2019年跨音速流与混合型偏微分方程高级研讨班

日程安排

 报告摘要

(1)Speaker: Prof. Xianpeng Hu

Department of Mathematics, City University of Hong Kong,

Title: Concentration phenomenon for compressible Naiver-Stokes equations

Time:NOV.18,16:30-18:00

Location: 661

Abstract: We will discuss the concentration phenomenon of the kinetic energy ρ|u|2, associated to isentropic compressible Navier-Stokes equations, in Rn with n = 2,3 and the adiabatic constant γ ∈ [1, n/2]. Except a space-time set with Hausdorff dimension less than or equal to Γ(n)+1, no concentration phenomenon occurs. Some recent development will be also discussed.

Speaker Introduction:  胡先鹏教授已经在可压的 Navier-Stokes 方程、粘弹性流体力学方程和 MHD 方程的数学理论上取得了相当不错的结果和进展，已在Communications on Pure and Applied Mathematics, SIAM Journal on Mathematical Analysis, Archive for Rational Mechanics and Analysis, Discrete and Continuous Dynamical Systems - Series A, Communications on Mathematical Physics 等国际知名数学杂志上发表多篇论文。

联系人：谭忠教授

(2)Speaker: Prof. Xianpeng Hu

Department of Mathematics, City University of Hong Kong,

Title: On compressible viscoelasticity with zero shear viscosity.

Time: NOV.19,09:30-11:30

Location: 661

Abstract: In this talk, the multi-dimensional compressible viscoelastic system with zero shear viscosity is discussed. With the help of potentials of solutions and the Green's function, we construct the global existence of solutions for small perturbations around the equilibrium by combining the energy and decay estimates based on the vector fields.

Speaker Introduction:  胡先鹏教授已经在可压的 Navier-Stokes 方程、粘弹性流体力学方程和 MHD 方程的数学理论上取得了相当不错的结果和进展，已在Communications on Pure and Applied Mathematics, SIAM Journal on Mathematical Analysis, Archive for Rational Mechanics and Analysis, Discrete and Continuous Dynamical Systems - Series A, Communications on Mathematical Physics 等国际知名数学杂志上发表多篇论文。

联系人：谭忠教授

以下为博士生汇报：

Time: NOV.19,15:30-18:00

Location: 661

报告人：戴祎琛

题目：A class of global large solutions to the magnetohydrodynamic equations with fractional dissipation 

摘要：The global existence and regularity problem on the magnetohydrodynamic (MHD) equations with fractional dissipation is not well understood for many ranges of fractional powers. We examines this open problem from a different perspective. We construct a class of large solutions to the d-dimensional (d=2,3) MHD equations with any fractional power. The process presented here actually assesses that an initial data near any function whose Fourier transform lives in a compact set away from the origin always leads to a unique and global solution.

报告人：贾翠满

题目：Well-posedness of compressible magneto-micropolar fluid equations 

摘要：We are  concerned with compressible magneto-micropolar fluid equations (\ref{1.1})-(\ref{1.2}). The global existence and large time behaviour of

solutions near a constant state to the magneto-micropolar-Navier-Stokes-Poisson (MMNSP) system is investigated in $\mathbb{R}^3$. By a refined energy

method, the global existence is established under the assumption that the $H^3$ norm of the initial data is small, but the higher order derivatives can

be large. If the initial data belongs to homogeneous Sobolev spaces or homogeneous Besov spaces, we prove the optimal time decay rates of the solution

and its higher order spatial derivatives. Meanwhile, we also obtain the usual $L^p-L^2$ $(1\leq p\leq2)$ type of the decay rates without requiring that

the $L^p$ norm of initial data is small.

报告人：吴文佩

题目：Decay of the viscoelastic Navier-Stokes-Poisson

摘要：We use an energetic variational approach to model the transport of compressible viscoelastic conductive fluids. Such a model can be called the three-dimensional compressible viscoelastic Navier-Stokes-Poisson equations. The global unique smooth solution to the Cauchy problem is obtained. In particular, we obtain the optimal time-decay rates of the solution and its higher-order spatial derivatives by using a pure energy method.

报告人：吴忠二

题目：Regularity and Local Energy Equation of Euler-korteweg Equation

摘要：In this paper, we mainly study the local energy equation of the weak solutions of the compressible isentropic and nonhomogeneous incompressible Euler-korteweg equation defined on $\mathbb{T}^{3}$. We prove that the regularity of the solution is sufficient to guarantee the balance of the total energy in the $B^{\alpha,\infty}_{3}((0,T)\times \mathbb{T}^d)$ space. We adopt a variant of Feireisl's method in \cite{Fei}.

报告人：谢明洪

题目：具有临界Sobolev迹嵌入指数的非线性分数阶反应扩散方程解的性质

摘要:本文考虑的是带临界指数的非线性分数阶反应扩散方程

我们运用Caffarelli-Silvestre延拓把非局部问题转化为可变分的局部问题，首先研究了低能量初值下方程的有限时间爆破行为以及全局解的衰减估计,其次运用Moser迭代得到了全局解的L^q(1≤q< ∞)估计,最后借助集中紧定理得到了在某时间序列t_n趋于无穷时解的渐近行为.