#### Short Programme on Free Boundary Problems

#### Short Programme on Free Boundary Problems

#### Content

Organizers: Gui-Qiang Chen, Feimin Huang, Dehua Wang, Yi Wang

Place: Room N202, South Building, AMSS

**July 16, 2017**

Chair: Feimin Huang

15:00-15:10 Opening Ceremony

Chair: Gui-Qiang Chen

15:10-16:00 Speaker: Charlie Elliott

16:10-16:30 Tea break

Chair: Mikhail Feldman

16:30-17:20 Speaker: Harald Garcke

17:30-18:20 Speaker: Shu Wang

18:30 Banquet

**July 17, 2017**

Chair: Charlie Elliott

9:00-9:50 Speaker: Michael Hinze

10:00-10:20 Tea break

Chair: Ming Mei

10:20-11:10 Speaker: Hyeong-Ohk Bae

11:20-12:10 Speaker: Beixiang Fang

12:20-14:00 Lunch

Chair: Dehua Wang

14:00-14:50 Speaker: Zhifei Zhang

15:00-15:50 Speaker: Vanessa Styles

16:00-16:20 Tea break

Chair: Zhifei Zhang

16:20-17:10 Speaker: Dehua Wang

17:20-18:10 Speaker: Mingjie Li

18:20 Dinner

**July 18, 2017**

Chair: Harald Garcke

9:30-10:20 Speaker: Mikhail Feldman

10:30-10:50 Tea break

Chair: Michael Hinze

10:50-11:40 Speaker: Hailiang Li

11:50-14:00 Lunch

Chair: Hailiang Li

14:00-14:50 Speaker: Ming Mei

15:00-15:50 Speaker: Myoungjean Bae

16:00-16:20 Tea Break

Chair: Feimin Huang

16:20-17:10 Speaker: Jose Francisco Rodrigues

17:20-18:10 Speaker: Gui-Qiang Chen

18:20 Dinner

**Titles and Abstracts**

Wake estimates of the incompressible Navier-Stokes equations in exterior domains

Hyeong-Ohk Bae

Ajou University, Korea

hyeongohk@hanmail.net

Abstract: In this paper, we intend to derive spatial-temporal decay estimates of solutions for the Navier-Stokes equations in the 3D exterior domains. We first linearize the Navier-Stokes system around the constant velocity v1 6= 0 at infinity to get the Oseen system which has a wake region (a paraboloid in the direction of v1). We then show how well solutions of the Navier-Stokes system approximate solutions of the system the steady Oseen system in the wake region ast → ∞. To this end, we study the asymptotic behavior of solutions to the system expressed as the difference between the solutions of the two systems by considering the weight of the form |x − tv1|.

Prandtl-Meyer reflection configurations, Transonic shocks, and Free boundary problems

Myoungjean Bae

POSTECH, Korea

mjbae@postech.ac.kr

Abstract: Prandtl (1936) first employed the shock polar analysis to show that, when a steady supersonic flow impinges onto a solid wedge whose angle is less than a critical angle (i.e., the detachment angle), there are two possible steady configurations: the steady weak shock solution and strong shock solution, and then conjectured that the steady weak shock solution is physically admissible since it is the one observed experimentally. The fundamental issue of whether one or both of the steady weak and strong shocks are physically admissible has been vigorously debated over several decades and has not yet been settled in a definite manner. In this talk, I will address this longstanding open issue and present the recent analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. This talk is based on a joint work with Gui-Qiang G. Chen and Mikhail Feldman.

Compressible Vortex Sheets and Free Boundary Problems

Gui-Qiang Chen

Oxford University, UK, and AMSS, CAS, China

chengq@maths.ox.ac.uk

Abstract: In this talk, we will discuss some recent developments in the analysis of nonlinear stability of vortex sheets, more generally characteristic discontinuities, and related free boundary problems arising in gas dynamics, relativistic fluid dynamics, and MHD.

PDES on evolving domains

Charlie Elliott

University of Warwick, UK

C.M.Elliott@warwick.ac.uk

Abstract: Many physical models give rise to the need to solve partial differential equations in time dependent regions. The complex morphology of biological membranes and cells coupled with biophysical mathematical models present significant computational challenges. In this talk we discuss the mathematical issues associated with the formulation of PDEs in time dependent domains in both flat and curved space. Here we are thinking of problems posed on time dependent d-dimensional hypersurfaces Gamma(t) in Rd+1. The surface Gamma(t) may be the boundary of the bounded open bulk region Omega(t). In this setting we may also view Omega(t) as (d+1)-dimensional sub-manifold in Rd+2. Using this observation we may develop a theory applicable to both surface and bulk equations. We will present an abstract framework for treating the theory of well- posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces using generalised Bochner spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hyper-surfaces. Our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data. Specifically, we study in turn a surface heat equation, an equation posed on a bulk domain, a novel coupled bulk-surface system and an equation with a dynamic boundary condition. We give some background to applications in cell biology. We describe how the theory may be used in the development and numerical analysis of evolving surface finite element spaces which unifies the discretisation methodology for evolving surface and bulk equations. We give some computational examples from cell biology involving the coupling of surface evolution to processes on the surface.

On Stability of Steady Transonic Shocks in Supersonic Flow

around a Wedge

Beixiang Fang

Shanghai Jiaotong University, China

bxfang@sjtu.edu.cn

Abstract: In this talk we are concerned with the stability of steady transonic shocks in supersonic flow around a wedge. When a uniform supersonic flow comes against a wedge with small vertex angle, a shock-front attached to the wedge appears. It has been indicated in the book Supersonic Flow and Shock Waves? by Courant and Friedrichs that there are two admissible shock solutions satisfying both Rankine-Hugoniot conditions and the entropy condition. The weaker one may be either supersonic or transonic, while the stronger one must be transonic. In this talk, we shall present stability results for 2-D and M-D flows. This work is based on the joint works with Prof. G.-Q. Chen and Prof. S.X. Chen.

Magnetic stabilization in the interface problems of ideal

incompressible MHD

Zhifei Zhang

Peking University, China

zfzhang@math.pku.edu.cn

Uniqueness for shock reflection problem

Mikhail Feldman

University of Wisconsin-Madison, USA

feldman@math.wisc.edu

Abstract: We discuss shock reflection problem for compressible gas dynamics, von Neumann conjectures on transition between regular and Mach reflections, and existence of regular reflection solutions for potential flow equation. Then we will talk about recent results on uniqueness of regular reflection solutions for potential flow equation in a natural class of self-similar solutions. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, and prove uniqueness by a version of method of continuity. A property of solutions important for the proof of uniqueness is convexity of the free boundary. This talk is based on joint work with G.-Q. Chen and W. Xiang.

Surfactants in two phase flow

Harald Garcke

University of Regensburgh, Germany

harald.garcke@mathematik.uni-regensburg.de

Abstract: The evolution of the free boundary in two phase flow can be significantly influenced by the presence of surfactants. A surfactant is a substance which lowers the surface tension when present on the free boundary. In the talk a sharp interface problem for two phase flow will be introduced which will allow for an energy estimate and this will make it possible to come up with a stable discretization using finite elements for the bulk and parametric finite elements for the interface. With the help of numerical simulations several phenomena caused by surfactants will be illustrated.

In a second part of the talk we introduce a phase field model approximating the free boundary problem. The new feature is that partial differential equations on a surface coupled to bulk partial differential equations are approximated in a phase field context such that an energy estimate is possible. This will allow it to show an existence result for the resulting highly nonlinear PDE system.

Optimal control of nonsmooth Cahn-Hilliard/Navier-Stokes systems

Michael Hinze

University of Hamburg, Germany

michael.hinze@uni-hamburg.de

Abstract: We consider the optimal control of a two-phase fluid governed by the thermodynamically consistent diffuse Cahn-Hilliard/Navier-Stokes interface model proposed in 2012 by Abels/Garcke/Grn. As key ingredient we present an energy stable simulation scheme proposed by the authors in 2016. It allows us to simulate two-phase fluids in an energy stable way and provides enough regularity to apply classic theory from optimal control. We prove existence of solutions to a semi-discrete in time optimal control problem, and present a convergence analysis for its finite element discretization. We illustrate the performance of our approach with some numerical examples. This also includes numerical simulations done with a posteriori error control based on the goal-oriented dualweighted residual method. (joint work with Harald Garcke, Uni Regensburg und Christian Kahle, TU Mnchen)

Behaviors of Navier-Stokes(Euler)-Fokker-Planck equations

Hailiang Li

Capital Normal University, China

hailiang li@cnu.edu.cn

Abstract: We consider the behaviors of global solutions to the initial value problems for the multi-dimensional compressible Navier-Stokes(Euler)-Fokker-Planck equations. It is shown that due the micro-macro coupling effects, the sound wave type propagation of this NSFP or EFP system for two-phase fluids is observed with the wave speed determined by the two-phase fluids. This phenomena can no be obsered for the pure Fokker-Planck equation.

Low Mach Number Limit of Multidimensional Steady Flows on the Air Foil Problem

Mingjie Li

Minzu University of China, China

lmjmath@163.com

Abstract: In this talk, the low Mach number limit of the steady irrotational Euler flows on the airfoil problem is considered. The limit is on the Holder space, which means the better uniform estimates. And the convergence rate is "2, which is higher than the Klainerman-Majda’s result in 1981, due to the irrotational property. This is a joint work with professors Tian-yi Wang and Wei Xiang.

Steady hydrodynamic model of semiconductors with sonic boundary

Ming Mei

McGill University and Champlain College, Canada

mei@math.mcgill.ca

Abstract: In this talk, we consider the well-posedness, ill-posedness and the regularity of stationary solutions to the hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary, and make a classification on these solutions. When the doping profile is subsonic, we prove that, the corresponding steady-state equations with sonic boundary possess a unique interior subsonic solution, and at least one interior supersonic solution; and if the relaxation time is large and the doping profile is a small perturbation of constant, then the equations admit infinitely many interior transonic shock solutions; while, if the relaxation time is small enough and the doping profile is a subsonic constant, then the equations admits infinitely many interior C1 smooth transonic solutions, and no transonic shock solution exists. When the doping profile is supersonic, we show that the system does not hold any subsonic solution; furthermore, the system doesn’t admit any supersonic solution or any transonic solution if such a supersonic doping profile is small enough or the relaxation time is small, but it has at least one supersonic solution and infinitely many transonic solutions if the supersonic doping profile is close to the sonic line and the relaxation time is large. The interior subsonic/supersonic solutions all are globally C12 H¨older-continuous, and the H¨older exponent 12 is optimal. The non-existence of any type solutions in the case of small doping profile or small relaxation time indicates that the semiconductor effect for the system is remarkable and cannot be ignored. The proof for the existence of subsonic/supersonic solutions is the technical compactness analysis combining the energy method and the phase-plane analysis, while the approach for the existence of multiple transonic solutions is constructed. The results obtained significantly improve and develop the existing studies. This is a joint work with Jingyu Li, Guojing Zhang and Kaijun Zhang.

On the obstacle-mass constraint problem for hyperbolic conservation laws

Jos Francisco Rodrigues

Universidade de Lisboa, Portugal

jfrodrigues@fc.ul.pt

Abstract: The obstacle-mass constraint problem for a multidimensional scalar hyperbolic conservation law is considered in a joint work with P. Amorim andW. Neves. We prove existence of an entropy solution by a penalisation/viscosity method. The mass constraint introduces a nonlocal Lagrange multiplier in the penalised equation, giving rise to a nonlocal parabolic problem. We introduce a compatibility condition relating the initial datum and the obstacle function which ensures global in time existence of solution. This is not a smoothness condition, but relates to the propagation of the support of the initial datum.

Numerical approximations of a tractable mathematical model for tumour growth

Vanessa Styles

University of Sussex, UK

v.styles@sussex.ac.uk

Abstract: We consider a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We derive sharp interface and diffuse interface finite element approximations of this model and present some numerical results.

Compressible Vortex Sheets in Elastodynamics

Dehua Wang

University of Pittsburgh, USA

dwang@math.pitt.edu

Abstract: The compressible vortex sheets in elastic fluids will be considered. The linear and nonlinear stability of vortex sheets will be discussed.

Boundary Layer Problem and Zero Viscosity-Diffusion Limit of the 2D/3D Incompressible Magnetohydrodynamic System with Dirichlet Boundary Conditions

Shu Wang

Beijing University of Technology, China

wangshu@bjut.edu.cn

Abstract: In this talk, we study the boundary layer problem, zero viscosity-diffusion limit and zero magnetic diffusion limit of the initial boundary value problem for the 2D/3D incompressible viscous and diffusive MHD system with Dirichlet boundary conditions. The main difficulties overcome here are to deal with the effects of the the difference between the viscosity and diffusion coefficient on the error estimates and to control the boundary layer resulted by the Dirichlet boundary condition for the velocity and magnetic field. Firstly, we establish the result on the stability of the Prandtl boundary layer of MHD system with a class of special initial data and 8 prove rigorously the solution of incompressible viscous and diffusive MHD system converges to the sum of the solution to the ideal inviscid MHD system and the approximating solution to Prandtl boundary layer equation by using the elaborate energy methods and the special structure of the solution to inviscid MHD system, which yields that there exists the cancellation between the boundary layer of the velocity and the one of the magnetic field. Next, we obtain the stability result on the boundary layer for the magnetic field and zero magnetic diffusion limit of viscous and diffusive MHD system with the general initial data when the magnetic diffusion coefficient goes to zero. Finally, for general initial data, we consider the boundary layer problem of the incompressible viscous and diffusive MHD system with the different horizontal and vertical viscosities and magnetic diffusions, when they go to zero with the different speeds. We prove rigorously the convergence to the ideal inviscid MHD system and the anisotropic inviscid MHD system from the incompressible viscous and diffusion MHD system by constructing the exact boundary layers and using the elaborate energy methods.