Junsik Bae (Kyungpook National University)
Title: Spectral stability of steady solutions in some shallow water models
Abstract: We consider some shallow water models that admit various steady solutions depending on the prescribed boundary conditions and bottom topography. In particular, discontinuous steady solutions arise when the flow undergoes a transition from a supercritical to a subcritical regime, whereas continuous transcritical solutions occur in transitions from subcritical to supercritical flow. We investigate the linear stability of these steady flows. This talk is based on the ongoing project with Miguel Rodrigues (University of Rennes) and Nina Aguillon (Sorbonne University).
Renjun Duan (The Chinese University of Hong Kong)
Title: On a free boundary problem of the Boltzmann equation
Abstract: We concern a free boundary value problem for the Boltzmann equation that models the interaction of a rarefied gas with a moving wall --- a classical piston problem in kinetic theory. While free boundary problems for fluid dynamic equations have been extensively investigated, their counterparts for kinetic models remain largely open due to the intrinsic difficulties caused by limited regularity and the strong coupling between particle transport and boundary motion. In the talk, we report a recent result on the topic with focus on the one-dimensional stability. Joint with Shengchuang Chang and Shuangqian Liu.
Sungho Han (Korea Advanced Institute of Science and Technology)
Title: Large-Time Behavior towards Composite Waves of Degenerate Shock and Rarefaction Wave for Modified KdV—Burgers Equation
Abstract: In this talk, we study the large-time behavior of solutions to the modified Korteweg-de Vries-Burgers equation $u_t+(u^3)_x=\mu u_{xx}-\kappa u_{xxx}$ with $\mu>0$ and $\kappa>0$. Since the flux is non-convex, the inviscid Riemann problem may be solved by a composite wave of a degenerate Oleinik shock and a rarefaction wave. First, we establish structural properties of the degenerate viscous-dispersive shock profile in the monotone regime. Then, by using these properties, we prove the time-asymptotic stability of the corresponding viscous-dispersive composite wave, up to a time-dependent shift, under small $H^1$ perturbations. The shock strength $\delta_S$ and the rarefaction strength $\delta_R$ need not be small, and are restricted only by the ratio $\delta_R\le\frac{5}{18}\delta_S$ and a smallness assumption on $\kappa\delta_S^2/\mu^2$. Our approach relies on the method of $a$-contraction with shifts developed by Kang and Vasseur. This is joint work with Namhyun Eun and Jeongho Kim.
In-Jee Jeong (Korea Institute for Advanced Study)
Title: Stability of vortex dipoles
Abstract: We consider the existence, uniqueness, and geometric properties of vortex dipoles, which are pairs of steadily-translating and counter-rotating vortices. First, we make precise the notion that the Lamb dipole, an explicit traveling wave solution described by Lamb back in 1895, is the dipole with "maximal mass" under enstrophy and impulse constraints. Then we discuss rigidity of dipoles near the Lamb dipole. Lastly, we discuss stability of multiple Lamb dipoles, allowing for dipoles with mixed signs. This is based on joint works with Ken Abe, Kyudong Choi, Guolin Qin, Yao Yao, and Weicheng Zhan.
Yoshiyuki Kagei (Institute of Science Tokyo)
Title: Decay properties of the compressible Navier-Stokes system on the half space
Abstract: In this talk I will consider the large time behavior of small strong solutions of the compressible Navier-Stokes system on a half space under the nonslip boundary condition around the motionless state. It is shown that the L^\infty norm of the spatial derivatives of the perturbation of the motionless state decays slower than that in the case of the Cauchy problem on the whole space when the initial perturbation is sufficiently small in some L^2- Soblev space. The decay rate is optimal in a certain sense. I also consider the asymptotic expansion of the perturbation if, in addition, the initial perturbation belongs to L^1. This talk is based on joint works with T. Kobayashi (Univ. Osaka), R. Nakasato (Shinshu Univ.) and M. Okita (Kurume College).
Kyungkeun Kang (Yonsei University)
Title: Separation oints and pressure gradients for the nonstationary Stokes system in the hlaf-space with localized boundary data
Abstract: We consider the Stokes system in the half-space with localized boundary data. We show that there exist boundary influxes for which the resulting flow exhibits flow reversal and boundary-layer separation. In the case of boundary-layer separation, we further investigate the dynamics and asymptotic behavior of the separation point. Moreover, through a perturbative argument, we construct solutions to the Navier–Stokes equations in the half-space that exhibit the same qualitative behavior as in the Stokes case. This is a joint work with T.-K. Chang and C.-H. Min.
Yachun Li (Shanghai Jiao Tong University)
Title: Global well-posedness and asymptotic behavior of large strong solutions to compressible fluids with variable coefficients
Abstract: In this talk, we discuss the 3D compressible Navier-Stokes equations with density-dependent or temperature-dependent coefficients in the whole space. We establish the global existence of large strong solution when the initial density and the initial temperature are linearly equivalent to some large constant states. Moreover, the optimal decay rates of the solution to its associated equilibrium are established. This is a joint work with Peng Lu and Zhaoyang Shang.
Yasunori Maekawa (Kyoto University)
Title: On stationary flow around a rotating cylinder with free boundary
Abstract: In this talk I will present some results from a joint project with Chun Liu (City University of Hong Kong) and Zhirun Zhan (Kyoto University) about the PDE system related to the stationary flow around a rotating cylinder in the presence of a free boundary.
Hideyuki Miura (Institute of Science Tokyo)
Title: On propagation and loss of regularity for inviscid SQG in the half-plane
Abstract: We consider the SQG equation without dissipation on the half-plane with the Dirichlet boundary condition, and prove local existence and uniqueness of solutions in C^{2,\beta} for any 0<\beta<1. We also show that for a class of smooth initial data, the unique corresponding solution does not belong to W^{3,\infty}. This is joint work with In-Jee Jeong and Junha Kim.
Miho Murata (Shizuoka University)
Title: Global solvability and the zero activity limit for the active nematic liquid crystal model
Abstract: Active nematics are orientationally ordered materials whose particles consume internal energy and generate internal stresses, leading to spontaneous flows. In this talk, we consider the active Beris-Edwards model describing the motion of active nematic liquid crystals in the whole space. We prove the global solvability of the system for sufficiently small initial data. Our approach is based on the maximal Lp-Lq regularity and decay estimates for analytic semigroup, which yield suitable time-weighted estimates for the nonlinear problem. As a consequence of the global estimates, we also discuss the active-to-passive limit: as the activity parameter tends to zero, the global solution of the active system converges to that of the corresponding passive Beris-Edwards model.
HyeonSeop Oh (Korea Advanced Institute of Science and Technology)
Title: Long-time dynamics toward a generic composite wave for the inflow problem of the Navier-Stokes-Fourier system
Abstract: In this talk, we study the asymptotic stability of solutions to the inflow problem for the one-dimensional Navier-Stokes-Fourier system on the half-line. Unlike in the whole-space problem, a boundary layer solution appears as an asymptotic profile due to the presence of the boundary. We consider the generic wave pattern: a superposition of a degenerate boundary layer, a rarefaction wave, a viscous contact wave, and a viscous shock. We prove that this wave pattern is asymptotically stable up to a time-dependent shift for the shock, provided that the initial perturbation and the strengths of the boundary layer, contact wave, and shock are sufficiently small. The rarefaction wave, however, is allowed to have arbitrarily large strength. This talk is based on joint work with Xushan Huang (Yonsei University), Moon-Jin Kang (KAIST), and Hobin Lee (Seoul National University).
Jaemin Park (Yonsei University)
Title: Rigidity and flexibility for 2D inviscid Boussinesq system in a periodic channel
Abstract: We study steady solutions of the two-dimensional inviscid Boussinesq system in a periodic channel. Our main interest is to understand when steady states near shear flows are rigid, and when nontrivial nearby structures can occur. We present several rigidity and flexibility results, highlighting the roles of stable stratification and stagnation points. In particular, we discuss situations in which laminar steady states are forced to be shear flows, as well as regimes where non-shear and even non-laminar steady states can bifurcate from shear profiles. This is a joint work with Marc Nualart (ICMAT) and Javier Penafiel-Tomas (EPFL).
Wanyong Shim (Korea Advanced Institute of Science and Technology)
Title: Self-similar singularity formation: a general framework and applications to nonlocal transport equations
Abstact: Finite-time gradient blow-up in nonlinear evolution equations can produce cusp singularities with different sharp H\"older regularities. In various models, these regularities appear to be determined by the leading-order equations governing the dynamics near blow-up. Motivated by this observation, we develop a general framework for self-similar singularity formation governed by a parametrized family of canonical leading-order equations. A central difficulty is that the self-similar scaling and corresponding profiles vary with the parameter, so standard approaches based on a fixed profile do not directly apply. To overcome this, we establish parameter-dependent profile estimates and combine them with a generalized maximum principle in a unified bootstrap argument. This yields a general theorem on singularity formation under suitable model-specific hypotheses. As an application, we prove finite-time gradient blow-up with sharp $C^{(5-b)/(b+3)}$ H\"older regularity for a class of nonlocal transport equations extending the $b$-family throughout the range $1<b<5$.
Ryo Takada (The University of Tokyo)
Title: Lower bounds for the lifespan of solutions to the Euler--Korteweg equations
Abstract: We consider the initial value problem for the Euler--Korteweg equations in the class of irrotational flows. In space dimensions two and higher, we establish a quantitative lower bound for the lifespan of solutions that explicitly exhibits its dependence on the Weber number, the Mach number, and the Sobolev norm of the initial perturbation. In particular, this lower bound can be taken arbitrarily large in a low-Weber-number regime under a suitable algebraic relation between the Mach and Weber numbers. This talk is based on joint work with Haruhiro Kurabe and Tomoaki Yoshizawa (The University of Tokyo).
Chunjing Xie (Shanghai Jiao Tong University)
Title: Some classifications of solutions for the Euler system and their applications
Abstract: The steady solutions for the Euler system are not only used to describe the possible asymptotic states of unsteady flows, but also used to describle some important physical process. Since the Euler system is translation and rotation invariant, a natural question is whether steady solutions inherit the symmetry when the domain has certain symmetry. We will discuss our recent results on classifications of these solutions in terms of the flow angles. Furthermore, we also show that the solutions are rotation invariant when the vorticity is differentiable, bounded, and integrable. Finally, as the steady solutions of the Euler system may not be unique in a given domain, we investigate the selection principle of the steady solutions via the vanishing viscosity limit.
Tong Yang (The Hong Kong Polytechnic University)
Title: Theories for kinetic equations with scaling invariance
Abstract: In this talk, we will first discuss the well-posedness of the Boltzmann without angular cutoff and Landau equations in critical function spaces—a fundamental open problem in kinetic theory. The main observation lies in the introduction of a novel anisotropic norm adapted to the intrinsic scaling invariance of the equations, which provides precise control over the high-frequency behavior of solutions. Then we will present a study on the Green’s function in short time with a nonnegative Hölder background profile. Finally, we will study the global forward self-similar solutions to Boltzmann equation with cutoff Maxwell molecules. The talk is based on some recent joint work with Ke Chen, Quoc-Hung Nguyen and Jiaqi Yang.
Zhao Yang (Chinese Academy of Sciences)
Title: Recent Progress in Inclined Thin-Film Flows
Abstract: The inviscid Saint-Venant equations serve as a canonical model for inclined thin-film flows. These equations support two distinct types of traveling-wave solutions: Dressler’s periodic roll waves and hydraulic shocks. Periodic roll waves emerge when uniform laminar flows become unstable, while hydraulic shocks develop from the sudden release of a dam separating fluids of different depths. In both scenarios, the maximum fluid heights can greatly exceed those of the corresponding steady laminar flows with the same discharge, leading to potentially severe water overflow. Hence, the existence and stability of these waves attract attention in both hydraulic engineering and mathematics. In this talk, we review recent longitudinal stability results for both types of waves, as well as multidimensional stability results for planar waves considered as solutions to the two-dimensional Saint-Venant equations. We will also discuss current difficulties encountered when constructing permanent flows in two dimensions and other related aspects.