Thursday (November 6th)
Yoshiyuki Kagei (Institute of Science Tokyo)
Title : A remark on the asymptotic behavior of solutions of the compressible Navier-Stokes system on a half space
Abstract : In this talk I will consider the asymptotic behavior of strong solutions of the compressible Navier-Stokes system on a half space under the nonslip boundary condition. It is known that the asymptotic leading term of the solution is given by a solution of the linearized system. I will show that the second order term in the asymptotics is given by a term caused by a nonlinear effect, which does not appear in the asymptotics of solutions on the whole space. This talk is based on a joint work with T. Kobayashi (Univ. Osaka), R. Nakasato (Shinshu Univ.) and M. Okita (Kurume College).
Jan Brezina (Kyushu University)
Title : On physically grounded boundary conditions for MHD
Abstract : We consider a general compressible MHD system, where the fluid occupies a bounded domain with a solid object immersed in it namely, Ω_F = Ω\Ω_int, and the induction equation holds also outside the fluid domain. In addition, the transport coefficients in the system possibly take different values within or without the fluid domain, i.e., the magnetic field propagates in a heterogeneous medium. Using suitable penalization in terms of the transport coefficients we perform several singular limits to show the following:
1. A rigorous justification of physically grounded boundary conditions for the compressible MHD system on a bounded domain.
2. Existence of weak solutions for arbitrary finite energy initial data in the situation the Maxwell induction equation holds also outside the fluid domain.
3. A suitable theoretical platform for numerical experiments on domains with geometrically complicated boundaries
Bongsuk Kwon (UNIST) + Masahiro Suzuki (Nagoya Institute of Technology)
Title : Stationary flows for viscous heat-conductive fluid in a perturbed half-space
Abstract : In this talk, we consider the non-isentropic compressible Navier–Stokes equation in a perturbed half-space with an outflow boundary condition as well as the supersonic condition. This equa tion models a compressible viscous, heat-conductive, and Newtonian polytropic fluid. We show the unique existence of stationary solutions for the perturbed half-space. The stationary solution depends on all directions and has multidirectional flow. We also prove the asymptotic stability of this stationary solution. This talk is based on a joint work with Professor Mingjie Li (Minzu Univ. of China) and Professor Katherine Zhiyuan Zhang (Northeastern Univ.).
[1] M. Li, K. Z. Zhang, and M. Suzuki, Stationary flows for viscous heat-conductive fluid in a perturbed half-space, arXiv:2410.14511.
Masashi Ohnawa (Tokyo University of Marine Science and Techonology)
Title : On the stability of steady gas flows in nozzles
Abstract : This study addresses the stability issue of steady gas flows in nozzles of finite length. The model is a quasi-one-dimensional Euler system assuming a barotropic equation of state. For convergent divergent nozzles typically used in industries, various types of steady flows appear depending on the pressure at the exit. Our main results claim that each type of stationary solution is asymptotically stable to small perturbations regardless of the magnitude of the derivative of the cross-section of the nozzle and the value of the pressure at the exit. This research is based on joint work with Masahiro Suzuki (Nagoya Institute of Technology).
I-Kun Chen (National Taiwan University)
Title : On the existence and regularity of weakly nonlinear stationary Boltzmann equations: A Fredholm alternative approach
Abstract : The celebrated Fredholm alternative theorem works for the setting of identity compact operators. This idea has been widely used to solve linear partial differential equations. In this article, we demonstrate a generalized Fredholm theory in the setting of identity power compact operators, which was suggested in Cercignani and Palczewski to solve the existence of the stationary Boltzmann equation in a slab domain, [1]. We carry out the detailed analysis based on this generalized Fredholm theory to prove the existence theory of the stationary Boltzmann equation in bounded three-dimensional convex domains. To prove that the integral form of the linearized Boltzmann equation satisfies the identity power compact setting requires the regularizing effect of the solution operators. Once the existence and regularity theories for the linear case are established, with suitable bilinear estimates, the nonlinear existence theory is accomplished. This talk is based on a collaborative work with Daisuke Kawagoe and Chun-Hsiung Hsia, [2].
[1] Cercignani, Carlo; Palczewski, Andrzej : Existence and uniqueness for nonlinear boundary value problems in kinetic theory. J. Statist. Phys. 46 (1987), no.1-2, 273-281.
[2] I-Kun Chen, Chun-Hsiung Hsia, Daisuke Kawagoe: On the existence and regularity of weakly nonlinear stationary Boltzmann equations : a Fredholm alternative approach arXiv:2501.02419.
HyeonSeop Oh (KAIST)
Title : Uniqueness and weak-BV stability for the isentropic Euler system: inflow and outflow problems
Abstract : In this talk, we consider initial-boundary value problems for the one-dimensional isentropic Euler system in the half-space, focusing on both inflow and outflow problems. We prove that small BV solutions in the subsonic region are unique and stable within a wild class of weak solutions satisfying the so-called strong trace property. In particular, we establish quantitative Hölder-type stability estimates in the L^2 norm. While the small BV solutions, being in the subsonic region, are associated with non-characteristic boundaries, the wild solutions admit characteristic boundaries. Our analysis is based on the a-contraction with shift method, although the stability results do not depend on the shift. This talk is based on a joint work with Moon-Jin Kang, Jiayun Meng, and Alexis Vasseur.
Namhyun Eun (KAIST)
Title : Uniqueness and stability of Riemann shocks to the full Euler system
Abstract : In this talk, we will discuss the stability of a Riemann shock solution to the compressible Euler system, which is a self-similar entropy shock connecting two distinct constant states, in a physical class of vanishing viscosity limits. We focus on the one-dimensional full Euler system and consider the Brenner-Navier-Stokes-Fourier system, proposed as an amendment of the Navier-Stokes-Fourier system, to describe the physical perturbation class. The proof is based on the method of a-contraction with shiftws, and we will also comment on future directions. This is a joint work with Moon-Jin Kang (KAIST) and Saehoon Eo (Stanford University), [1].
[1] Saehoon Eo, Namhyun Eun, Moon-Jin Kang. Stability of a Riemann shock in a physical class: from Brenner-Navier-Stokes-Fourier to Euler. arXiv:2501.02419.
Yoshihiro Ueda (Kobe University)
Title : Stability of the composite wave for the scalar viscous conservation law
Abstract : In this talk, we consider the stability of the composite wave for the scalar viscous conservation laws. Especially, we focus on the case that the flux function is non-convex. Then the corresponding Riemann problem admits a Riemann solution which consists of an Oleinik shock and a rarefaction wave. In this situation, we will show the asymptotic stability of the composite wave of the viscous Oleinik shock and the rarefaction wave. This is a joint research with Masaya Kageura from Kobe University.
Friday (November 7th)
Bum Ja Jin (Mokpo National University)
Title : Analysis of Fluid–Structure Operators in Exterior Flows with Navier Slip Conditions
Abstract : We investigate the linearized dynamics of a rigid body immersed in an incompressible viscous fluid filling the entire space, subject to Navier slip boundary conditions. While semigroup regularity and decay estimates are well established for the classical no-slip case, analogous results under slip conditions have been largely confined to bounded domains. In this work, we establish strong L^q -regularity, bounded analyticity, and optimal L^q − L^r decay estimates for the associated fluid-structure operator in the exterior setting. These findings lay a rigorous foundation for constructing strong W^(2,1)_q solutions and Kato-type solutions to the corresponding nonlinear fluid-structure interaction system.
Hobin Lee (KAIST)
Title : Long-time behavior of composite wave of planar viscous shocks for the 3D Barotropic Navier-Stokes equations
Abstract : We will present the latest result on long-time behavior of solutions to the 3D compressible Navier-Stokes equations with initial H2-small perturbation of Riemann data. Especially, we consider the Riemann data generating Riemann solution composed of planar shocks with small amplitudes. We prove that the solution of Navier-Stokes system converges, uniformly in space, towards a composition of two planar viscous shock waves as time goes to infinity, up to dynamical shifts.
Xushan Huang (KAIST)
Title : Stability of viscous shock for the Navier-Stokes-Fourier system: outflow and impermeable wall problems
Abstract : We investigate the time-asymptotic stability of solutions to the one-dimensional Navier-Stokes-Fourier system in the half space, focusing on the outflow and impermeable wall problems. When the prescribed boundary and far-field conditions form an outgoing viscous shock, we prove that the solution converges to the viscous shock profile, up to a dynamical shift, provided that the initial perturbation and the shock amplitude are sufficiently small. In order to obtain our results, we employ the method of a-contraction with shifts. Although the impermeable wall problem is technically simpler to analyze in Lagrangian mass coordinates, the outflow problem leads to a free boundary in that framework. Therefore, we use Eulerian coordinates to provide a unified approach to both problems. This is the first result on the time-asymptotic stability of viscous shocks for initial-boundary value problems of the Navier-Stokes-Fourier system for the outflow and impermeable wall cases.
[1] Xushan Huang, Hobin Lee and HyeonSeop Oh. Stability of viscous shock for the Navier-Stokes-Fourier system: outflow and impermeable wall problems. arXiv:2509.02215.
Saturday (November 8th)
Shinya Nishibata (Institute of Science Tokyo)
Title : Spherically symmetric stationary solutions to the heat-conductive compressible Navier-Stokes equation for outflow problem
Abstract : In the present talk, we discuss the properties and the asymptotic stability of spherically symmetric stationary solutions to the heat-conductive compressible Navier-Stokes equations in the exterior domain of a unit ball with out-flowing boundary condition. The existence of the stationary solution is proved by Professor Akitaka Matsumura and Itsuko Hashimoto. We derive several properties of the stationary solution, especially convergence rates towards the far field. By making use of these rates, we derive a-priori estimate of the perturbation from the stationary solution in the suitable weighted Sobolev space. It shows the asymptotic stability of the stationary solution. In this derivation, the relative energy form plays an essential role. We also discuss several results on the isentropic flow.
Yoonjung Lee (Yonsei University)
Title : Global smooth solutions to the irrotational Euler-Riesz system in 3D
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 [1] 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 and allows us to construct a global irrotational solution for the Euler–Poisson system in 3D. In this talk, we are interested in the Euler system with the Riesz potential as a generalization to the Poisson case. 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 [2] for ion dynamics model and present the recent result of the global irrotational solution to the 3D Euler–Riesz system. This work is a joint work with Young-Pil Choi and Jinwook Jung.
[1] Y. Guo, Smooth irrotational flows in the large to the Euler-Poisson system in R^3+1, Comm. Math. Phys., 195, (1998), 249-265.
[2] Y. Guo and B. Pausader, Global Smooth Ion Dynamics in the Euler-Poisson System, Comm. Math. Phys., 303, (2011), 89-125.
In-Jee Jeong (Seoul National University)
Title : On Keller–Segel systems of consumption type
Abstract : We consider the Keller-Segel system of consumption type, potentially coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up in the C^2 norm ([1]).
[1] In-Jee Jeong and Kyungkeun Kang. Wellposedness and singularity for inviscid Keller–Segel–fluid system of consumption type, Comm. Math Phys., 390, 1175–1217 (2022)
Jeongho Kim (Kyung Hee University)
Title : On the time-asymptotic stability of viscous-dispersive shock wave for the capillary flow
Abstract : In this talk, I will introduce basic concepts of capillary fluids, called the Navier–Stokes–Korteweg system, and present time-asymptotic stability of it. In particular, we are interested in a stability of viscous-dispersive shock wave, and its composition with a rarefaction wave. We will show that, under the smallness assumption of the shock strength and initial perturbation, the solution to the Navier–Stokes–Korteweg system converges to the viscous-dispersive shock wave with dynamic shift (and its composition with the rarefaction wave). Our method is based on the celebrated a-contraction with shift method developed by Kang and Vasseur. This talk is based on the joint works [1, 2] with Prof. Moon-Jin Kang, Dr. Sungho Han, and Dr. Hobin Lee.
[1] S. Han, M.-J. Kang, J. Kim, and H. Lee, Long-time behavior towards viscous-dispersive shock for Navier– Stokes equations of Korteweg type. J. Differential Equations 426 (2025) 317–387.
[2] S. Han and J. Kim, Time-asymptotic stability of composite wave for the one-dimensional compressible fluid of Korteweg type. To appear in SIAM. J. Math. Anal.
Naoto Deguchi (Institute of Science Tokyo)
Title : Existence and stability of time-periodic solutions of the compressible Navier-Stokes equations in a 3D exterior domain
Abstract : We study the time-periodic problem for the compressible Navier-Stokes equation in a three dimensional exterior domain, subject to an external force which decays at spatial infinity. We obtain the existence result of the time-periodic solution when an external force is small enough. We also prove the global existence and the time decay estimate for a solution of the initial value problem of the perturbation equation around the time-periodic solution, under the smallness assumption for the initial perturbation. The uniform-in-time estimate in Besov spaces on an exterior domain plays a crucial role in the proof.
Wanyong Shim (KAIST)
Title : Stability of shock profiles for the Navier–Stokes–Poisson system
Abstract : We consider the Navier–Stokes–Poisson (NSP) system, which describes the dynamics of positive ions in a collision-dominated plasma. The NSP system admits a one-parameter family of smooth traveling wave solutions, known as shock profiles. In this talk, I will present my research on the stability of these shock profiles. Our analysis is based on the pointwise semigroup method, which combines spectral analysis with Green’s function estimates. We first establish the spectral stability of the shock profiles. Building upon this, we derive pointwise bounds on the Green’s function for the associated linearized problem, which yield linear and nonlinear asymptotic orbital stability.
Sungho Han (KAIST)
Title : Stability of Viscous–Dispersive Shock Waves via the Method of a-Contraction with Shifts
Abstract : In this talk, we present a detailed analysis of the time-asymptotic stability of viscous–dispersive shock waves for the one-dimensional compressible Navier–Stokes–Korteweg system. Building upon the general overview by Prof. Jeongho Kim, we focus on the proof based on the method of acontraction with shifts. We will explain how the weighted relative entropy structure is utilized to establish global-in-time convergence toward the viscous-dispersive shock and its composite wave structure. A distinctive feature compared to the classical Navier–Stokes system is the necessity of higher-order control in the energy estimates. The standard L^2 -level relative-entropy estimate fails to close due to the persistence of an H^2 term, which requires us to propagate and close uniform H^2-estimates for the velocity component to complete the energy estimate. This talk is based on the joint works [1, 2] with Prof. Moon-Jin Kang, Prof. Jeongho Kim, and Dr. Hobin Lee.
[1] S. Han, M.-J. Kang, J. Kim, and H. Lee, Long-time behavior towards viscous–dispersive shock for Navier–Stokes equations of Korteweg type, J. Differential Equations 426 (2025), 317–387.
[2] S. Han and J. Kim, Time-asymptotic stability of composite wave for the one-dimensional compressible fluid of Korteweg type, to appear in SIAM J. Math. Anal.
Yunjoo Kim (UNIST)
Title : A Comparative Analysis of Blow-up Dynamics in the Camassa-Holm and Burgers’ Equations
Abstract : The Camassa-Holm equation shares a structural similarity with scalar conservation laws, such as the Burgers’ equation, in that its solution exhibits a C1 blow-up governed by a Riccati-type differential equation. However, a fundamental difference in their particle dynamics under the Lagrangian framework leads to starkly contrasting outcomes in their blow-up profiles and post-blow-up behavior. In this talk, we will explain why their blow-ups are so different. This is joint work with Bongsuk Kwon (UNIST).
[1] Yunjoo Kim, Bongsuk Kwon and Jeongsik Yoon. Sharp regularity of gradient blow-up solutions in the Camassa-Holm equation. arXiv:2412.00558