In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.

In the past several years, the study of singular stochastic PDEs has significantly progressed by two groundbreaking theories: the theory of regularity structures by Hairer and the paracontrolled calculus by Gubinelli, Imkeller, and Perkowski.

The first half of this talk is a short introduction to these theories. In the remaining time, we prove an equivalence statement between two theories under some general assumptions, which are satisfied by the regularity structures introduced by Bruned, Hairer, and Zambotti for the study of a large class of singular stochastic PDEs. The latter part of the talk is based on a joint work with Ismaël Bailleul (Université Rennes 1).

The incompressible 3D Euler equations have total kinetic energy conservation for smooth (spatially periodic) solutions. In the recent resolution of the Onsager conjecture by Isett, on the other hand, below certain threshold Hölder regularity, Euler flows with total kinetic energy dissipation have been constructed. In this talk, I'll discuss a strong Onsager conjecture: the existence of Hölder continuous Euler flows with total kinetic energy dissipation and satisfying the local energy inequality. I'll also talk about an analogous question for the isentropic compressible Euler equations. The talk will be based on joint works with Camillo De Lellis and Vikram Giri.

In this talk, we consider the Boltzmann equation outside of convex ball with specular reflection boundary condition. Unlike previous results for convex domains, the trajectory which undergoes specular reflection is not differentiable anymore and we prove local C^{1/2}_{x,v} regularity. This is in stark difference to other boundary conditions (e.g. BV regularity of diffuse BC case). In particular, we introduce novel ‘specular singularly’ and ‘shift method’ to measure singularity.

In this talk I will review some of our recent results on SPDEs driven by multiplicative noise of transport type. Under a suitable scaling of the noise to high Fourier modes, these equations converge to the corresponding deterministic parabolic equations; moreover, larger intensity of transport noise leads to stronger dissipation in the limit equations. I will discuss some consequences of such scaling limit, including asymptotic uniqueness of stochastic 2D Euler equations and suppression of blow-up for stochastic 3D Navier-Stokes equations in vorticity form. The recent results on large deviations and Gaussian fluctuations underlying the scaling limit will also be mentioned. The talk is based on joint works with Franco Flandoli and Lucio Galeati.

We consider the vorticity equation for the viscous incompressible fluid on the 2D unit sphere and study the linear stability of the Kolmogorov type flows. The Kolmogorov type flows are stationary solutions to the vorticity equation given by the spherical harmonics and can be seen as a spherical version of the plane Kolmogorov flow in the 2D flat torus. For the plane Kolmogorov flow, it is observed that a perturbation decays faster than the usual viscous decay rate when the viscosity coefficient is sufficiently small. Such a phenomenon is called the enhanced dissipation and was rigorously verified by several authors with different methods recently. In this talk, we show that the enhanced dissipation occurs for the two-jet Kolmogorov type flow on the unit sphere which is given by the zonal spherical harmonic function of degree two. We also give a somewhat new idea for the spectral analysis of the perturbation operator arising in the study of the enhanced dissipation, which makes use of the mixing property of the perturbation operator expressed by a recurrence relation for the spherical harmonics. This talk is partly based on a joint work with Yasunori Maekawa (Kyoto University).

The purpose of this talk is to give an introduction to stochastic PDEs of fluid, especially to young scholars who may not be familiar with this research direction. I will also discuss some (due to limitation of time) specific recent developments and remaining problems that I am aware of.

In this talk, I will discuss some results on radial symmetry property for stationary and uniformly-rotating solutions for the 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.