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

主講人：黎野平 南通大學(xué)教授

時(shí)間：2024年5月20日8：30

地點(diǎn)：三號(hào)樓332室

舉辦單位：數(shù)理學(xué)院

主講人介紹：黎野平，南通大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院教授、博士研究生導(dǎo)師、湖北“楚天學(xué)者”特聘教授。先后在湖北大學(xué)、武漢大學(xué)和香港中文大學(xué)獲教育學(xué)學(xué)士學(xué)位、理學(xué)碩士學(xué)位和博士學(xué)位。主要致力于非線性偏微分方程的研究，尤其是來(lái)自物理、材料、生物和醫(yī)學(xué)等自然科學(xué)中的各類(lèi)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(guó)際、國(guó)內(nèi)的重要學(xué)術(shù)期刊雜志上發(fā)表論文100余篇，其中SCI90余篇。同時(shí)，主持完成國(guó)家自然科學(xué)基金3項(xiàng)和教育部博士點(diǎn)博導(dǎo)專(zhuān)項(xiàng)、上海市教委創(chuàng)新項(xiàng)目以及江蘇省自然科學(xué)基金等省部級(jí)科研項(xiàng)目10余項(xiàng)；現(xiàn)在正主持國(guó)家自然科學(xué)基金面上項(xiàng)目1項(xiàng)和參加國(guó)家自然科學(xué)基金重點(diǎn)項(xiàng)目1項(xiàng)。

內(nèi)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.