Free Boundary Problem related to Euler-Poisson system. Some topics on subsonic potential flows

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Leon Wilkins
5 years ago
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1 ABSTRACTS

2 Free Boundary Problem related to Euler-Poisson system BAE, Myoungjean Pohang University of Science and Technology (POSTECH), Korea One dimensional analysis of Euler-Poisson system shows that when incoming supersonic flow is fixed, transonic shock can be represented as a monotone function of exit pressure. From this observation, we expect well-posedness of transonic shock problem for Euler-Poisson system when exit pressure is prescribed in a proper range. In this talk, I will present recent progress on transonic shock problem for Euler-Poisson system, which is formulated as a free boundary problem with mixed type PDE system. This talk is based on collaboration with Ben Duan, ChujingXie and Jingjing Xiao. Some topics on subsonic potential flows CHEN, Chao Fujian Normal University, China chenchao@fjnu.edu.cn In this talk, we present the existence and uniqueness of subsonic potential flows in general smooth bounded domains when the normal component of the momentum on the boundary is prescribed. It is showed that if the Bernoulli constant is given larger than a critical number, there exists a unique subsonic potential flow. Moreover, as the Bernoulli constants decrease to the critical number, the subsonic flows converge to a subsonic-sonic flow.

3 Smooth transonic Euler-Poisson flows in finitely long nozzles DUAN, Ben Dalian University of Technology, China In this talk, the unique existence of smooth transonic Euler-Poisson flow in a flat given nozzle will be discussed. Moreover, under physically acceptable boundary conditions, we prove that the flow is stable in sense of small perturbation of data at the entrance and exit. Also, we prove the unique existence of sonic curve. This is a joint work with MyoungjeanBae and ChunjingXie. On the free boundary problem of compressible Euler equations in physical vacuum with general initial densities HAO, Chengchun Institute of Mathematics, AMSS, Chinese Academy of Sciences, China hcc@amss.ac.cn In this talk, I will show a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0 \in H^5$. I derive a mixed space-time interpolation inequality which play a vital role in the energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$, which is different from the known results.

4 On the Boltzmann and Landau equations in a unified framework HE, Lingbing Tsinghua University, China lbhe@math.tsinghua.edu.cn In this talk, we will show some recent progress on the well-posedness of Boltzmann and Landau equations and the asymptotics from Boltzmann to Landau equation in the so-called grazing collisions limit. Initial values for the Navier-Stokes equations in spaces with weights in time Hsu, Pen-Yuan Waseda University, Japan pyhsu@ms.u tokyo.ac.jp We consider the nonstationarynavier-stokes system in a smooth bounded domain $\Omega \subset {\R}^3$ with initial value $u_0\in L^{2}_{\sigma}(\Omega)$. It is an important question to determine the optimal initial value condition in order to prove the existence of a unique local strong solution satisfying Serrin's condition. In this work, we introduce a weighted Serrin condition that yields a necessary and sufficient initial value condition to guarantee the existence of local strong solutions $u(\cdot)$ contained in the weighted Serrin class $\int_0^t (\tau^\alpha \ u(\tau)\ _q)^s\dd\tau<\infty$ with $\frac{2}{s} + \frac{3}{q} =1-2\alpha$, $0<\alpha<\frac12$. Moreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class. This talk is based on a joint work with ReinhardFarwig (TU Darmstadt) and Yoshikazu Giga (U. Tokyo).

5 Well-posedness of Prandtl equation with Robin boundary condition in analytic class and inviscid limit JIANG, Ning Tsinghua University, China Wang-Wang-Xin proposed a program in 2010 on the inviscid limit of Navier-Stokes equations with Navier-slip boundary condition, in particular, the slip length depends on the viscosity. For different slip length, the correction term of Euler equations are Prandtle equation with Dirichlet or Robin boundary condition. We prove the well-posedness of Prandtl equation with Robin boundary condition and justify one inviscid limit in analytic class. This is a joint work with Yutao, Ding. Asymptotic limits of the full Navier-Strokes-Fourier-Poisson system JU, Qiangchang Institute of Applied Physics and Computational Mathematics, China qiangchang_ju@yahoo.com The quasi-neutral limit of the full Navier-Stokes-Fourier-Poisson system for the plasmas and semiconductors in the multi-dimensional torus is considered in the framework of both smooth and weak solutions as the scaled Debye length goes to zero. In particular, the effects of large temperature variations are taken into account.

6 Nonuniqueness of weak solutions to the Riemann problem for compressible Euler equations in 2D KREML, Ondrej Academy of Sciences of the Czech Republic, Czech Using the method developed by Camillo De Lellis and Laszlo Szekelyhidi for the incompressible Euler system we prove that in the case of compressible isentropic Euler equations it holds that for every Riemann initial data such that the corresponding self-similar solution consists of 2 shocks there exists also infinitely many other admissible weak solutions. Moreover, for some of these initial data the self-similar solution is not entropy rate admissible in the sense of Dafermos. Blow up for initial boundary value problem of critical semilinear wave equation outside 3-D ball LAI, Ningan Lishui University, China hyayue@gmail.com In this talk we first give a survey of the Strauss conjecture, and then show the blow up result for the initial boundary value problem of critical semilinear wave equation with small data outside 3-D ball.

7 Large-Time Behavior of Solutions to 1D Compressible Navier-Stokes System in Unbounded Domains with Large Data LI, Jing AMSS, Chinese Academy of Sciences, China We are concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. The temperature is proved to be bounded from below and above independently of both time and space. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity.note that the initial data can be arbitrarily large. This result is proved by using elementary energy methods. On lifespan of smooth axi-symmetric solution to 2-D Euler system LI, Jun Nanjing University, China lijun@nju.edu.cn In this talk, I will introduce a recent result on lifespan of smooth axi-symmetric solutions to 2-D compressible non-isentropic Euler system. Here small smooth initial data is compactly supported. In this case, the smooth solution will blow up in finite time and the blow up point is near the outward light-cone. The lower bound of lifespan is obtained by careful analysis on the related profile equations and the control on the influence of entropic. Furthermore, by the characteristics decomposition and the analysis of the solution near the light-cone, the upper bound of the lifespan is also obtained. Some discussion will be given in the final.

8 Regularity of Traveling Free Surface Water Waves with Vorticity LI, Wei-Xi Wuhan University, China wei We present in this talk the real analyticity of all the streamlines, including the free surface, of a steady flow of water over a flat bed, with a Hölder continuous vorticity function. Furthermore, if the vorticity possesses some Gevrey regularity of index s, then the stream function admits the same Gevrey regularity throughout the fluid domain. Stability of Wave Patterns of the Navier-Stokes-Poisson System LIU, Shuangqian Jinan University, China tsqliu@jnu.edu.cn In this talk, we are concerned with the nonlinear stability of wave patterns of the one dimensional Navier-Stokes-Poisson system. On account of the quasineutral assumption, we first construct a non-trivial solution profile through the quasineutral Euler equations, and then prove that such a non-trivial profile is time-asymptotically stable. The problem of stability of both the rarefaction wave and viscous contact wave will be discussed in the talk.

9 Existence of weak solutions to 3D Euler equation under helical symmetry NIU, Dongjuan Capital Normal University, China In this talk we will establish the existence of a helical weak solutions to 3D incompressible Euler equations in full space for an initial velocity without helical swirl and whose initial vorticity is compactly supported and belongs to L^p, p>1. This result is an extension of the result of Bronzi, Lopes Filho and Nussenzveig Lopes, who studied the existence of Euler equations with the vorticity belonging to p>4/3. KdV limit of the Euler-Poisson system PU, Xueke Chongqing University, China xuekepu@cqu.edu.cn In this talk, we will discuss the KdV limit of the Euler-Poisson system in the long wavelength limit. We will discuss this limit both formally and mathematically rigorously.

10 On Multi-dimensional Riemann problems for the Chaplygin gas QU, Aifang Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, China In this talk, we review some results on multi-dimensional Riemann problems for the Chaplygin gas. We mainly focus on the construction as well as the classification of the solution. Some open problems are present. One-Dimensional Non-isentropic Euler Equations with Periodic Initial Data QU, Peng Fudan University, China The non-isentropic Euler system with periodic initial data in $\mathbb{r}^1$ is discussed by analyzing wave interactions in a framework of specially chosen Riemann invariants and applying the method of approximate conservation laws and approximate characteristics. An $\mathcal{o}(\varepsilon^{-2})$ lower bound is established for the life span of the entropy solutions with initial data that possess $\varepsilon$ variation in each period.

11 Some results on radiating gas model and related model RUAN, Lizhi Central China Normal University, China In this talk, we are concerned with the global wellposedness, stability and convergence rate of smooth solutions to the Cauchy problem of radiating gas model and related model near some existing equilibria state, such as constant states, the planar rarefaction waves and diffusion waves. Compressible Navier-Stokes Equations: Relative Entropy Inequality and Applications SUN, Yongzhong Nanjing University, China We show that the weak solutions to the isentropic Navier-Stokes equations satisfy the relative entropy inequality. As applications, we report some results on weak-strong uniqueness, incompressible inviscid limit on expanding domains and singular limit problem in MHD. These are joint works with E. Feireisl, A. Novotny and S. Necasova.

12 Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations WANG, Chengbo Zhejiang University, China In this talk, we will talk about our recent work on the long time existence of small solutions, with sharp lifespan, for a class of semi-linear wave equations with two nonlinearities, which is in relation with both the Strauss conjecture and Glassey conjecture. This is joint work with Kunio Hidano and Kazuyoshi Yokoyama. Inviscid Limit and Zero Surface Tension Limit for the Compressible Navier-Stokes Equations with Free Surface WANG, Yong Institute of Mathematics, AMSS, Chinese Academy of Sciences, China yongwang@amss.ac.cn In this paper, we consider the inviscid limit and zero surface tension limit for the compressible Navier-Stokes equations on the free boundary problem with surface tension. We obtain uniform estimates for the solutions of the compressible Navier-Stokes in the Sobolev conormal spaces in a time interval $[0,T_0]$ which is independent of the viscosity and surface tension coefficient. Based on the uniform estimates, the vanishing viscosity and zero tension limit is also established by the strong compactness argument.

13 Bifurcation for a free boundary problem modeling tumor growth with inhibitors WANG, Zejia Jiangxi Normal University, China zejiawang@jxnu.edu.cn In this talk, we deal with a free boundary problem modeling tumor growth with inhibitors. This problemhas a unique radially symmetric stationary solution with radius $r=r_s$. The tumor aggressiveness is modeled bya positive tumor aggressiveness parameter $\mu$.it is shown that there exist a positive integer $m^{**}\in\mathbb R$ and a sequence of $\mu_m$,such that for each $\mu_m(m>m^{**})$, symmetry-breaking solutions bifurcatefrom the radially symmetric stationary solutions. Competing dual gradient system in chemotaxis WANG, Zhian The Hong Kong Polytechnic University, Hong Kong zhi.an.wang@polyu.edu.hk Most of past and current researches on chemotaxis models deal with attraction and repulsion separately. But in some biological processes (or experiments), the repulsive process often follows the attractive process for balance in order to accomplish some biological objects. Hence an attraction and repulsion chemotaxis model will be more realistic than a sole attraction or repulsion chemotaxis model in this scenario. In this talk, we shall present recent mathematical progress an attraction-repulsion dual gradient chemotaxis model and show the interplay of these two competing biological processes. Pattern formations will be shown and various open questions will be presented.

14 Shock diffraction problems XIANG, Wei City University of Hong Kong, Hong Kong In this talk, I would like to present one of our current research projects, that is on the mathematical analysis of shock diffraction by convex cornered wedges for both the nonlinear wave equation and 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. Decay estimates of the MHD equations without magnetic diffusion and of the related models XIANG, Zhaoyin University of Electronic Science and Technology of China, China zxiangmath@gmail.com We present in this talk the decay estimates of small smooth solution to the MHD equations without magnetic diffusion. Our result confirms the numerical observation that the energy of the MHD equations is dissipated at a rate independent of the ohmic resistivity. We will also show that our argument can be applied to a hyperbolic-parabolic system modeling chemotaxis.this is a joint work with X. Ren, J. Wu and Z. Zhang.

15 Global well/ill-posedness for rotating shallow water system XIE, Chunjing Shanghai Jiao Tong University, China In this talk we will discuss recent progress on global well/ill-posedness for rotating shallow water system. On the global classical solutions of the 3-D compressible Euler flows in infinitely expanded balls XU, Gang Jiangsu University, China This paper concerns with the global existence of a smooth compressible viscous fow in an infinitely expanded ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to the compressible Euler flow with time dependent boundaries and vacuum states at infinite time. The flow is described by the 3-D compressible isentropic irrotational Euler equations. From the physical point of view, because of the mass conservation of gases, the moving gases in the expansive ball will gradually become rarefactive and eventually tend to a vacuum state with the development of time. We will confirm this interesting physical phenomenon by the rigorous mathematical proof and simultaneously show that there are no appearances of vacuum domains in any part of the expansive ball.

16 Some recent results on the global well-posedness of Boussinesq and MHD systems XU, Xiaojing Beijing Normal University, China In this talk, we will give some recent results on the global well-posedness of Boussinesq and MHD systems with some kinds of dissipation terms, including the subcritical and critical fractional Laplacian, damping, partial viscosity and viscosity depending on temperature. Partial Regularity for the fractional Navier-Stokes equation YU, Yong The Chinese University of Hong Kong, Hong Kong We are going to talk about the partial regularity for the fractional Navier-Stokes equation. The results generalize the famous result of Caffarelli-Kohn-Nirenberg and F.H.Lin for the standard Navier-Stokes equation in 3D.

17 Transonic shocks in steady non-isentropic compressible Euler flows YUAN, Hairong East China Normal University, China We review some results on transonic shocks in steady non-isentropic compressible Euler flows, and introduce a new result on stability of spherical transonic shocks based on a decomposition of the three dimensional Euler system and some tools from differential geometry, which is obtained together with Li Liu and Gang Xu. Two-Dimensional Steady Supersonic Exothermically Reacting Euler flow past a bending wall ZHANG, Yongqian Fudan University, China yongqianz@fudan.edu.cn In this paper, we study the two-dimensional steady supersonic Euler flow moving over a Lipschitz bending wall which is a small perturbation of a convex one and establish the global existence of entropy solution when the total variation of the initial data and the slope of the boundary are sufficiently small. The flow is governed by an ideal polytropic gas and undergoes a one-step exothermic chemical reaction under the reaction rate function $\phi(t)$ is assumed to have a positive lower bound. We employ the modified wave front tracking scheme to construct the approximate solution and develop a Glimm-type functional by incorporating the approximated large rarefaction wave and Lipschitz bending wall to obtain the uniform bounds on the total variation of the approximate solutions. Then we use these bounds to prove the convergence of the approximate solutions to a global entropy solution that containing a large rarefaction wave issuing from the bending wall.in addition, the asymptotic behavior of the entropy solutions in the flow direction are also established.

Stability of Mach Configuration

Stability of Mach Configuration Suxing CHEN Fudan University sxchen@public8.sta.net.cn We prove the stability of Mach configuration, which occurs in moving shock reflection by obstacle or shock interaction