会议名称：第五届偏微分方程青年学术论坛

会议时间：4月21日-22日

会议地点：海韵园数学物理大楼117报告厅

会议日程：（见下表）

4月21日（星期六）

4月22日（星期日）

报告题目与摘要

On Admissible Locations of Transonic Shock-Fronts for Full Euler Flows in an Almost Flat Finite Nozzle

方北香（上海交通大学）

In this talk, we are concerned withthe inviscid 2-D steady Euler flow pattern with a single shock front in an almost flat finite nozzle, which enters the nozzle with a supersonic state and leave with a subsonic one. The nozzle is almost flat in the sense that its boundary is a general perturbation of a flat one. We are going to determine the whole flow pattern, including the location of the shock front, the supersonic flow field ahead of it and the subsonic flow field behind it, with the prescribed receiver pressure at the exit of the nozzle. The key difficulty is the information of the approximate location of the shock front. We are going to overcome this difficulty by designing a free boundary problem for the linearized Euler system which will yield useful information on the initial approximation location of the shock front. With these information, we can then further apply a nonlinear iteration scheme to determine the whole flow patten in the nozzle, including the location of the shock front. This is a joint work with Prof. Zhouping Xin.

Heat flow for Dirichlet-to-Neumann operator with critical growth

方飞（北京工商大学）

In this talk, we will discuss the heat flow equation for Dirichlet-to-Neumann operator with critical growth. By assuming that the initial value is lower-energy, we obtain the existence, blowup and regularity. On the other hand, a concentration phenomenon of the solution when the time goes to infinity is proved.

Convergence rates of solutions for the damped Compressible Euler Equations with Vacuum

耿世锋（湘潭大学）

We study the asymptotic behavior ofacompressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. By a intensive entropy analysis,we prove that,the any L^\infty weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges strongly in the  natural L^1 topology with decay rates to the Barrenblatt's solution of porous medium equation while the momentum is described by the Darcy's law. Our results improve and extend the existing convergence rates to the Barrenblatt's solution of porous medium equation.

Some studies on the relaxation models in hydrodynamics

胡玉玺（中国矿业大学（北京））

The compressible Navier-Stokes equations are to describe the dynamics of compressible fluid. The system are composed of mass equation, momentum equation and energy equation. Mathematically, the system are underdetermined and one need to give constitutive equations to close the system. Newtonian law in describing the relation of stress tensor and velocity, and Fourier law in describing the heat conductivity are the two main constitutive relations in fluid dynamics. In this talk, we shall investigate some other constitutive relations, namely, Cattaneo's law for heat conductivity and Maxwell's law for the stress tensor and velocity, which can be considered as some kind of relaxation of the classical system. The importance of these two constitutive relations are demonstrated by many physicists and biologists recently, in the field such as skin burns, nano fluids , biological materials, nanoscale mechanical devices vibrating in simple fluid, etc. However, the corresponding mathematical results are still very few. We shall present some related mathematical results and give some new results.

On stochastic primitive equations of the large-scale ocean

黄代文（北京应用物理与计算数学研究所）

In this talk, we give some results on stochastic primitive equations of the large-scale ocean. Firstly, we recall the global well-posedness and long-time dynamics for the viscous primitive equations describing the large-scale oceanic motion with white noise. Secondly, we introduce some results on ergodicity of thestochastic primitive equations driven by degenerate noise.

Entropy-bounded solutions of the full compressible Navier-Stokes equations with vacuum

李进开（香港中文大学）

The entropy is one of the fundamental states of a fluid and, in the viscous case, the equation that it satisfies is both degenerate and singular in the region close to the vacuum. In spite of its importance in the gas dynamics, the mathematical analyses on the behavior of the entropy near the vacuum region, were rarely carried out; in particular, in the presence of vacuum, either at the far field oron the physical boundaries, it was unknown if theentropy remains its boundedness. It will be shown in this talk that the ideal gases retain their uniformboundedness of the entropy, locally or globally in time, for both the Cauchy problem and the initial-boundary value problems, if the vacuum occurs only at the far field or on the physical boundary, as long as the initial density behaves well at the far field or near the boundary. For the Cauchy problem, the density is required to decay slowly enough at the far field, while for the initial-boundary value problem, the $(\gamma-1)$-th power of density is required to be equivalent to the distance to the boundary, near the boundary.

Global existence and exponential stability for the compressible Navier-Stokes equations with discontinuous data

梁闯闯（重庆大学）

We consider the viscous compressible barotropicNavier-Stokes equations with discontinuous initial data in the bounded domain. The small global piecewise strong solution is obtained and exponentially decays to the constant stationary state. The jump of the fluid density across the discontinuous surface is exponentially decay in time.This work is joint with H.Kong, Prof. H.Li and G.Zhang.

The relaxation limit of thermal non-equilibrium flows

刘成杰（上海交通大学）

The talk is devoted to the relaxation limit of thermal non-equilibrium flows in half space with general initial data. We construct the initial layers and study the interaction between initial layers and possible boundary layers. Then, it is shown that the solution is uniformly bounded in a conormalSobolev space in the vanishing relaxation limit. This is a joint work with Prof. Tao Luo from City University of Hong Kong.

The existence and limit behavior of the shock layer for 1D stationary compressible non-Newtonian fluids

刘进静（西北大学）

We give a definition of shock layer for a class of stationary compressible non-Newtonian fluids in one dimension. Then the existence and uniqueness of the shock layer are established. In addition, the limit behaviorof the shock layer are analyzed. It is shown that, as the viscosity coefficient and the heat conductivity coefficient vanish, the shock layer to the non-Newtonian fluids tends to the shock wave of the corresponding Euler equations.

Nonlinear Instability of Non-isentropic Fluid Flows

秦绪龙（中山大学）

Nonlinear dynamic instabilityof steady smooth profile is rigorously demonstrated fornon-isentropic, compressible inviscid fluid flows when the convection is present.

Zero-viscosity limit of Navier-Stokes equations with Navier-slip boundary condition

陶涛（山东大学）

In this talk, we discuss the zero-viscosity limit problem of the Navier-Stokes equations in the half space with the Navier friction boundary condition where the slip length is a power of the viscosity:                                               -|_{y=0}=0. We rigorously justify the convergence process in sense from Navier-Stokes equations to Euler equations and some Prandtl equations for Gervey data when  and is a rational number. This is ajoint work with Wendong Wang and Zhifei Zhang.

Quantitative Pointwise Estimate of the Solution of the LinearizedBoltzmann Equation

王海涛（上海交通大学）

Westudy the quantitative pointwise behavior of the solutionsof the linearizedBoltzmannequationfor hard potentials, Maxwellian molecules and soft potentials, with Grad’s angular cutoff assumption. More precisely, for solutions inside the finite Mach number region, we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region, we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. This is a joint work with Yu-Chu Lin and Kung-Chien Wu.

Global solution to 3D spherically symmetric compressibleNavier-Stokes equations with large data

王梅（西安理工大学）

In this paper, we prove the global well-posedness of the solution to 3D spherically symmetric, compressible and isentropic Navier-Stokes equations in the whole space with arbitrarily large initial data when the shear viscosity $\mu$ is a positive constant and the bulk viscosity $\lambda(\rho)=\rho^\beta$ with $0\leq \beta\leq \gamma$ and $\gamma$ being the adiabatic exponent in the $\gamma$-law pressure. First, the global classical solution is obtained away from the symmetry center $r=0$ with arbitrarily large and non-vacuum data. In particular, it is shown that the solution will not develop the vacuum states in any finite time away from the symmetry center if the initial density does not contain vacuum states. Then the global weak solutions with the symmetry center $r=0$ are obtained as the limit of the classical solutions in the exterior domain of a ball $B_\varepsilon(0)$ with the center at the origin and the radius $\varepsilon>0$ when the ball shrink to the origin, that is, $\varepsilon\rightarrow0+$, for any fixed total mass $h>0$ and then let $h\rightarrow0+$.

Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowskispacetime

王涛（武汉大学）

We study vortex sheets for the relativistic Euler equations in three-dimensional Minkowskispacetime. A necessary and sufficient condition for the weakly linear stability is obtained by analyzing roots of the Lopatinskii determinant associated to the constant coefficient linearized problem. Under this stability condition, we prove the existence and nonlinear stability of relativistic vortex sheets by a Nash-Moser iterative scheme. This talk is based on a joint work with Prof. Gui-Qiang Chen and Prof. Paolo Secchi.

On Steady Euler Flows with Large Vorticity and Characteristic Discontinuities

王天怡（武汉理工大学）

In this talk, we want to present the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. A new approach is introduced for the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets/entropy waves by the compensated compactness argument. This is the first result on the global existence of solutions of the multidimensional steady compressible full Euler equations with free boundaries, which are not necessarily small perturbations of piecewise constant background solutions. The subsonic-sonic limit of the solutions is also shown. Finally, through the incompressible limit, the existence and uniqueness of incompressible Euler flows is established in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solution with vortex sheets. This is the joint work with Gui-Qiang G. Chen, Fei-Min Huang, and Wei Xiang.

Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model

汪文军（上海理工大学）

In this talk, we consider a compressible two-fluid model with constant viscosity coefficients and unequal pressure functions $P^+\neq P^-$. We obtain the global solution and its optimal decay rate (in time) with some smallness assumptions. In particular, capillary pressure is taken into account in the sense that $\Delta P=P^+-P^-=f\neq 0$ where the difference function $f$ is assumed to be a strictly decreasing function near the equilibrium relative to the fluid corresponding to $P^-$. This is a joint work with Prof. Steinar Evje and Prof. Huanyao Wen.

The Cahn-Hilliard Equation with Dynamic Boundary Conditions

吴昊（复旦大学）

The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In order to account for possible short-range interactions of the material with the solid wall, various dynamic boundary conditions have been proposed in the literature. In this talk, we introduce a new class of dynamic boundary conditions for the Cahn-Hilliard equation. The derivation is based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation. Then under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary problem with or without surface diffusion. Furthermore, we establish the uniqueness of asymptotic limit as time goes to infinity and characterize the stability of local energy minimizers for the system.

Global small solution to the MHD systems and some related problems

吴奕飞（天津大学）

In this talk, we present some results and proofs on the global existence, uniqueness and the explicit large-time decay rates of smooth solutions to the compressible MHD system when the initial data is close to an equilibrium state. In addition, we will discuss some related topics.

Some properties of an integral equation with the axis symmetric kernel

许建开（湖南农业大学）

This paper is devoted to exploring the properties of positive solutions for a class of nonlinear integralequation(s) involving the axis-symmetric, which raises from weak type convolution-Young's inequalityand the stationary Magnetic compressible fluid stars. With the help of Moving plane, we conclude that all ofpositive solutions is decreasing on the symmetric axis. Meanwhile, the integrableinterval are also obtained.

Wellposedness and waves phenomena in MHD

徐丽（中国科学院数学与系统科学研究院）

I will talk about my work on the wellposedness theory and wave phenomena in MHD. First, I will give a short introduction and survey on MHD. Then I will show the main results and key ideas of their proof.

On the Free Surface Motion of Highly Subsonic Heat-conducting Inviscid  Flows

曾惠慧（清华大学）

In this talk, I will present a recent result joint with Tao Luo on the free surface problem of a highly subsonic heat-conducting inviscid flow. Adopting a geometric approach developed by Christodoulou and Lindblad in the study of the free surface problem of incompressible  inviscid flows, we give the a priori estimates of Sobolev norms in 2D and 3-D under the Taylor sign condition by identifying a suitable higher order energy functional. The estimates for some geometric quantities such as the second fundamental form and the injectivity radius of the normal exponential map of the free surface are also given. I will discuss the issues of the strong coupling of large variation of temperature, heat-conduction, compressibility of fluids and the  evolution of free surface, loss of symmetries of equations, and loss of derivatives in closing the argument which is a key feature compared with Christodoulou and Lindblad's work.

Local well-posedness of the free-surface inviscid two phase flow model

张映辉（湖南理工学院）

We investigate a free boundary problem for the inviscid two-phase flow model with moving physical vacuum boundary, which is a degenerate and characteristic hyperbolic system of conservation laws. Based a suitable transformation, a special degenerate parabolic approximation and delicate uniform energy estimates, we prove the existence and uniqueness of the local weak solutions. This can be regarded as a generalization of the results in [Coutand, D., Shkoller, S.,  Commun. Pure Appl. Math.,64: 328-366,2011] from single-phase gas model to two-phase gas-liquid model.