Kenji Nakanishi (Kyoto University) Global wellposedness of general PDE on the Fourier half space
We consider the Cauchy problem of PDE with general linear part of constant coefficients and complex analytic nonlinearity. Restricting the Fourier support of the initial data to the half space, we obtain global wellposedness, with no condition of smallness or other structure of the equation, in a large space of Schwartz distributions on the Fourier side. This is joint work with Baoxiang Wang (Jimei/Peking).
Kyungkeun Kang (Yonsei University) On porous medium equation with divergence type of drift term and its applications
We establish weak solutions of porous medium equations with a divergence type of drift term, provided that the drift belongs to a scaling invariant classes. Constructed weak solutions satisfy not only an energy inequality but also moment and speed estimates. As applications, we consider some Keller-Segel equations of porous medium type whose known existence and regularity results are consequently improved.
Hiroki Ohyama (Kyoto University) Fast rotation limit for the magnetohydrodynamics system in a 3D layer applications
We consider the initial value problem for the incompressible magnetohydrodynamics system with the Coriolis force in a three-dimensional infinite layer. We prove the unique existence of global solutions for initial data in the scaling invariant space when the speed of rotation is sufficiently high. Moreover, we show that its global solution converges to that of the coupled system of the 2D incompressible magnetohydrodynamics equations and the 3D induction equations as the rotation speed tends to infinity. This is based on a joint work with Keiji Yoneda.
Junha Kim (Ajou Univ.) Convergence and Nonconvergence of the Euler-Maxwell Equations as the Speed of Light Tends to Infinity
We consider the convergence of smooth solutions of the Euler-Maxwell equations to the corresponding smooth solutions of the magnetohydrodynamics as the speed of light tends to infinity. As far as we know, the strong convergence of solutions to the Euler-Maxwell equations has not been studied before. We employ a proper auxiliary linear system of the Euler-Maxwell system to obtain the convergence and nonconvergence of the Euler-Maxwell equations to the MHD equations.
Tatsuya Miura (Kyoto University) Classification and stability theory of planar p-elasticae
Euler's elastica is a critical point of the bending energy under the fixed length constraint, and its $L^p$-counterpart is called p-elastica. In this talk I will discuss classification and stability theory of planar p-elasticae, with the emphasis on how the non-quadraticity p \neq 2 of the energy as well as the resulting nonlinear diffusion term in the Euler--Lagrange equation leads to several challenges and new phenomena compared to the classical case p = 2.
This talk is based on joint work with Kensuke Yoshizawa (Nagasaki University).
Yohei Yamazaki (Kyushu U.) Center stable manifold for ground states of nonlinear Schrödinger equations with internal modes
In this talk, we discuss the dynamics of solutions of nonlinear Schr\"odinger equation near unstable ground states.
Under the assumption of the existence of internal modes, we show the asymptotic stability for the solutions on a center stable manifold around unstable ground states. To show the time integrability of the internal component of solutions, we assume Fermi Golden Rule and apply the refined profile by Cuccagna and Maeda. This is the joint work with Masaya Maeda in Chiba University.
Kihyun Kim (Seoul National Univ.) On classification of global dynamics for energy-critical equivariant harmonic map heat flows
We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices D≥3; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $\dot{H}^1$-bounded radial solutions to energy-critical heat equations in dimensions N≥7, building upon soliton resolution for such solutions. This is a joint work with Frank Merle (IHES and CY Cergy-Paris University).
Keisuke Takasao (Kyoto University) Brakke's mean curvature flow with obstacles
In this talk, we show the global existence of Brakke's mean curvature flow with obstacles and with a right-angle condition, when the obstacles have $C^{1,1}$ boundaries. We use the Allen-Cahn equation with forcing term representing obstacles and we prove the convergence of the Radon measures given by the energy of the equation to Brakke's mean curvature flow in the sense of varifolds. We also show simple sub- and supersolutions that correspond to the obstacles. This talk is based on a joint work with Dr. Katerina Nik (TU Delft).
Soonsik Kwon (KAIST) Finite time blow-up construction of Calogero-Moser derivative nonlinear Schrödinger equations
I will present a finite time blow-up result of the Calogero–Moser derivative nonlinear Schrödinger equation (CM-DNLS). It is an L^2-critical nonlinear Schrödinger equation with explicit solitons, self-duality, and pseudo-conformal symmetry. More importantly, this equation is known to be completely integrable in the Hardy space and the solutions in this class are referred to as chiral solutions. A rigorous PDE analysis of this equation with complete integrability was recently initiated by Gérard and Lenzmann. Our main result constructs smooth, chiral, and finite energy finite-time blow-up solutions with mass arbitrarily close to that of soliton, answering the global regularity question for chiral solutions raised by Gérard and Lenzmann. Our proof also gives a construction of a codimension one set of smooth finite energy initial data (but without addressing chirality) leading to the same blow-up dynamics. Our blow-up construction in the Hardy space might also be contrasted with the global well-posedness of the derivative nonlinear Schrödinger equation (DNLS), which is another integrable L^2-critical Schrödinger equation. This talk is based on a joint work with Kihyun Kim and Taegyu Kim, arXiv:2404.09603
Nobu Kishimoto (Kyoto University) Modified scattering of the kinetic derivative nonlinear Schrödinger equation
We consider asymptotic behavior of small solutions to the kinetic derivative NLS. This is a one-dimensional NLS with a non-local cubic derivative nonlinear term, which has dissipative effect. In the periodic setting, it is known that dissipation becomes prominent and the initial value problem is ill-posed backward in time even for small data. In contrast, on the real line we can show global well-posedness and modified scattering in both time directions for small solutions in weighted Sobolev spaces. This is joint work with Kiyeon Lee (KAIST).
Kiyeon Lee (KAIST) The global dynamics for the Maxwell-Dirac system
In this talk, we study the (1+3) dimensional massive Maxwell-Dirac system in the context of global existence and asymptotic behavior of solutions under the Lorenz gauge condition, as well as the modified and linear scattering phenomena for the Dirac spinor and the electromagnetic potential, respectively. The primary ingredients of this talk are a vector fields energy method combined with a detailed analysis of the space-time resonance argument. This approach allows us to establish decay estimates and energy bounds crucial for proving the main theorems. Especially, we provide the explicit phase correction arising from the strong nonlinear resonances.
Yasunori Maekawa (Kyoto University) Local rigidity of the Couette flow for the stationary Triple-Deck equations
The Triple-Deck equations are a classical boundary layer model which describes the asymptotics of a viscous flow near the separation point, and the Couette flow is an exact stationary solution to the Triple-Deck equations. In this talk we prove the local rigidity of the Couette flow in the sense that there are no other stationary solutions near the Couette flow in a scale invariant space. This provides a stark contrast to the well-studied stationary Prandtl counterpart, and in particular offers a first result towards the rigidity question raised by R. E. Meyer in 1983. This talk is based on a joint work with Sameer Iyer (University of California, Davis).
Senjo Shimizu (Kyoto University) Local solvability of free boundary problems for the compressible Navier-Stokes equations in critical spaces
In this talk, we consider local well-posedness of the compressible Navier-Stokes equations in the scaling critical homogeneous Besov spaces. Our proof is based on end-point maximal $L^1$-regularity for the initial-boundary value problem of the corresponding linear problem for velocity in the half-space. This is a joint work with Takayoshi Ogawa (Waseda University).
Jaemin Park (U. Basel) Stability of stratified density under incompressible flows
In this talk, I will discuss asymptotic stability in the incompressible porous media equation in a periodic channel. It is well known that a stratified density, which strictly decreases in the vertical direction, is asymptotically stable under sufficiently small, smooth perturbations. We achieve improvements in the regularity assumptions on the perturbation and in the convergence rate. We apply a similar idea to the Stokes transport system. Instead of relying on the linearized equations, we directly address the nonlinear problem, and the decay of solutions will be obtained from the gradient flow structure of the equation.
Yohei Tsutsui (Kyoto University) Two-weight inequality for the heat flow and solvability of Hardy-Henon parabolic equation
We provide two-weight inequalities for the heat flow on the whole space by applying the sparse domination. Then, we consider the local and global existence of solutions to Hardy-Henon parabolic equation, which has a potential belonging to a Muckenhoupt class.
Jihoon Lee (Chung-Ang University) Local and global well-posedness of Navier-Stokes-Maxwell and Hall-MHD
In this talk, wel discuss various results for the local-in-time existence of strong solutions to the Navier-Stokes-Maxwell equations and global-in-time existence of strong solutions to the Naier-Stokes-Maxwell equations with the smallness assumptions on the initial data. We also consider the local well-posedness of a strong solution to the Hall-MHD equations. This talk is based on the joint work with Kyungkeun Kang(Yonsei U.) and Dinh-Duong Nguyen(CAU and Yonsei U.).
Masayuki Hayashi (Kyoto University) The Cauchy problem for the logarithmic Schrödinger equation
We revisit the Cauchy problem for the logarithmic Schrödinger equation. The main difficulty for this equation is that the nonlinear term has a singularity at the origin and breaks the local Lipschitz continuity. In this talk, we mainly focus on constructing strong solutions in the higher energy space. This is based on a joint work with Tohru Ozawa (Waseda University).