Title & abstract

This talk will revisit the 2D Euler patch dynamics on the plain. In the presence of the relatively smooth exterior velocity, the merging of the central pair of patches can be shown. The rate of that process is double-exponential, which is the best possible due to Yudovich's theory. Our talk will also touch on the subject of V-states that are related to the merging problem. 

We consider the surface quasi-geostrophic equation with critical dissipation on a smooth bounded domain to investigate the existence of smooth global solutions in Besov spaces. The existence of such solutions in the two-dimensional whole space is already known through the use of the Fourier transform. In this presentation, we assume the Dirichlet boundary condition and establish estimates that are related to a maximum principle involving spectral localization for the fractional Laplacian and a commutator estimate. 

We will report recent illposedness results for patch solutions of Euler and α-SQG equations. 

In this talk, we discuss some PDEs that describe fluid motion under the influence of gravity, including the incompressible porous media equation and incompressible Boussinesq equation in two dimensions. Using an interplay between various monotone and conserved quantities, we construct rigorous examples of small scale formations as time goes to infinity. These growth results work for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth. (Based on joint works with Alexander Kiselev and Jaemin Park).

In this talk, we discuss Leray-Hopf solutions to the two-dimensional Navier-Stokes equations with vanishing viscosity. We aim to demonstrate that when the initial vorticity is only integrable, the Leray-Hopf solutions in the vanishing viscosity limit do not exhibit anomalous dissipation. Moreover, we extend this result to the case where the initial vorticity is merely a Radon measure, assuming its singular part maintains a fixed sign. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. This is a joint work with Luigi De Rosa (University of Basel). 

We consider the vorticity form of the Navier-Stokes equations on the 2D unit sphere and study the nonlinear stability of the two-jet Kolmogorov type flow which is a stationary solution given by the zonal spherical harmonic function of degree two. Due to the structure of the viscous term, a perturbation contains a nondissipative part whose velocity form is an infinitesimal rotation around some axis. Our interest is in studying the effect of the nondissipative part on the evolution of the whole perturbation. We show that the nondissipative part is preserved in time for all initial perturbation, which is natural in view of conservation of angular momentum. Moreover, we obtain the convergence of the dissipative part towards a nontrivial equilibrium, which can be expressed explicitly in terms of the nondissipative part of the initial perturbation. To prove these results, we make use of some properties of Killing vector fields on a manifold. 

In the 3D Navier-Stokes turbulence, the energy cascade is a process of the creation of smaller-scale coherent vortices by larger ones. In this talk, we reformulate the energy cascade in terms of this stretching process and derive the Kolmogorov's −5/3 power law  (in the internal range) under physically reasonable assumptions. In the dissipation range, Elsinga-Ishihara-Hunt (2020) have extensively studied the extreme dissipation and  they concluded that significant shear layers are crucial to explain and quantify the dissipation rate statistics and extremes. On the other hand, in mathematics,  Jeong-Yoneda (2021,2022) considered this extreme dissipation (in other words, enhanced dissipation) by using short time smooth solutions to the 3D Navier-Stokes equations. More precisely, we discovered specific sequences of smooth solutions under the following 2+1/2 dimensional situation: small scale vertical vortex blob being stretched by large scale flows: two-dimensional shear flow or anti-parallel pairs of horizontal vortex tubes, and proved these solutions are expressing extreme dissipation. I also briefly explain these mathematical results.

We consider the incompressible Euler equations and related PDEs in scaling critical Sobolev spaces, which are also critical for local well-posedness. We show various ill/well-posedness results for the initial value problem at critical regularity. Then, we discuss some applications of understanding critical dynamicsc, including singularity formation and enhanced dissipation for the dissipative counterparts. 

We consider the incompressible Euler equations on the whole space. Our problem is to characterize solutions supported on finitely many Fourier modes; i.e., solutions represented by the sum of finitely many vector fields of the form $\cos(n\cdot x) a_n(t)$ or $\sin(n\cdot x) b_n(t)$. In the 2D case, Elgindi, Hu, and \v{S}ver\'{a}k (2017, CMP) gave an answer to this question: such a solution must be stationary, and its Fourier support must be a subset of either a circle centered at the origin or a line passing through the origin. In this talk, we give an answer to the problem in 3D: roughly speaking, such a solution must be stationary and either a 2D-like flow or a Beltrami flow. This is based on a joint work with Tsuyoshi Yoneda (Hitotsubashi University).

Convex integration is a technique with which one can prove non-uniqueness and stochastic PDEs are PDEs forced by random noise. In the past several years, many works were devoted to applying such convex integration on stochastic PDEs. We review such developments. 

Let us consider the three-dimensional Navier-Stokes flow past a moving obstacle with prescribed translational and angular velocities, where the boundary of the obstacle is Lipschitz continuity. In order to analyze the nonlinear problem, the $L^q$-$L^r$ estimates of the solution to the linearized problem play an important role, and such estimates  in the exterior of the quiescent body were established by Tolksdorf and Watanabe (2020) and Watanabe (2023).  The aim of this talk is to extend their results to the case of the moving body. This talk is based on a joint work with Keiichi Watanabe (Suwa University of Science).