勷勤数学•专家报告

题      目：Hilbert Expansion for the Boltzmann equation with frictional force

报  告  人： 姜在红 教授  (邀请人：喻洪俊)

                                               浙江师范大学

时      间： 9月12日  16：00-17:00

地     点：数科院东楼401

报告人简介:

          姜在红，浙江师范大学数学科学学院教授，博士生导师。2010 年博士毕业于中国科学技术大学与香港城市大学（联合培养），主要从事流体力学及动理学等偏微分方程相关理论的学习和研究。在Archive for Rational Mechanics and Analysis、Nonlinearity、Journal of Differential Equations、Journal of Nonlinear Science等国际期刊上发表论文 50 余篇。

摘      要：

           Although the weak diffusive limit from the Boltzmann equation to the incompressible Navier-Stokes-Fourier system was previously established for the Maxwell boundary condition within the framework of renormalized solutions [SaintRaymond2009] [Jiang-Masmoudi2017], the corresponding strong diffusive limit has remained open except for the special scaling $\alpha \sim \epsilon^{1/2}$ [Jiang-Masmoudi2017]. In this work, we resolve this problem by proving the strong diffusive limit for the full range of accommodation coefficients $\alpha \in [0,1] $ within the framework of strong solutions. The principal innovations of our proof include: (1) an $\epsilon$-stretching method for reduction to a single-bounce $L^\infty$ estimate; and (2) a dissipative decomposition based on a carefully constructed rotating Maxwellian to handle the challenging near-specular regime $\alpha \ll \epsilon$.In this talk, we consider the Hilbert expansion for the Boltzmann equation with frictional force. We introduce a modified micromacro decomposition to resolve the nonlinear coupling between the macroscopic velocity and the distribution density, and establish a $L^\infty-L^2-W^{1,\infty}-\dot{H}^ {1}$ bootstrap argument to close the energy estimates. We prove that, near a local Maxwellian, any solution of the Boltzmann equation with a frictional force converges globally to the corresponding solution of the Euler system with a damping term.

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