Taoufik Hmidi
Title: Application of Bifurcation Theory to Vortex Motion
Abstract: In this mini-course, I aim to explore some key aspects of bifurcation theory in infinite-dimensional spaces. The first part will focus on fundamental tools, with a particular emphasis on the Crandall-Rabinowitz theorem. In the second part, I will apply these concepts to vortex dynamics and prove Burbea's result on the construction of rotating vortices for 2d Euler equations. Finally, in the third part, I will extend the discussion to prove recent results on general active scalar equations involving completely monotone kernels.
Haroune Houamed
Title: Yudovich’s theory in the context of plasmas
Abstract: It is well known, due to Yudovich, that the two-dimensional Euler equations are globally well-posed for an initial vorticity that is solely bounded and integrable. Although justifying the sharpness of this result is still open, generally speaking, we tend to believe that Yudovich’s theory lies in the borderline between well-posedness and ill-posedness of the Euler equations. On the other hand, extending this theory to more general models, involving strong coupling with Euler’s vorticity, is more challenging and sometimes even fails to be true. In this talk, I will highlight a few aspects of positive and negative results in these directions before I move on to sketching the main ingredients behind the validity of Yudovich’s theory for the incompressible Euler—Maxwell system, which governs the evolution of an ideal plasma in a relativistic context. Moreover, it is to be emphasized that our approach allows us to analyze, in a rigorous way, the behavior of the global solutions within the non-relativistic regime, thereby deriving a classical MHD model from the Euler—Maxwell system.
Emeric Roulley
Title : Periodic vortex patch motion in bounded simply-connected domains
Abstract : We consider the Euler equations within a simply-connected bounded domain. The dynamics of a single point vortex are governed by a Hamiltonian system, with most of its energy levels corresponding to time-periodic orbits. We show that, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches. We provide concrete examples of these non-degeneracy conditions, which are satisfied by a broad class of domains, including convex ones. The proof uses Nash-Moser scheme and KAM techniques combined with complex geometry tools. Additionally, we will present a vortex duplication mechanism to generate synchronized time-periodic motion of multiple vortices.
Omar Lazar
Title: On the Muskat problem with surface tension.
Abstract: The Muskat problem is a mathematical model used to describe the dynamics of the interface between two immiscible fluids in a porous medium. I will discuss a recent result regarding the existence of possibly large solutions for the Muskat problem when gravity and surface tension are taken into account.
Kyudong Choi
Title: Existence of a touching dipole as a maximizer of kinetic energy
Abstct: The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical computations of Sadovskii in [J. Appl. Math. Mech., 1971], and has gained significant interest due to its relevance in the inviscid limit of planar flows via Prandtl--Batchelor theory and as the asymptotic state for vortex ring dynamics. In this talk, I will explain the motivation why to study a Sadovskii vortex patch, and sketch its proof of the existence of such a vortex, by solving the energy maximization problem under the exact impulse condition and an upper bound on the circulation. This is joint work with In-Jee Jeong(SNU) and Youngjin Sim(UNIST).
Jihoon Lee
Title : Singular Limit Problem from Euler-Maxwell to Inviscid Magnetohydrodynamics
Abstract : In this talk, we consider the difference estimates for a class of Euler-Maxwell system and those of magnetohydrodynamics system(MHD) in three dimensions. We decompose the Euler-Maxwell system into two parts-auxiliary linear system and error part. We obtain the exact convergence rate from the solution of the Euler-Maxwell system to that of MHD as the speed of light tends to infinity. This is the joint work with Dongha Kim(Yonsei U.) and Junha Kim(Ajou U.).
Jeaheang Bang
Title: Self-Similar Solutions to the Stationary Navier-Stokes Equations in a Higher Dimensional Cone
Abstract: Self-similar solutions play an important role in understanding the regularity and asymptotic behavior of solutions to the Navier-Stokes equations. We recently showed that axisymmetric self-similar solutions to the stationary Navier-Stokes equations in an $n$-dimensional cone with the no-slip boundary condition except at the origin must be trivial when $n\geq 4$. It rules out this particular scenario of boundary singularity, which has finite Dirichlet energy when $n\geq 5$. The main idea is to apply ODE techniques along with a sign property of the head pressure. This is a joint work with Changfeng Gui, Chunjing Xie, Yun Wang, and Hao Liu.
Zineb Hassainia
Title: Time Quasi-Periodic Vortex Patches
Abstract: In this talk, I will discuss recent results on the emergence of quasi-periodic vortex patch solutions for the planar Euler equations. These structures are identified near steady-state patch solutions, provided a suitable parameter is selected from a Cantor-type set of almost full Lebesgue measure. The proofs rely on the Nash-Moser implicit function scheme and KAM theory.
Jinwook Jung
Title: Modulated energy estimates for singular kernels and their applications
Abstract: In this talk, we provide modulated interaction energy estimates for the kernel $K(x) = |x|^{-\alpha}$ with $\alpha \in (0,d)$, and their applications. The proof relies on a dimension extension argument for an elliptic operator and its commutator estimates. For the applications, we first discuss the quantified asymptotic limit of kinetic equations with singular nonlocal interactions. We show that the aggregation equations and the isothermal or pressureless Euler system with singular interaction kernels are rigorously derived. Second, we employ the estimates to establish the well-posedness theories in H\"older spaces for the kinetic and fluid equations involving singular interaction kernels, mainly about inviscid $\alpha$-surface quasi-geostrophic ($\alpha$-SQG) equations and kinetic Cucker-Smale model. If time permits, we also present the ill-posedness theory for $\alpha$-SQG equations in H\"older spaces. This talk is based on the collaboration with Y.-P. Choi (Yonsei Univ.) and J. Kim (Ajou Univ.).
Ken Abe
Title: Stationary self-similar profiles for the two-dimensional inviscid Boussinesq equations
Abstract: We consider (-alpha)-homogeneous solutions (stationary self-similar solutions of degree -alpha) to the two-dimensional inviscid Boussinesq equations in a half-plane. We show their non-existence and existence with both regular and singular profile functions. This talk is based on a joint work with D. Ginsberg (Brooklyn College of CUNY) and I.J. Jeong (Seoul National University).
Javier Gomez-Serrano