Talks (titles and abstracts)
Jacob Bedrossian: tba
Yann Brenier: tba
Alfred Bruckstein: tba
Russel Caflisch: tba
José Antonio Carrillo: Global minimizers of Interaction Energies
I will review the existence and uniqueness of global minimizers for interaction energy functionals. Euler-Lagrange equations in the infinity wasserstein distance will be discussed. Based on linear convexity/concavity arguments, qualitative properties of the global minimizers will also be treated. Anisotropic singular potentials appearing in dis- locations will be shown to have rich qualitative properties with loss of dimension and ranges of explicit minimizers. A large part of the course will be based on several works in collaboration with Ruiwen Shu (University of Oxford).
Tony Chan: tba
Alina Chertock: tba
Albert Cohen: tba
Pierre Degond: tba
Ronald DeVore: tba
Qiang Du: tba
Björn Engquist: tba
Benjamin Gess: tba
Helge Holden: tba
Thomas Hou: Potentially singular behavior of 3D incompressible Navier-Stokes equations
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, I will present some numerical evidence that the 3D Navier-Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of \(10^7\). This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations. Unlike the Hou-Luo blowup scenario, the potential singularity of the 3D Euler and Navier-Stokes equations occurs at the origin. We have applied several blowup criteria to study the potentially singular behavior of the Navier-Stokes equations. The Beale-Kato-Majda blow-up criterion, the blowup criteria based on the growth of enstrophy and negative pressure, the Ladyzhenskaya-Prodi-Serrin regularity criteria all seem to imply that the Navier-Stokes equations develop potentially singular behavior. Finally, we present some new numerical evidence that a variant of the axisymmetric Navier-Stokes equations with time dependent fractional dimension develops nearly self-similar blowup with maximum vorticity increased by a factor of \(10^{30}\).
Shi Jin: tba
Alexander Kiselev: Regularity of vortex and SQG patches
Patch solutions are an important class of special singular solutions to the 2D Euler or surface quasi-geostrophic (SQG) equations that model evolution of regions of vorticity with sharp boundaries (like hurricanes) or sharp temperature fronts in atmosphere. I will discuss recent progress on regularity properties of vortex and SQG patches. In particular, I will present an example of a vortex patch with continuous initial curvature that immediately becomes infinite but returns to \(C^2\) class at all integer times only without being time periodic. The proof involves derivation of a new system describing the patch evolution in terms of arc-length and curvature. A similar approach leads to discovery of strong ill-posdness of the SQG patches in all but \(L^2\) based spaces, or spaces of infinitely smooth functions. The talk is based on a work joint with Xiaoyutao Luo.
Alexander Kurganov: tba
Qin Li: tba
Pierre-Louis Lions: tba
Hailiang Liu: Global dynamics and photon loss in the Kompaneets equation
The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We prove several results concerning the long-time convergence of solutions to Bose-Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose-Einstein condensation. We provide two conditions that guarantee that photon loss occurs, and show that once loss is initiated then it persists forever. We prove that as times tends to infinity, solutions necessarily converge to equilibrium and we characterize the limit in terms of the total photon loss. Additionally, we provide a few results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is a contraction in \(L^1\). None of these results impose a boundary condition at the zero-energy boundary. This is a joint work with Joshua Ballet, Gautam Iyer, David Levermore, and Robert Pego.
Stéphane Mallat: tba
Pierangelo Marcati: tba
Antoine Mellet: tba
Helena Nussenzveig Lopes: tba
Benoît Perthame: tba
Denis Serre: Compensated integrability on tori ; a priori estimate for space-periodic gas flows
We extend our theory of Compensated Integrability of positive symmetric tensors, to the case where the domain is the product of a linear space \({\mathbb R}^k\) and of a torus \({\mathbb R}^m/\Lambda\), \(\Lambda\) being a lattice of \({\mathbb R}^m\). We apply our abstract results in two contexts, in which \(k=1\) is associated with a time variable, while \(m=d\) is a space dimension. On the one hand to \(d\)-dimensional gas dynamics, governed by the Euler equations, when the initial data is space-periodic~; we obtain an a priori space-time estimate of our beloved quantity \(\rho^{1\over d}p\). On the other hand to hard spheres dynamics in a periodic box \(L{\mathbb T}_d\). We obtain a weighted estimate of the average number of collisions per unit time, provided that the "linear density" \(Na/L\) (\(N\) particles of radius \(a\)) is below some threshold.
Roman Shvydkoy: tba
Edriss Titi: tba
Emil Wiedemann: Non-Deterministic Solution Concepts in Fluid Dynamics
As more and more ill-posedness results have been shown for fluid PDEs (not only by convex integration!), the idea to solve the Cauchy problem by some unique weak or entropy solution has become questionable. Instead, non-deterministic solution concepts such as measure-valued or statistical have sparked much recent research interest. They also seem to be more in line with well-known theories of turbulence, which are typically statistical. I will give an overview of such generalised solution concepts, including their weak-strong stability, their relation to more conventional solutions, and questions of existence.