Second Frontiers in Fluid and Kinetic Partial Differential Equations

Tobias Barker (University of Bath)

In the first part, I will discuss results concerning absence of (interior) local anomalous dissipation in the 2D inviscid limit in the presence of boundaries. The implications for local balance laws for the limiting 2D Euler solution will also be discussed. Joint work with Milton C. Lopes Filho and Helena J. Nussenzveig Lopes (Universidade Federal do Rio de Janeiro).

In the second part, I will discuss quantitative classification of potentially singular 3D Navier-Stokes solutions at any given time in the region of potential blow-up times. The quantitative lower bounds prior to any potential blow-up time (and in the open vicinity of it) are in principle amenable to numerical testing.

Dongfen Bian (Beijing Institute of Technology)

This talk addresses the problem of proving the nonlinear instability of an equilibrium starting from its linear instability when there is no existence theory for the corresponding equations. We design a general method based on the use of analytic functions to overcome this difficulty and apply it to the classical problem of the stability of a plasma column in magnetohydrodynamics (MHD) as an illustration.

Yann Brenier (Université Paris-Saclay)

I will present a way to solve the Cauchy problem for nonlinear evolution PDEs by space-time convex optimization based on their weak formulation. One of the simplest examples is the quadratic porous medium equation, for which the Aronson-Bénilan inequality is sharply used to prove that the strategy works for arbitrarily long time intervals. A similar result holds true for the Burgers equation. For the more challenging Euler equations of incompressible fluids, the concept of subsolution (in the sense of convex integration theory) plays a crucial role. Finally, I will mention how the Einstein equations in vacuum can be considered in that framework.

Alexandru Ionescu (Princeton University)

I will talk about several results, in collaboration with Benoit Pausader, Xuecheng Wang, and Klaus Widmayer, on the global nonlinear asymptotic stability of certain homogeneous equilibria among solutions of the Vlasov-Poisson system.

In particular, I will discuss a proof of nonlinear Landau damping in optimal Gevrey-3 spaces in the confined case (x,v)∈T^d×R^d, and a result on the existence and injectivity of nonlinear scattering operators for this system.

Zhen Lei (Fudan University)

The Prandtl equation is fundamental in boundary layer theory. In this talk we present recent progress on global regularity of the unsteady equation and sharp asymptotic stability of the steady equation. Based on joint work with Prof. Hao Jia (University of Minnesota) and Dr. Cheng Yuan (Fudan University).

Vlad Vicol (New York University)

The compressible Euler equations are the fundamental model of gas dynamics. These PDEs admit a remarkable class of singularities known as implosions: a subset of the primary flow variables (density, velocity, pressure) become unbounded at a single point in space-time. Unlike shock singularities, where the solution remains bounded but develops infinite gradients, implosion singularities represent a strong blowup of the solution itself. Understanding the existence and stability of such singularities is a fundamental problem in gas dynamics. In this talk, we survey a number of recent results on Euler implosions.

Yi Wang (Academy of Mathematics and Systems Sciences, CAS)

The talk is based on our recent developments on the time-asymptotic stability of generic Riemann profiles, which are the viscous version of generic inviscid Riemann solutions including the composite of viscous shock profile, centered rarefaction wave and even viscous contact discontinuity, to some typical viscous conservation laws, such as barotropic/full compressible Navier-Stokes equations, and non-convex scalar viscous conservation laws/viscoelastic system.

Yao Yao (National University of Singapore)

In this talk, I will discuss two results on infinite-in-time growth in 2D and 3D incompressible Euler equations. For the 2D Euler equation on the whole plane, we construct the first example giving superlinear growth of the vorticity gradient for smooth compactly supported vorticity (joint with In-Jee Jeong and Tao Zhou). For the 3D axisymmetric Euler equation without swirl, we establish some upper and lower bound for the radial moment of vorticity, and prove that under some sign and symmetry conditions, all solutions must have their vorticity L^p norm growing to infinity with some power-law rate for all p≥1. To the best of our knowledge, this is the first result to establish power-law L^p-norm growth for smooth, compactly supported initial vorticity in R³. Joint work with Khakim Egamberganov.

Zhifei Zhang (Peking University)

We study the asymptotic stability of shear flows in the high Reynolds number regime. In particular, we consider Sobolev perturbations of canonical shear profiles such as Couette and Kolmogorov flows and investigate their optimal asymptotic stability threshold. The analysis reveals a delicate competition between inviscid mechanisms—such as phase mixing and inviscid damping—and viscous effects, including enhanced dissipation driven by shear-induced mixing.

Weiren Zhao (NYU Abu Dhabi – New York University)

In this talk, I will introduce the recent result about the optimal stability threshold for the Vlasov-Poisson equation with weak Fokker-Planck collision. We prove that if the initial perturbation is of size ν^1/2 in the critical weighted space H_x^logL_v²(<v>^m), then the solution remains the same size in the same space. Moreover, a space-time type Landau damping holds, namely, ||E||_Lt²Lx² ≲ ν^1/2; and a point-wise type Landau damping holds, namely, ||E(t)||_L² ≲ ν^1/2<t>^−N for any N>0 for t≥ν⁻¹.

We also prove that there exists initial perturbation in H_x¹L_v²(<v>^m) with size ν^{1/2−3ε₀/2} for any ε₀>0, such that the enhanced dissipation fails to hold in the following sense: there is 0<T≪ν^−1/3 such that

||f_≠(T)||_Lx²Lv² ≳ ν^−δ₁ ||f_≠(0)||_Hx¹Lv²

with some δ₁>0.

We solve the open problem raised in Bedrossian, arXiv:2211.13707, about the sharp stability threshold in lower regularity spaces. The main idea is to construct a wave operator D with a very precise expression to absorb the nonlocal term, namely,

D[∂_tg+v·∇_xg+E·∇_vμ] = (∂_t+v·∇_x)D[g].