# ON DEMAND Oxbridge PDE Conference 2022 - DAY TWO

**Prof. Clément Mouhot**** - University of Cambridge**

**Quantitative Hydrodynamic Limits of Stochastic Lattice Systems**

1.CM_.SLIDES.Oxbridge PDE22.pdf

In this talk, I will present a simple abstract quantitative method for proving the hydrodynamic limit of interacting particle systems on a lattice, both in the hyperbolic and parabolic scaling. In the latter case, the convergence rate is uniform in time. This "consistency-stability" approach combines a modulated Wasserstein-distance estimate comparing the law of the stochastic process to the local Gibbs measure, together with stability estimates a la Kruzhkov in weak distance, and consistency estimates exploiting the regularity of the limit solution. It avoids the use of “block estimates” and is self-contained. We apply it to the simple exclusion process, the zero range process, and the Ginzburg-Landau process with Kawasaki dynamics. This is a joint work with Daniel Marahrens and Angeliki Menegaki (IHES).

**Dr Michele Coti Zelati – Imperial College London**

**Quantitative stability results in fluid mechanics and kinetic theory**

2.MCZ_.SLIDES.Oxbridge PDE22.pdf

We discuss recent results on the nonlinear asymptotic stability of certain stationary solutions to fluids and kinetic equations. We show how various modifications of classical techniques involving vector fields and hypocoercivity can be used to obtain sharp quantitative stability results. Examples include the Navier-Stokes equations, the Boltzmann equations and models describing active particles and suspensions.

**Christopher Irving - ****University of Oxford**

**Quasiconvexity in the general growth setting **

3.CI_.SLIDES.Oxbridge PDE22.pdf

The quasiconvexity condition of Morrey plays a fundamental role in the calculus of variations, both in establishing the existence and partial regularity of minima in the vector-valued setting. I will discuss some recent and ongoing work on variational problems satisfying a quasiconvexity and general growth condition, which will include the degenerate case when the integrand grows ‘almost linearly.’

**Assoc. Prof. Jingwei Hu - INI / University of Washington**

**Fast Fourier Spectral Methods for Nonlinear Boltzmann Kinetic Equations — Algorithm and Analysis **

4.JH_.SLIDES.Oxbridge PDE22.pdf

Boltzmann equation is one of the central equations in kinetic theory and finds applications in many science and engineering disciplines. Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. I will discuss recent progress on development of fast Fourier-Galerkin spectral methods for Boltzmann type collisional kinetic equations. I will also present a new stability and convergence result of the method.

**Assoc. Prof. Cyril Imbert - INI / École Normale Supérieure Paris**

**Decay estimates for large velocities in the Boltzmann equation without cut-off **

5.CI_.SLIDES.Oxbridge PDE22.pdf

We consider solutions to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions for hard and moderately soft potentials *without the angular cutoff assumption*, and under the *a priori* assumption that the main hydrodynamic fields, namely the local mass and local energy and local entropy, are controlled along time. We establish quantitative estimates of *propagation*in time of "pointwise polynomial moments", i.e. sup_{x}_{,v}* f*(*t*,*x*,*v*)(1+|*v*|)^{q}, *q*≥0. In the case of hard potentials, we also prove *appearance* of these moments for all *q*≥0. In the case of moderately soft potentials we prove the *appearance* of low-order pointwise moments. Joint work with Clément Mouhot and Luis Silvestre.

**Assoc. Prof. Maria Pia Gualdani - ****INI / The University of Texas at Austin**

**Global regularity estimates for the homogeneous Landau equation. **

6.MPG_.SLIDES.Oxbridge PDE22.pdf

Kinetic equations describe evolution of interacting particles. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936, Lev Landau derived a new mathematical model for motion of plasma. One of the main features of the Landau equation is nonlocality, meaning that particles interact at large, non-infinitesimal length scales. In this talk I will focus on the homogeneous Landau equation, and on the question of global regularity vs finite-time blow-up for smooth initial data. This is a joint work with Nestor Guillen.

**Prof. François Golse - INI / ****École ****Polytechnique**

**From the N-body Schrödinger equation to the Euler-Poisson system**

7.FG_.SLIDES.Oxbridge PDE22.pdf

This talk discusses the validity of the joint mean-field and classical limit of the bosonic quantum N-body dynamics leading to the pressureless Euler-Poisson system for any factorized initial data whose first marginal has a monokinetic Wigner measure. The interaction is given by the repulsive Coulomb potential. The validity of this derivation is limited to finite time intervals on which the Euler-Poisson system has a rapidly decaying at infinity, smooth solution. One key ingredient in the proof is a remarkable inequality proved by S. Serfaty (*Duke Math. J. *169 (2020), 2887–2935).

Joint work with Thierry Paul (CNRS & Sorbonne université)

**Prof. Emer. Claude Bardos – INI / Universidad de París VII Denis Diderot**

**Zero viscosity limit for solutions of the Navier Stokes Equations in a domain with curved boundary and no slip boundary condition. **

8.CWB_.SLIDES.Oxbridge PDE22.pdf

This is a report on a work done with Toan Nguyen, Trinh Nguyen and Edriss Titi. Starting from the Kato theorem and using recent contributions of Mayekawa, Nguyen and Nguyen and Kukavica, Vicol and Wang we prove the convergence to the solution of the Euler equation for short time and analyticity hypothesis concerning the boundary and the initial data. Proofs can be adapted both to interior , exterior or non simply connected domain.