In this talk I will outline recent work in collaboration with
Pierre Germain, Zaher Hani and Jalal Shatah regarding a rigorous
derivation of the kinetic wave equation. The proof presented will
rely of methods from PDE, statistical physics and number...
Due to its importance in materials science where it models the
slow relaxation of grain boundaries, multiphase mean curvature flow
has received a lot of attention over the last decades.
In this talk, I want to present two theorems. The first one
is...
For the Obstacle Problem involving a convex fully nonlinear
elliptic operator, we show that the singular set of the free
boundary stratifies. The top stratum is locally covered by a
$C^{1,\alpha}$-manifold, and the lower strata are covered by
$C^{1...
I will explain how coisotropic submanifolds of symplectic
manifolds can be distinguished among all submanifolds by a
criterion ("local rigidity") related to the Hofer energy necessary
to disjoin open sets from them. This criterion is invariant
under...
In this talk I will describe joint work with D. Alonso-Orán and
A. Córdoba where we extend a result, proved independently by
Kiselev-Nazarov-Volberg and Caffarelli-Vasseur, for the critical
dissipative SQG equation on a two dimensional sphere. The...
The well-posedness of the incompressible Euler equations in
borderline spaces has attracted much attention in recent years. To
understand the behavior of solutions in these spaces, the
logarithmically regularized Euler equations were introduced.
In...
In the optimal transport problem, it is well-known that the
geometry of the target domain plays a crucial role in the
regularity of the optimal transport. In the quadratic cost case,
for instance, Caffarelli showed that having a convex target
domain...
In a recent result, Buckmaster and Vicol proved non-uniqueness
of weak solutions to the Navier-Stokes equations which have bounded
kinetic energy and integrable vorticity.
We discuss the existence of such solutions, which in addition
are regular...
We address the inviscid limit for the Navier-Stokes equations in
a half space, with initial datum that is analytic only close to the
boundary of the domain, and has finite Sobolev regularity in the
complement. We prove that for such data the...
