We prove a desingularization theorem for analytic vortex sheets of the 3D incompressible Euler equations. Starting from an analytic solution of the corresponding Birkhoff-Rott system, we construct, for every sufficiently small thickness parameter $\varepsilon>0 $, an exact Euler vorticity supported on a tubular neighborhood of width $ O(\varepsilon) $ around the sheet, and defined on a time interval that does not shrink to~0 as $\epsilon\to0$. We show that these vorticities converge, in the sense of distributions, to the prescribed vortex sheet. In particular, we conclude that analytic 3D vortex sheet motions arise as limits of exact Euler flows with lifespan bounded from below independently of~$ \varepsilon $. The proof hinges on the study of vorticities defined in terms of a time-dependent foliation by almost parallel surfaces and of divergence-free vector fields tangent to these surfaces.

Positive solutions $u$ to the fractional fast diffusion equation $\partial_t u + (-\Delta)^s (u^\frac{N-2s}{N+2s}) = 0$ on $(0, T^*) \times \mathbb R^N$ are known to extinguish in finite time $T^* < \infty$ provided the initial datum is regular enough. Moreover, as $t \to T^*$, the extinction is governed by a certain profile $U_{T_*, z, \lambda}$.

In this talk, we strengthen the above assertions by giving optimal quantitative rates of convergence. More precisely, we prove the bound $\frac{u(t,\cdot)}{U_{T_*, z, \lambda}(t,\cdot)} - 1 = \mathcal O( (T_*-t)^\frac{N+2s}{N-2s+2})$, in a natural weighted energy norm. The main point here is that the exponent $\frac{N+2s}{N-2s+2}$ is sharp. This is the analogue of recent results by Bonforte and Figalli (CPAM, 2021) and Akagi (ARMA, 2023) valid for $s = 1$ and bounded domains $\Omega \subset \mathbb R^N$. The additional difficulty, besides including the fractional setting, is the degeneracy of the linearized operator stemming from the symmetries of the limit equation on $\mathbb R^N$, which is not present in the domain case. As a consequence, our result is new also in the local case $s = 1$.

We obtain similar results for the fractional fast diffusion equation $\partial_t u + (-\Delta)^s (u^m) = 0$ on bounded domains $\Omega \subset \mathbb R^N$, with subcritical exponent $m \in (\frac{N-2s}{N+2s}, 1)$.

This is joint work with Meng Yu (Goethe-Universität Frankfurt).

We study the ergodic behaviour of McKean–Vlasov equations driven by common, divergence-free transport noise. In particular, we show that in dimensions greater or equal to 2, if the noise is mixing and sufficiently strong, it can enforce the uniqueness of invariant probability measures, even if the deterministic part of equation has multiple steady states. This is joint work with Benjamin Gess and Rishabh S. Gvalani.