2014 Workshop on Nonlinear Evolutionary Partial Differential Equations. Abstracts

Transcription

1 11 Abstracts

2 12 Global existence of weak shocks past a solid ramp Myoungjean Bae Pohang University of Science and Technology, Korea mybjean@gmail.com, mjbae@postech.ac.kr Prandtl (1936) first employed the shock polar analysis to show that, when a steady supersonic flow impinges a solid wedge whose angle is less than a critical angle (i.e., the detachment angle), there are two possible configurations: the weak shock solution and the strong shock solution, and conjectured that the weak shock solution is physically admissible. In this talk, I address this longstanding open issue and present recent analysis to establish the stability theorem for steady weak shock solutions as the long-time limits of unsteady flows for all the physical parameters up to the detachment angle for potential flow. This talk is based on collaboration with Gui-Qiang G. Chen (Univ. of Oxford) and Mikhail Feldman (UW-Madison). Global smooth solutions in 3 R to short wave-long wave interactions systems for viscous compressible fluids Hermano Frid Instituto de Matemática Pura e Applicada-IMPA, Brazil hermano@impa.br The short wave-long wave interactions for viscous compressible heat conductive fluids is modeled, following Dias \& Frid (2011), by a Benney-type system coupling Navier-Stokes equations with a nonlinear Schr\"odinger equation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in ${\mathbb R}^3$ when the initial data are small smooth perturbations of an equilibrium state. We also consider the case where the long waves represent the magnetohydrodynamics flow. The talk is based on two joint works with \textsc{ronghua Pan and Weizhe Zhang} and \textsc{junxiong Jia and Ronghua Pan}.

3 13 On the free boundary problem of incompressible inviscid MHD flows Chengchun Hao Chinese Academy of Sciences, China In this talk, I will show the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid MHD equations in all physical spatial dimensions n=2 and 3 by adopting a geometrical point of view used in Christodoulou & Lindblad CPAM 2000, and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids. This is based on a joint work with Professor T. Luo. Compensated compactness in multi-dimension steady Euler equations Femin Huang Chinese Academy of Sciences, China fhuang@amt.ac.cn In this talk, I will present some new results in the multi-dimension steady Euler equations by the theory of compensated compactness. On Liouville Theorems and Global Regularity Criteria to the 3D Axisymmetric Navier-Stokes Equations Quansen Jiu Capital Normal University, China jiuqs@mail.cnu.edu.cn In this talk, we will present some recent results on Liouville theorems and global regularity to the 3-D axisymmetric Navier-Stokes equations with swirl under some assumptions on $u_\theta$ and $u_z$ or some weighted estimates of the solutions which are scaling invariant.

4 14 Hamiltonian evolution of Monokinetic measures with rough momentum profile Peter Markowich University of Cambridge, UK Consider a monokinetic probability measure on phase space. In this lecture, we study the structure of the monokinetic measure and the associated position density when transported by a regular Hamiltonian flow. In particular, we give estimates on the number of folds on which the obtaned measure is concentrated. We explain how our results can be applied to investigate the classical limit of the Schrödinger equation by using the formalism of Wigner measures. Our formalism includes initial momentum profiles with much lower regularity than required by the WKB method. Finally, we discuss a few examples showing that our results are sharp. Regularity of the Mumford-Shah minimizers at the crack-tip Hayk Mikayelyan Xi an Jiaotong Liverpool University, China Hayk.Mikayelyan@xjtlu.edu.cn We consider the Mumford-Shah functional in the plain and study the asymptotics of the solution near the crack-tip. It is well-known that the leading term in the asymptotics can be given by the imaginary part of the complex square root function with a certain coefficient related to the stress intensity factor in the fracture mechanics. We calculate higher order terms in the asymptotic expansion, where the homogeneity orders of those terms appear to be solutions to a certain trigonometric relation. We also develop new numerical methods to compute the minimizers. (joint work with John Andersson and Zhilin Li)

5 15 Structure of nodal and singular sets for solutions to semilinear PDEs Henrik Shahgholian The Royal Institute of Technology, Sweden I shall discuss issues related to the level surfaces and singular sets for solutions to semilinear problems of the type $\Delta u = f(u)$, where $f$ admits discontinuities. Across such discontinuity points the PDE changes qualitatively and therefore one may see such nodal sets as free boundaries. The kind of free boundary we obtain depends on the type of discontinuity of $f$, or more exactly it depends on the right and left limit values for $f$ at such discontinuity points. I shall present partial results as well as a program for a possible analysis of this problem. Elementary wave interactions for a Chaplygin gas in two-dimensions Wancheng Sheng Shanghai University, China mathwcsheng@shu.edu.cn Abstract: In order to construct global piecewise smooth solutions to the two-dimensional (2D) Riemann problems for the compressible Euler equations, it is important to investigate 2D elementary wave interactions. In a recent paper, Serre [Arch. Rat. Mech. Anal., 191 (2008), ] investigated 2D shock interaction for a Chaplygin gas equations and constructed global solutions to several 2D Riemann problems. In this paper, we study 2D elementary wave interactions for the Chaplygin gas. These elementary waves include shock waves, simple waves, and delta waves. Using these elementary wave interactions, we also construct global piecewise smooth solutions to several 2D Riemann problems.

6 16 Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers Endre Süli University of Oxford, UK We prove the existence of global-in-time large-data weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a bounded domain in two or three space dimensions for the density $\rho$, the velocity $\ut$ and the pressure $p$ of the fluid, with an equation of state of the form $p(\rho) = c_p \rho^\gamma$, where $c_p$ is a positive constant and $\gamma>\frac{3}{2}$. The right-hand side of the Navier-Stokes momentum equation includes an elastic extra-stress tensor, which is the sum of the classical Kramers expression and a quadratic interaction term. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. This is joint work with John W. Barrett (Department of Mathematics, Imperial College London). Partial regularity of suitable weak solutions to the Navier-Stokes equations Yanqing Wang Capital Normal University, China wangyanqing20056@gmail.com In this talk, we will first present a unified proof on partial regularity of suitable weak solutions to non-stationary and stationary Navier-Stokes equations proved by previous authors such as L. Caffarelli, R. Kohn and L. Nirenberg, M. Struwe, H. Dong and D. Du, H. Dong and R. Strain. Particularly, we obtain the partial regularity of the suitable weak solutions to the 4D time-dependent Navier-Stokes equations. Our proof relies on the De Giorgi iteration recently developed by A. Vasseur and some elementary observation of these equations. Then we show some anisotropic regularity conditions for the suitable weak solutions to the 3D Navier-Stokes equations. Finnally, we are concerned with possible time singular points and eventual regularity of weak solutions to the Navier-Stokes equations with fractional dissipation $(-\Delta)^{\alpha}$. My co-authors are Professor Quansen Jiu and Dr. Gang Wu.

7 17 Stability of wave patterns to the bi-polar Vlasov-Poisson-Boltzmann system Yi Wang Chinese Academy of Sciences, China We first present a new decomposition around the local Maxellian to the bipolar Vlasov-Poisson-Boltzmann system. Then as an application of the decomposition, the time-asymptotic stability of rarefaction waves are proved for the 1D bipolar Vlasov-Poisson-Boltzmann system. Then the stability of viscous shock profile and the viscous contact wave will also be concerned. Note that these results imply that the elementary wave patterns are still stable even with the effect of the electronic fields through bilopar Vlasov-Poisson couplings with Boltzmann equation. This is joint with Hailiang Li, Tong Yang and Mingying Zhong. Shock Diffraction Problem to the Potential Flow Wei Xiang City University of Hong Kong, China weixiang@cityu.edu.hk In this talk, I would like to present one of our current research projects, the shock diffraction by convex cornered wedges for the potential flow. The existence of the regular configuration for the potential flow is established up to the critical wedge angle, which would be the criterion of the transition between the regular configuration and the Mach configuration. A new difference scheme for strong detonation waves and Chapman-Jouguet detonation waves Xiaozhou Yang Wuhan Institute of Physics and Mathematics, CAS, China xzyang@wipm.ac.cn In this talk, we will present a new difference scheme for combustion computation, using this scheme, we can cover the cases which previous schemes can not cover. we also prove the convergence of this new scheme.

8 18 On Supersonic Flow past an Infinite Delta Wing Chao Yi Fudan University, China In this talk, we study some mathematical aspects on supersonic flow past a delta wing. Here the wing is assumed to be infinite along its edges, so we need only to seek self-similar solutions to this supersonic shock problem, which is formulated as a free boundary value problem of nonlinear mixed-type equation. In the case that the wing is of zero-thickness, the mixed-type equation then reduces to a degenerate elliptic equation, and thus an existence result can be established by using the framework of Chen-Feldman for regular shock reflection problem. Moreover, some differences between the regular shock reflection model and the general delta wing model are discussed. This talk is based on a joint work with my advisor Prof. Shuxing Chen. Existence of regular solutions to compressible Navier-Stokes equations with density-dependent viscosities Shengguo Zhu Shanghai Jiao Tong University, China zhushengguo@sjtu.edu.cn We survey some of our recent progress on the local well-posedness problem for compressible Navier-Stokes Equations with density dependent viscosity when initial density does not have a uniform positive lower bound. When viscosity coefficients are constant multiple of a power of density, we identify the class of initial data admitting a local classical solution local in time when the power is greater or equal to 1. The questions on the singularity formation will also be addressed. This talk is mainly based on joint works with Y. Li,and R. Pan.