Abstract : This mini-symposium is aimed to bring together the leading experts as well as promising young researchers to present their recent results in partial differential equations with applications in fluid dynamics. Key topics focus on the most challenging open problems in the area such as global regularity, uniqueness of solutions, singular limits, boundary layers behavior, and free boundary problems, etc. It also provides a premier interdisciplinary forum for senior and junior researchers to exchange their experiences in the study of partial differential equations. The talks will span from analysis through modeling and computation to applications of partial differential equations.
Organizer(s) : Yachun Li, Ming Mei, Shinya Nishibata, Ronghua Pan
[04512] Two-Dimensional Riemann Problems: Transonic Shocks and Free Boundary Problems
Format : Online Talk on Zoom
Author(s) :
Gui-Qiang George Chen (University of Oxford)
Abstract : We are concerned with global solutions of multidimensional Riemann problems for nonlinear hyperbolic systems of conservation laws, focusing on their global configurations and structures. We present some recent developments in the rigorous analysis of two-dimensional Riemann problems involving transonic shock waves and free boundary problems through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations.
[05347] Global stability of steady supersonic flow for 1D Compressible Euler system
Format : Talk at Waseda University
Author(s) :
Jianli Liu (Shanghai University)
Abstract : It is more important to consider the stability of compressible flow with some phyical effects. In this talk, we will give the global nonlinear stability of steady supersonic flows for one dimensional unsteady compressible Euler systems with physical effect, such as a nonlinear damping representing frictions, heat transfer term or mass addition.
[04192] Scaling limit of vortex dynamics on the filtered-Euler flow
Format : Talk at Waseda University
Author(s) :
Takeshi Gotoda (Tokyo Institute of Technology)
Abstract : We consider weak solutions of the 2D filtered-Euler equations, which describe a regularized Euler flow. We show that, in the limit of the filtering scale, filtered weak solutions converge to weak solutions of the 2D Euler equations and an energy dissipation rate for the filtered weak solution converges to zero for initial vorticity in a certain class.
[05331] Characteristic Decomposition for Hyperbolic System
Format : Talk at Waseda University
Author(s) :
Wancheng Sheng (Shanghai University)
Abstract : In this talk, we show the method of characteristic decompositions for hyperbolic conservation laws. By this methods, we give some results on the multidimensional Riemann problems of compressible Euler equations.
[04072] Global Finite-Energy Solutions of the Compressible Euler-Poisson Equations for General Pressure Laws with Spherical Symmetry
Format : Talk at Waseda University
Author(s) :
Feimin Huang (Academy of Mathematics and Systems Science, Chinese Academy of Science)
Abstract : We are concerned with global finite-energy solutions of the three-dimensional compressible Euler-Poisson equations with gravitational potential and general pressure law,
especially including the constitutive equation of white dwarf stars. In this paper, we construct a global finite-energy solution with spherical symmetry of the Cauchy problem for the Euler-Poisson
equations as the vanishing viscosity limit of the corresponding compressible Navier-Stokes-Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through
the compensated compactness analysis and uniform estimates in $L^p$ via several new main ingredients. A new key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum ($\rho=0$) and far-field ($\rho=\infty$) are carefully analyzed. Owing to the generality of the pressure law,
only the $W^{-1,p}_{\mathrm{loc}}$-compactness of weak entropy dissipation measures
with $p\in [1,2)$ can be obtained; this is rescued by the equi-integrability of weak entropy pairs
which can be established by the estimates obtained above, so that the div-curl lemma still applies.
Finally, based on the above analysis of weak entropy pairs, the $L^p$ compensated compactness framework for the compressible Euler equations with general pressure law is established.
This new compensated compactness framework and the techniques developed in this paper
should be useful for solving further nonlinear problems with similar features.
[04789] Global-in-time quasi-neutral limit for a two-fluid Euler-Poisson system
Format : Talk at Waseda University
Author(s) :
Yue-Jun Peng (Université Clermont Auvergne)
Abstract : We consider Cauchy problem for a two-fluid Euler-Poisson system where
the single parameter is the Debye length.
When the initial data are sufficiently close to constant
equilibrium states, we show the uniform global existence of smooth solutions
and justify the convergence of the system to compressible Euler equations
with damping as the Debye length tends to zero. We also establish global
error estimates of the solutions. A key step of the proof is to control
the quasi-neutrality of the velocities by using a projection operator.
[03623] A compressible two-fluid model with unequal velocities: existence and uniqueness
Format : Talk at Waseda University
Author(s) :
Huanyao Wen (South China University of Technology)
Abstract : In this talk, we will introduce our recent works on the existence and uniqueness theory of a compressible two-fluid model with unequal velocities. The viscosity coefficients depend on the density functions, which can be degenerate.
[04539] On the Stability of Outflowing Compressible Viscous Gas
Format : Talk at Waseda University
Author(s) :
Yucong Huang (University of Edinburgh)
Shinya Nishibata (Tokyo Institute of Technology)
Abstract : I will discuss the long-time stability of a spherically symmetric motion of outflowing isentropic and compressible viscous gas. The fluid occupies unbounded exterior domain, and it is flowing out from an inner sphere centred at the origin. In this talk, I will show that, for a large initial data, the solution will converge to the stationary solution as time goes to infinity. This is a joint work with S. Nishibata.
[04434] Hyperbolic Cattaneo-Approximation of the compressible Navier-Stokes-Fourier system
Format : Talk at Waseda University
Author(s) :
Jiang Xu (Nanjing University of Aeronautics and Astronautics)
Abstract : We will talk about the Cattaneo-Chistov approximation of the compressible non-isentropic Navier-Stokes system in whole space. First, we establish a global well-posedness of the Navier-Stokes-Cattaneo-Christov system uniformly with respect to the relaxation parameter. Then we justify the strong convergence of the solution toward that of the compressible Navier-Stokes system, and the explicit convergence rates are also exhibited.
[04767] Local regularity conditions on initial data for local energy solutions of the incompressible Navier-Stokes equations
Format : Talk at Waseda University
Author(s) :
Hideyuki Miura (Tokyo institute of technology)
Kyungkeun Kang (Yonsei university)
Tai-Peng Tsai (University of British Columbia)
Abstract : We study the regular sets of local energy solutions to the Navier-Stokes equations in terms of conditions on the initial data. It is shown that if a weighted L2 norm of the initial data is finite, then all local energy solutions are regular in a region confined by space-time hypersurfaces determined by the weight. This result refines and generalizes Theorems C and D of Caffarelli, Kohn and Nirenberg (1982).
[04554] Sharp non-uniqueness of weak solutions to viscous fluids
Format : Talk at Waseda University
Author(s) :
Yachun Li (Shanghai Jiao Tong University)
Zirong Zeng (Shanghai Jiao Tong University)
Deng Zhang (Shanghai Jiao Tong University)
Peng Qu (Fudan University)
Abstract : In this talk, I will present our recent results about non-uniqueness of weak solutions to some viscous fluid models. For incompressible Navier-Stokes equations , we proved the sharp non-uniqueness of weak solutions at two endpoints of the Ladyžhenskaya-Prodi-Serrin (LPS) criteria even in hyper-viscous regime. For MHD equations, we prove the sharp non-uniqueness near one endpoint of the LPS condition. Furthermore, the strong vanishing viscosity and resistivity result is obtained, it yields the failure of Taylor’s conjecture along some sequence of weak solutions. For hypo-viscous compressible Navier-Stokes equations, we prove that there exist infinitely many different weak solutions with the same initial data. This provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Our proof are based on the spatial-temporal intermittent convex integration scheme. These are joint works with Yachun Li, Peng Qu and Deng Zhang.
[05340] On controllability of the incompressible MHD system
Format : Talk at Waseda University
Author(s) :
Yaguang Wang (Shanghai Jiao Tong University)
Abstract : In this talk, we shall introduce our recent study on the controllability of the initial boundary value problem for the incompressible magnetohydrodynamic systems. For the two-dimensional ideal incompressible MHD system, we obtained the global exact controllability by using the return method, and for the two- and three-dimensional viscous MHD systems with coupled Navier slip boundary condition, we deduced the global approximate controllability. This is a joint work with Manuel Rissel.
