计学学院204

报告人：黎野平教授

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.

报告人简介：黎野平，南通大学数学与统计学院二级教授、博士研究生导师。先后在湖北大学、武汉大学和香港中文大学获学士学位、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，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。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。