Special Session 25: Recent Progress on Mathematical Analysis of PDEs Arising in Fluid Dynamics

The global existence and low Mach number limit for full Navier-Stokes equations around the Couette flow in channels

Tuowei Chen School of Mathematics, South China University of Technology Peoples Rep of China Co-Author(s):

Abstract: This talk is concerned with the two-dimensional full compressible Navier-Stokes equations between two finite or infinite parallel isothermal walls, where the upper wall is moving with a horizontal velocity, while the lower wall is stationary, and there allows a temperature difference between the two walls. It is shown that if the initial state is close to the Couette flow with a temperature gradient, then the global strong solutions exist, provided that the Reynolds and Mach numbers are low and the temperature difference between the two walls is small. The low Mach number limit of the global strong solutions is also shown for the case that both walls maintain the same temperature.

Qualitative and quantitative homogenization of some non-Newtonian flows in perforated domains

Yong Lyu Nanjing University Peoples Rep of China Co-Author(s):    Richard M. Hofer, Florian Oschmann; Zhengmao Qian, Chenchen Zhang

Abstract: We consider the homogenization of some viscous incompressible non-Newtonian flows in perforated domains. With certain general assumptions on the nonlinear stress tensor, we show the limit system is the Darcy's law for the case of `small holes', and system remain unchanged for the case of `large holes'. Quantitative convergence rates are also given for both the velocity field and the pressure.

Convergence of a finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system

Bangwei She Capital Normal University Peoples Rep of China Co-Author(s):    Eduard Feireisl, Maria Luk\`a\v{c}ov\`a-Medvid`ov\`a, Yuhuan Yuan

Abstract: We prove strong convergence of an upwind-type finite volume method to a weak solution of the Navier-Stokes-Fourier system with the Dirichlet boundary conditions. The limit solution satisfies a weak form of the mass and momentum equations, together with a weak form of the entropy and ballistic energy inequalities, and complies with the weak-strong uniqueness principle. The finite volume method uses piecewise-constant spatial approximations. The convergence proof is based on a combination of delicate consistency estimates with a careful analysis of the oscillations of numerical densities via renormalisation of the continuity equation.

Global well-posedness and decay rates of solutions to a P1-approximation model arising from radiation hydrodynamics

Wenjun Wang University of Shanghai for Science and Technology Peoples Rep of China Co-Author(s):

Abstract: This talk is concerned with the global existence and large-time behavior of solutions to the Cauchy problem for a P1-approximation radiation hydrodynamics model. The global existence result is established for small perturbations in the Sobolev space around a stable radiative equilibrium state. Moreover, the decay rates of the solution and its derivatives are obtained accordingly.

Global existence and uniqueness of strong solution to compressible Navier-Stokes equations with vacuum

Huanyao Wen South China University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will introduce our recent progress on global wellposedness of compressible Navier-Stokes equations with vacuum and smallness of initial data satisfying scaling invariance properties in three dimensions.

Hydrodynamic limit for compressible Navier-Stokes-Vlasov equations

Lei Yao Northwestern Polytechnical University Peoples Rep of China Co-Author(s):    Yunfei Su; Lei Yao

Abstract: We study the hydrodynamic limit of weak solutions to compressible Navier-Stokes-Vlasov equations in one dimension bounded domain. Due to the absence of dissipation terms in particle equation, it is difficult to study this problem. We take advantage of compactness result in one space dimension to obtain the convergence of macroscopic density of the particles in $C([0,T];H^{-1})$. The proof relies on relative entropy method to obtain the corresponding strong convergence of fluid density. At last, we give a recent result about hydrodynamic limit for multidimensional compressible Navier-Stokes-Vlasov-Poisson equations with local alignment force.

Universality in the Low Mach Number Limit via a Convex Integration Framework

Cheng Yu University of Florida USA Co-Author(s):    Ming Chen, Alexis Vasseur, Dehua Wang

Abstract: In this talk, I will discuss the low Mach number limit of the compressible Euler equations from the perspective of convex integration. Given any prescribed $L^2$ weak solution of the incompressible Euler equations, we construct a corresponding family of weak solutions to the compressible system via a refined convex integration scheme. We then show that, as the Mach number tends to zero, this family converges strongly to the prescribed incompressible solution. This is joint work with Ming Chen, Alexis Vasseur, and Dehua Wang.