Abstract : This mini-symposium will focus on the analysis of fluid dynamics and free boundary problems including the geometric evolution equations. We will put particular emphasis on the study of existence, uniqueness, regularity, global existence and stability, singularity formation of the modeling equations and the motion of free interfaces in Euclidean spaces or on manifolds. The study of fluid dynamics and free boundary problems have profoundly impacted many applied fields such as physics, biology and material sciences. Thus the analysis of these problems provides a critical and rigorous mathematical descriptions of the corresponding physical phenomena.
[03643] Transonic flows and free boundary problems in gas dynamics
Format : Talk at Waseda University
Author(s) :
Dehua Wang (University of PittsburghUniversity of Pittsburgh)
Abstract : In this talk, the transonic flows and free boundary problems in gas dynamics will be considered. The existence and stability of solutions will be presented for transonic flows past an obstacle and in a nozzle.
[03380] Regularity and asymptotics for porous medium equations in bounded domains
Format : Talk at Waseda University
Author(s) :
Tianling Jin (The Hong Kong University of Science and Technology)
Xavier Ros-Oton (Universitat de Barcelona)
Jingang Xiong (Beijing Normal University)
Abstract : We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time. This allows us to refine the asymptotics of solutions for large times. We establish faster rate of convergence and prove that the convergence holds in the regular topology.
[03966] Energy concentration and weak stability in fluid dynamics
Format : Talk at Waseda University
Author(s) :
Xianpeng Hu (City University of Hong Kong)
Abstract : The weak stability is an important issue in fluid dynamics. We will discuss the mathematical understudying of concentration phenomena in the framework of weak solutions with either critical or subcritical energy. Two typical examples, including two dimensional incompressible Euler equations and compressible Navier-Stokes equations, will be discussed.
[03949] On Ericksen-Leslie system with free boundary
Format : Talk at Waseda University
Author(s) :
Yong Yu (The Chinese University of Hong Kong)
Chenyun Luo (The Chinese University of Hong Kong)
Kaihui Luo (The Chinese University of Hong Kong)
Abstract : In this talk, we discuss a 3D simplified Ericksen-Leslie system subjected to the free boundary condition with surface tension. Dynamical stability of the classical planar wave solutions will also be addressed when the liquid crystal droplet is thin.
[00618] L1 maximalr regularity and its application to the Navier-Stokes equations
Format : Talk at Waseda University
Author(s) :
Yoshihiro Shibata (Department of Mathematics, Waseda University)
Abstract : I will talk about the L1 maximal regularity theorem to the Stokes equations and its application to free boundary problem for the Navier-Stokes equations.
Abstract : I will consider the motion of an incompressible viscous fluid on compact surfaces without boundary.
Local in time well-posedness is established in the framework of $L_p$-$L_q$ maximal regularity for initial values in critical spaces.
It will be shown that the set of equilibria consists exactly of the Killing vector fields. Each equilibrium is stable and any solution starting close to an equilibrium converges at an exponential rate to a (possibly different) equilibrium. In case the surface is two-dimensional, it will be shown that any solution with divergence free initial value in $L_2$ exists globally and converges to an equilibrium.
[04880] The Curve Shortening Flow for Immersed Curves
Format : Talk at Waseda University
Author(s) :
Patrick Guidotti (UC Irvine)
Abstract : We will revisit and study the curve shortening flow for immersed curves and its numerical computation.
[03690] On a thermodynamically consistent model for magnetoviscoelastic fluids in 3D
Format : Talk at Waseda University
Author(s) :
Hengrong Du (Vanderbilt University)
Yuanzhen Shao (University of Alabama)
Gieri Simonett (Vanderbilt University)
Abstract : In this talk, we consider a system of equations that model a non-isothermal magnetoviscoelastic fluid, which is thermodynamically consistent. The system is analyzed by means of the Lp-maximal regularity theory. First, we will discuss the local existence and uniqueness of a strong solution. Then it will be shown that a solution initially close to a constant equilibrium exists globally and converges to a (possibly different) constant equilibrium. Finally, we will show that that every solution that is eventually bounded in the topology of the natural state space exists globally and converges to the set of equilibria.
Abstract : In this talk, we consider some contact angle problems for the dynamic of fluids and discuss several topics such as existence and uniqueness of solutions as well as their qualitative behaviour.
[00258] The relativistic Euler equations with a physical vacuum boundary
Format : Talk at Waseda University
Author(s) :
Marcelo Mendes Disconzi (Vanderbilt University)
Abstract :
We consider the relativistic Euler equations with a physical vacuum boundary and an equation of state $p(\varrho)=\varrho^\gamma$, $\gamma > 1$. We establish the following results. i. local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; ii. low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; iii. stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions, in part by measuring the distance between their respective boundaries, is propagated by the flow; iv. we establish sharp, essentially scale invariant energy estimates for solutions; v. we establish a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in $L^1_t Lip_x$ and a suitable weighted version of the density is at the same regularity level. This is joint work with Mihaela Ifrim and Daniel Tataru.
