Workshop on Nonlinear PDEs in Fluid Mechanics

Whether the incompressible Euler equations in ℝ³ can develop a finite-time singularity from smooth initial data is a long-standing open problem in mathematical fluid mechanics. In the axisymmetric setting without swirl, global regularity is known for compactly supported C^α initial vorticity for all α ≥ 1/3.

Below this regularity threshold, for any α ∈ (0,1/3), we construct exact C^α self-similar blowup profiles for the vorticity of the 3D axisymmetric Euler equations without swirl, and build on them to prove asymptotically self-similar blowup from compactly supported C^α initial vorticity and C^1,α ∩ L² initial velocity. Our construction is based on lifting smooth blowup profiles for a 1D nonlocal model and exploits the anisotropy of the 3D self-similar flow.

I will discuss uniqueness and orbital stability of several families of traveling vortex structures for the incompressible Euler equations, including concentrated vortex rings, near-Lamb dipoles, and large-impulse dipoles. This talk is based on joint works with K. Abe, D. Cao, I.-J. Jeong, S. Lai, W. Zhan, and C. Zou.

We present recent progress on the global persistence of boundary regularity for temperature patches in the spaces W^k,∞ and C^k,γ for the two-dimensional whole-space nondiffusive Boussinesq system with subcritical dissipation. We also discuss the infinite-Prandtl-number limit for such temperature patch solutions.

Vortex structures are fundamental objects in the mathematical theory of incompressible fluids. Classical examples include vortex rings, vortex pairs, and rotating vortices, all of which exhibit rich dynamical and geometric behavior.

I will present several recent developments in the qualitative theory of vortex structures in ideal incompressible fluids, with particular emphasis on their existence, uniqueness, stability, and rigidity. The presentation is mainly based on a series of joint works with collaborators carried out over the past several years.

This talk concerns the incompressible limit for free-surface inviscid fluids, with particular emphasis on the compressible Euler equations. In contrast to fixed-domain problems, the motion of the free interface introduces new and substantial difficulties. We will present methods for establishing this singular limit in various settings, depending on the scale of the initial singular perturbation and the boundary conditions, including whether surface tension and vortex sheets are taken into account.

We establish the nonlinear stability threshold O(ν^3/2) for the three-dimensional Couette flow governed by the compressible Navier–Stokes equations. While stability thresholds are well understood in two dimensions for both compressible and incompressible flows, and in three dimensions for incompressible flows, the three-dimensional compressible case remains open due to additional structural features, strong mode interactions, and wave coupling.

The proof is based on a refined frequency-space approach. For zero modes, we separate and estimate the main contributions from diffusion waves, acoustic waves, and the lift-up mechanism. For non-zero modes, new multiplier estimates and a structure-based decomposition track the interaction between dissipation and acoustic effects.

Mean-field limits and propagation of chaos for interacting particle systems with singular interaction forces have recently seen major advances, driven by new quantitative stability techniques beyond the classical Lipschitz regime.

This talk surveys developments based on relative entropy, modulated energy, modulated free energy, and duality methods, highlighting how they deliver explicit quantitative control for singular kernels. I will then present recent progress on particle approximations of the Landau equation, non-exchangeable particle systems, and selected links to PDE numerical algorithms, before concluding with open problems and future directions.

We establish local existence of strong solutions for Lipschitz initial density and L² ∩ VMO⁻¹ initial velocity, and global existence for sufficiently small BMO⁻¹ data when the density is close to a positive constant. The proof relies on a refined freezing-coefficient method, Koch–Tataru type estimates, and new time-weighted energy estimates, yielding the first borderline BMO⁻¹/VMO⁻¹ theory for the inhomogeneous Navier–Stokes equations.

I will discuss recent results on fluid equations obtained with the assistance of AI. I will first outline the relationship between the mixing properties of incompressible flows and the diffusion process, including dissipation enhancement. I will then focus on the so-called unmixing phenomenon and on the existence of Batchelor scales for certain flows, both of which are results recently proved using AI.

I will present two results concerning the spatially inhomogeneous non-cutoff Boltzmann and Landau equations with very soft potentials. The first is a scaling-critical global well-posedness theory near a Maxwellian. For sufficiently small initial perturbations measured in a weighted anisotropic Riesz-potential norm, we prove global existence and uniqueness in the whole space.

The second is a short-time pointwise Green-function theory for variable-coefficient kinetic equations. By constructing a frozen small-jump kernel along kinetic characteristics and applying a parametrix expansion, we obtain bounds that capture the anisotropic fractional Kolmogorov geometry and rapid decay away from the characteristic region.

Around a global smooth irrotational solution to the classical isentropic compressible Euler–Poisson system, we construct classical solutions to the one-species relativistic Vlasov–Maxwell–Boltzmann system on any finite time interval and rigorously justify the combined hydrodynamic and Newtonian limits to the Euler–Poisson system.

The analysis is based on a Hilbert expansion in ε for the relativistic Vlasov–Maxwell–Boltzmann system, an asymptotic expansion in c⁻¹ for the relativistic Euler–Maxwell system, and estimates that are uniform in c and ε for both the expansion coefficients and the remainder terms, without imposing any a priori relation between ε and c.