# Registered Data

## [00733] Compressible fluid dynamics and related PDE topics

**Session Date & Time**:- 00733 (1/4) : 3C (Aug.23, 13:20-15:00)
- 00733 (2/4) : 3D (Aug.23, 15:30-17:10)
- 00733 (3/4) : 3E (Aug.23, 17:40-19:20)
- 00733 (4/4) : 4C (Aug.24, 13:20-15:00)

**Type**: Proposal of Minisymposium**Abstract**: This mini-symposium is aimed to bring together the leading experts as well as promising young researchers to present their recent results in compressible fluid dynamics and related PDE topics. Key topics focus on the most challenging open problems in the compressible fluid dynamics such as existence of solutions, asymptotic stability of wave pattern and singular limits, etc. It also provides a premier interdisciplinary forum for senior and junior researchers to exchange their experiences in the study of partial differential equations coming from compressible fluid dynamics.**Organizer(s)**: Feimin Huang, Song Jiang, Takayuki Kobayashi, Yong Wang**Classification**:__35L65__,__35Q35__,__35Q85__**Speakers Info**:- Jing Li (Academy of Mathematics and Systems Science, Chinese Academy of Science)
- Yong Wang (Academy of Mathematics and Systems Science, Chinese Academy of Science)
- Hirokazu Saito (The University of Electro-Communications)
- Yong Lv (Nanjing University)
- Xiangdi Huang (Academy of Mathematics and Systems Science, Chinese Academy of Science)
- Itsuko Hashimoto (Kanazawa University)
- Feng Xie (Shanghai Jiaotong University)
- Qian Yuan (Academy of Mathematics and Systems Science, Chinese Academy of Science)
- Kai Koike (Tokyo Institute of Technology)
- Xiaoding Shi (Beijing University of Chemical Technology)
- Yi Wang (Academy of Mathematics and Systems Science, Chinese Academy of Science)
- Qiangchang Ju (Institute of Applied Physics and Computational Mathematics)

**Talks in Minisymposium**:**[01289] Time-asymptotic stability of Riemann solutions to the compressible Navier-Stokes equations****Author(s)**:**Yi Wang**(AMSS, Chinese Academy of Sciences)

**Abstract**: The talk is concerned with our recent developments on the time-asymptotic stability of generic Riemann solutions to the one-dimensional compressible Navier-Stokes equations including both isentropic and full cases.

**[01339] Stability analysis of Prandtl expansions for two dimensional MHD equations in Sobolev spaces****Author(s)**:**Feng Xie**(Shanghai Jiao Tong University)

**Abstract**: In this talk, I will review briefly the classical Prandtl boundary layer expansions method to analyze the structure of viscous flow under the high Reynolds number. And then I will discuss the vanishing viscosity limit of solution to initial boundary value problem for both incompressible and compressible magneto-hydrodynamics (MHD) equations in Sobolev spaces, where the velocity is always imposed the no-slip boundary condition. The effects of the normal component of magnetic field, tangential component of magnetic field, magnetic diffusion and the change of density on the mathematical analysis for stability of MHD boundary layer and vanishing viscosity limit of solution to MHD equations will be addressed.

**[01378] Global Strong and Weak Solutions to Compressible MHD System****Author(s)**:**Xiaoding Shi**(Beijing University of Chemical Technology)

**Abstract**: The barotropic compressible magnetohydrodynamic equations in a general two-dimensional bounded simply connected domain is considered here. For initial density allowed to vanish, we prove that the initial-boundary-value problem of 2D compressible MHD system admits the global strong and weak solutions without any restrictions on the size of initial data.

**[01436] Inviscid Limit Problem of radially symmetric stationary solutions for compressible Navier-Stokes equation****Author(s)**:**itsuko hashimoto**(kanazawa university)- Akitaka Matsumura (Osaka university)

**Abstract**: The present paper is concerned with an inviscid limit problem of radially symmetric stationary solutions for an exterior problem in R^n (n≧2) to compressible Navier-Stokes equation, describing the motion of viscous barotropic gas without external forces, where boundary and far field data are prescribed. For both inflow and outflow problems, the inviscid limit is considered in a suitably small neighborhood of the far field state. For the outflow problem, we prove the uniform convergence of the Navier-Stokes flows to the Euler flows in the inviscid limit. On the other hand, for the inflow problem, we show that the Navier-Stokes flows uniformly converge to a superposition of boundary layer solution and Euler flows in the inviscid limit. The estimates of algebraic rate toward the inviscid limit are also obtained.

**[01614] Convergence Rate Estimates for the Low Mach and Alfven Number Three-Scale Singular Limit of Compressible Ideal Magnetohydrodynamics****Author(s)**:**Qiangchang JU**(Institute of Applied Physics and Computational Mathematics, Beijing)

**Abstract**: Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfven numbers tend to zero at different rates. The proofs use a detailed analysis of exact and approximate fast, intermediate, and slow modes together with improved estimates for the solutions and their time derivatives, and the time-integration method. This is a joint work with Bin Cheng and Steve Schochet

**[01617] Nonlinear asymptotic stability of vortex sheets with viscosity effects****Author(s)**:**Qian Yuan**(Chinese Academy of Sciences)

**Abstract**: In this talk, we can see that although a vortex sheet is not an asymptotic attractor for the compressible Navier-Stokes equations, a viscous profile that approximates the vortex sheet can be computed explicitly. It is shown that if the strength of vortex sheet is weak, then its associated viscous profile is asymptotically stable in the $ L^\infty $-norm with small initial perturbations for the compressible Navier-Stokes equations.

**[01660] Time-asymptotic expansion with pointwise remainder estimates for 1D viscous compressible flow****Author(s)**:**Kai Koike**(Tokyo Institute of Technology)

**Abstract**: We consider solutions to 1D compressible Naiver−Stokes equations around a constant steady state. We construct a time-asymptotic expansion with pointwise remainder estimates. The leading-order term of the expansion is the well-known diffusion wave and the higher-order terms are newly introduced family of waves which we call higher-order diffusion waves. Thanks to the pointwise remainder estimates, we can show that the expansion is valid for a fixed point $x$ and also in any $L^p(\mathbb{R})$-norm including the case of $1\leq p<2$. The proof is based on pointwise estimates of Green’s function.

**[01694] Asymptotic stability for the two-phase Navier-Stokes equations with surface tension and gravity****Author(s)**:**Hirokazu Saito**(The University of Electro-Communications)

**Abstract**: We consider the motion of two immiscible, viscous, incompressible capillary fluids, fluid$_+$ and fluid$_-$, in the presence of a uniform gravitational field acting vertically downward in $\mathbf{R}^N$ for $N \geq 3$. At the initial time, fluid$_-$ occupies a half-space-like domain such as oceans of infinite depth, while the complement of its closure is filled with fluid$_+$. The asymptotic stability of the trivial steady state is proved if fluid$_-$ is heavier than fluid$_+$.

**[01715] Global solutions on compressible Euler and Euler-Poisson equations****Author(s)**:**Yong Wang**(Academy of Mathematics and System Sciences, CAS, China)

**Abstract**: I will talk some results on the global existence of solutions to compressible Euler and Euler-Poisson of large intial data with spherical symmetry.

**[01716] Global solutions on compressible Euler and Euler-Poisson equations****Author(s)**:**Yong Wang**(Academy of Mathematics and System Sciences, CAS, China)

**Abstract**: I will talk some results on the global existence of solutions to compressible Euler and Euler-Poisson of large intial data with spherical symmetry.

**[01776] Some recent results on compressible Navier-Stokes equations****Author(s)**:**Jing Li**(AMSS, Chinese Academy of Sciences)

**Abstract**: We investigate the barotropic compressible Navier-Stokes equations with slip boundary conditions in a three-dimensional (3D) simply connected bounded domain, whose smooth boundary has a finite number of two-dimensional connected components. For any adiabatic exponent bigger than one, after obtaining some new estimates on boundary integrals related to the slip boundary conditions, we prove that both the weak and classical solutions to the initial-boundary-value problem of this system exist globally in time provided the initial energy is suitably small. Moreover, the density has large oscillations and contains vacuum states. Finally, it is also shown that for the classical solutions, the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially.

**[01785] Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable****Author(s)**:**Yong Lu**(NANJING UNIVERSITY)- Ping Zhang (Academy of Mathematics and System Sciences, CAS)

**Abstract**: We prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which is slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to 2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of constant state nor of small energy.

**[03229] Hidden structures behinds the compressible Navier-Stokes equations and its applications to the corresponding models****Author(s)**:**Xiangdi Huang**(Academy of Mathematics and Systems Science)

**Abstract**: In this talk, we will review the past developments on the solutions of the compressible Navier-Stokes equations and reveal the three hidden structures which linked the weak solution to the strong one. Based on these observations, we proved the Nash's conjecture in 1958s and establish global exsitence theory for both isentropic and heat-conductive compressible Navier-Stokes equations. Moreover, for the 3D compressible Navier-Stokes equations, we will show the existence of local weak solutions with higher regularity and local strong solutions with lower regularity. Also, we will mention the recent results on the blowup of the local strong solutions to the MHD equations in finite time and global existence of weak solutions of the compressible Navier-Stokes equations in bounded domains under Dirichlet boundary conditions.