Title & abstract

The study of the interaction of coaxial vortex rings dates back to the pioneering paper by Helmholtz (1858). Helmholtz considered vortex motion in a incompressible and inviscid fluid based on the Euler equations. His study includes the motion of circular vortex filaments, and he observed that motion patterns such as head-on collision may occur. Since then, many researches have been done on head-on collision of coaxial vortex rings, and interaction of coaxial vortex rings in general. Most of these research are either experiments conducted in a laboratory or numerical simulations of the Navier–Stokes equations, and the rigorous mathematical treatment of head-on collision of vortex rings are very scarce.

In light of this, we consider the head-on collision of two coaxial vortex rings, which have circulations of opposite sign, described as the motion of two coaxial circular vortex filaments under the localized induction approximation. A vortex filament is a space curve on which the vorticity of the fluid is concentrated. In our present work, we approximated thin vortex structures, such as vortex rings, by vortex filaments, and described the motion as the motion of a curve in the three-dimensional Euclidean space. In this formulation, a vortex ring is a space curve in the shape of a circle. We prove the existence of solutions to a system of nonlinear partial differential equations modelling the interaction of two vortex filaments proposed by the speaker which exhibit head–on collision. We also give a necessary and sufficient condition for the initial configuration and parameters of the filaments for head-on collision to occur. Our results suggest that there exists a critical value γ∗ > 1 for the ratio γ of the magnitude of the circulations satisfying the following. When γ ∈ [1,γ∗], two approaching rings will collide, and when γ ∈ (γ∗,∞), the ring with the larger circulation passes through the other and then separate indefinitely. As far as the speaker knows, the existence of such threshold γ∗ is only indirectly suggested via numerical investigations of the head-on collision of coaxial vortex rings, for example by Inoue, Hattori, and Sasaki (2000). Hence, our result is the first to obtain the threshold in a way that is possible to numerically calculate γ∗, as well as prove that the threshold exists in a framework of a mathematical model.

We visit results on stability of exact vortex solutions of the incompressible Euler equations including circular vortex patches, Lamb's dipoles, Hill's spherical vortices. They are solutions minimizing energy under natural constraints so that their stability comes from calculus of variations. This talk is based on joint works with K. Abe (Osaka City Univ.) and with D. Lim (UNIST).

I will discuss various issues related to the blow-up problem for the 3D Euler equation.

We study the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus when the differential order of the main term, the fractional Laplacian, is larger than 7/4. No renormalization has to be done for this model. (This is a joint work with Yoshihiro Sawano)

We consider the Burgers equation with the critical dissipation.

∂_t u + (- ∂ _x^2 ) ^{1/2} + u ∂ _x u = 0, t > 0 , x ∈ ℝ

u(0,x) = u_0(x)

We impose that the initial data is smooth and integrable, and it is known that the solution exists globally in time. In this talk, it will be shown that every solution behaves like the Poisson kernel as time tends to infinity. Analyticity in spacetime will be also discussed.

We consider the incompressible fluid equations including the Euler and SQG equations in critical Sobolev spaces, which are Sobolev spaces with the same scaling as the Lipschitz norm of the velocity. We show that initial value problem for the equations are ill-posed at critical regularity. This is based on joint works with Tarek Elgindi, Tsuyoshi Yoneda, and Junha Kim.

We are concerned with the question of well-posedness of stochastic three dimensional incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak--strong uniqueness; (iii) non-uniqueness in law; (iv) existence of a strong Markov solution; (v) non-uniqueness of strong Markov solutions; all hold true within this class. Moreover, as a byproduct of (iii) we obtain existence and non-uniqueness of probabilistically strong and analytically weak solutions defined up to a stopping time and satisfying an energy inequality. This talk is based on joint work with Martina Hofmanova and Xiangchan Zhu.