Title & Abstract

Abstract: This talk will discuss a parabolic-elliptic type repulsive chemotaxis-consumption model for (u,v) in an n-dimensional ball, where n is three or higher. Under no-flux for u and Dirichlet(v=M) boundary conditions, the model is characterized by a diffusion coefficient D(u) that extends the porous diffusion coefficient u^{m-1}. When m>1, global boundedness is shown for any M>0, and this result is extended to the case 0<m<=1 for sufficiently small M. On the other hand, when 0<m<2/n the system has blow-up for large M. This talk is based on joint work with K. Kang and D. Kim.

Abstract: A smart fluid is a fluid whose properties (e.g. viscosity) can be changed by an external stimulus (e.g. electric field). In this talk, we present existence results for smart fluids, particularly electrorheological fluids and chemically reacting fluids, which can be described as generalized non-Newtonian fluids with a variable power-law index.

Abstract: We study the stability of the time-periodic solution of the compressible Navier-Stokes equation in a 3D exterior domain, subject to time-periodic external forces that decay at spatial infinity. We first establish the existence of the time-periodic solution when an external force is small enough. We prove the global existence result of the initial value problem for the perturbation around the time-periodic solution, provided that the initial perturbation is sufficiently small. Furthermore, we derive the time decay estimate of the perturbation when an initial perturbation is small and belongs to some Lebesgue space. The uniform in time estimate of the semigroup in some Besov spaces on an exterior domain plays a crucial role in the proof.

Abstract: The three-dimensional incompressible Euler equations under axisymmetry have been widely studied. While the “no-swirl” assumption makes the system very similar to the two-dimensional vorticity equations, it is still possible for solutions to have unbounded vortex stretching. After reviewing classical confinement results in 2D and 3D, we report some progress on the issue of vortex stretching for axisymmetric Euler equations.

Abstract: In this talk, we are concerned with the Liouville type theorems for the steady incompressible magnetohydrodynamics (MHD) equations. We establish that the solution to the steady MHD equations is identically zero under the integrability assumptions on (v, b). We show that, in particular, a combination of a strong integrability condition on the velocity of a fluid and a weak integrability condition on the magnetic field gives a sufficient condition on the Liouville type theorems. Furthermore, we show that the combination of the growth condition of the potential for the fluid velocity and the integrability condition for the magnetic field leads to the triviality of the solution.  

Abstract: We study the singularity formation in a one-dimensional hyperbolic–parabolic chemotaxis model, which describes vascular network formation in early vasculature. We show that, while the cell density and velocity remain bounded, their gradients blow up in finite time. The chemoattractant concentration stays smooth. 

Abstract: The compressible Euler system is one of physical models which obeys the hyperbolic conservation law with no dissipation. It is known that the pure compressible Euler flows generally blows up in a finite time though, a remarkable work of Y. Guo discovered that the Poisson interaction force makes some oscillation and leads to some dispersion such as the Klein-Gordon effect. Such dispersion prevents the singularity formation phenomenon for the Euler--Poisson system and allows us to construct a global irrotational solution in the three dimensional case. In this talk, we consider the Euler system with the Riesz potential in 3D. The Riesz interaction serves as a generalization to the Poisson case and has been extensively studied in the physics literature. We investigate a dispersion feature of the linearized Euler--Riesz system. Unlike the Euler-Poisson case, the main difficulty in constructing the global irrotational solution arises from the singularity of the nonlinearity. We would like to explain a strategy to control the singularity, motivated by the work of Y. Guo and B. Pausader for ion dynamics model and present the recent result of the global irrotational solution to the 3D Euler--Riesz system.

Abstract: We consider the nonstationary Stokes system with zero initial data and nonzero localized boundary data in the half-space. We prove that there exist certain types of space-time localized influx through the boundary such that the resulting flow exhibits certain flow reversal phenomena in the sense that at least one of the components for the corresponding velocity field changes its sign away from the boundary. The boundary data are chosen to be only in the normal direction and the data is taken to be a product of nonnegative functions of the space variables and the time variable, the latter which loses its smoothness at the time when it vanishes. Using such boundary data, we are able to show flow reversal phenomena by the careful analysis of the solution formula, including its pointwise estimates, of the Stokes system in the half space based on the Green tensors introduced by Solonnikov(1977). To be more precise, under our construction, the tangential components of the velocity field exhibit at least one sign change, while the normal component exhibits at least two sign changes. In addition, the normal component has the opposite sign to the tangential components near the boundary, while it has the same sign to the tangential components sufficiently far from the origin. This is a joint work with Kyungkeun Kang and Tongkeun Chang.

Abstract: In this talk, we present a quantified analysis of the hydrodynamic limit of a relativistic quantum system coupled with self-consistent electromagnetic fields, focusing on the Maxwell–Klein–Gordon (MKG) system. We first establish global well-posedness of the MKG system under the Coulomb gauge, even in the presence of a self-interaction potential. To derive the hydrodynamic limit, we introduce a single scaling parameter that simultaneously governs the non-relativistic and semi-classical regimes by controlling both the speed of light and Planck’s constant. As this parameter tends to zero, we rigorously obtain quantified estimates for the asymptotic convergence of the electrostatic MKG system to the classical Euler–Poisson system, based on the modulated energy method.  

Abstract: In the present talk, we discuss properties and the asymptotic stability of spherically symmetric stationary solutions to the compressible Navier-Stokes equations in the exterior domain of a unit ball. We study two types of boundary conditions, i.e., inflow and outflow conditions. For both problems, we derive several properties of stationary solutions, especially convergence rates toward far fields. By making use of these rates, we derive a-priori estimate of the perturbation from the stationary solution in the suitable Sobolev space. It shows the asymptotic stability of the stationary solution. In this derivation, the relative energy form plays an essential role. For the outflow problem, we do not have to assume any smallness assumptions on the initial data for the outflow problem if it belongs to the suitable weighted Sobolev space. This result is shown by using the representation formula of the density. We also discuss the asymptotic stability with the non-symmetric initial perturbation from the stationary solution.

Abstract: Bernoulli's free boundary problem is an overdetermined boundary value problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. It will be shown that a family of free boundaries for variable boundary data is characterized as a non-local geometric flow of hypersurfaces. I will also talk about recent progress on a conjecture of Flucher and Rumpf that asserts the existence of a family of free boundaries concentrating at non-degenerate local minima of the Robin function.

Abstract: In this talk, we will discuss Leray-Hopf solutions to the incompressible Navier-Stokes equations with vanishing viscosity. We explore important features of turbulence, focusing around the anomalous energy dissipation phenomenon. As a related result, I will present a recent result proving that for two-dimensional fluids, assuming that  the initial vorticity is merely a Radon measure with nonnegative singular part, there is no anomalous energy dissipation. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. We will also discuss possible extensions to the viscous SQG equation in the context of Hamiltonian conservation and existence of weak solutions for a rough initial data. This is a joint work with MiKael Latocca (Univ. Evry) and Luigi De Rosa (GSSI).

Abstract: In this talk, we will discuss the well-posedness and some good structures in the Hall magnetohydrodynamics (Hall MHD) system based on joint works with Hantaek Bae (UNIST) and Kyungkeun Kang (Yonsei University). In the first part of the talk, we introduce the concept of the magneto-vorticity field and its regularity. Using this regularity, we show that the Hall MHD with 2D variables is globally well-posed, provided that only the vertical component of the initial current density is sufficiently small. In the second part, we discuss double Beltrami states in the Hall MHD. We first examine the double Beltrami states as a special class of steady solutions to the ideal Hall MHD, closely related to Beltrami flows in incompressible fluid dynamics. We then extend our analysis to time-dependent double Beltrami states in the viscous and resistive Hall MHD, exploring their exact form and stability properties.

Abstract: We review some of the concepts behind the hydrodynamic limit theory. Then we present some recent results, which include (1) the convergence of the ionic-Vlasov-Poisson equations to the pressureless/isothermal ionic Euler--Poisson equations, and (2) the convergence of a multi-species BGK-model to an isentropic two-phase fluid system. We also discuss some limitations of our method, and subjects for future study. The presentation is based upon joint works with Young-Pil Choi, Dowan Koo, and SeungYeon Cho, Byung-Hoon Hwang.

Abstract: We study the behavior of the scaling critical Lebesgue norm of finite time blow-up solutions to a nonlinear heat equation (the Fujita equation). It was proved in Mizoguchi-Souplet (2019) that the critical norm also diverges at the blow-up time under the assumption that the blow-up is of type I. Here, the blow-up is called type I if the blow-up rate is bounded by a spatially homogeneous solution up to the coefficient. In this talk, we focus on the case where the nonlinearity belongs to the Sobolev supercritical range, and then we show the critical norm blow-up without assuming that the blow-up is of type I. The proof is based on the blow-up argument developed by Escauriaza-Seregin-Šverák (2003) and Seregin (2012) for the three dimensional Navier-Stokes equations. Moreover, we also give estimates of the blow-up rate for the critical norm. The proof is based on the quantitative epsilon-regularity and quantitative Carleman inequalities. This kind of strategy originates from Tao (2021). This talk is based on a joint work with Tobias Barker (University of Bath) and Hideyuki Miura (Institute of Science Tokyo).

Abstract: Starting with the definition of Brakke flow, which is a generalized notion of mean curvature flow, I describe some of the recent results I obtained on the existence and regularity aspect of the problem. Most works are joint work with Lami Kim and Salvatore Stuvard.