Partial Differential Equations and their Applications Seminar

Term 2 24-25

14th January - 21st January - Antonín Češík (University of Warwick) -- Variational methods for dynamical PDE: visco-elastic solids with collisions and fluid-structure interactions

Energy-based methods have often proven useful in analysis of evolutionary PDEs. They open the problems to powerful analytical tools from calculus of variations. In particular the celebrated De Giorgi's minimizing movements method, which can handle gradient flows and quasi-static problems. A recently developed "hyperbolic minimizing movements" method extends this to the case of fully dynamical problems (i.e. including inertia). It combines the variational approach to handle nonlinearities, and PDE estimates to handle the kinetic energy.

In this talk, we present some recent applications of this to applications to fully dynamical problems. First, we obtain the existence result for nonlinear viscoelastic solids in the large deformation regime with arbitrary collisions. For this we construct a physically correct measure-valued contact force. Next, we treat nonlinear viscoelastic bulk solid coupled to a Navier-Stokes equation with a full slip condition at the fluid-solid interface. We complement these results by studying the fully time-discrete version of the scheme, as a step towards future numerical implementations.

The talk is based on joint works with G. Gravina, M. Kampschulte, S. Schwarzacher.

28th January- No seminar 4th February - Klaus Widmayer (University of Zurich)-- Landau damping near the Poisson equilibrium in R^3

While "Landau damping" is regarded as an important effect in the dynamics of hot, collisionless plasmas, its mathematical understanding is still in its infancy. In particular, the terminology has evolved to include several types of (stabilizing) effects in diverse physical contexts, the mathematical description of which can differ markedly between various settings of relevance. This talk presents a recent nonlinear stability result for the homogeneous "Poisson" equilibrium of the Vlasov-Poisson equations on R^3, which relies on a combination of oscillatory and damping effects.

This is based on joint work with A. Ionescu, B. Pausader and X. Wang.

11th February - Amelie Loher (University of Cambridge)

Title: Decay estimates for weak solutions of the Boltzmann equation

Abstract: We introduce a suitable notion of weak solutions to the Boltzmann equation on bounded space domains. In case of moderately soft and hard potentials, we show that these solutions decay pointwisely at any polynomial rate in velocity, provided that the mass, energy and entropy of the solutions are bounded. This is joint work with Cyril Imbert.

18th February- Daniel Pezzi (Johns Hopkins University)--Sharp spectral projection estimates for the torus

It is a natural, but oftentimes difficult, question to deduce information about a periodic function from its Fourier series expansion. Spectral projection estimates are one way to explore this connection. If a function has its spectrum contained in an annulus, how large can the function be in an L^p norm? For unit width annuli this question is answered by the universal estimates of Sogge. If the projection operator projects onto an eigenspace, this is the Discrete Restriction conjecture of Bourgain. We present sharp estimates for intermediate cases for a large subset of the relevant parameters, which addresses a conjecture of Germain and Myerson. We will discuss the techniques used, similarities and differences to the case of general manifolds, and potential future avenues of attack.

25th February - Tom Sales (University of Warwick)--Topics on the Navier-Stokes equations on evolving domains

In physics and biology one may obtain models of physical phenomena involving partial differential equations posed on evolving domains, which may or may not be known a priori. In this talk we discuss some problems related to the Navier-Stokes equations posed on evolving domains. Firstly, we discuss recent results concerning the derivation and well-posedness of a Navier-Stokes-Cahn-Hilliard system posed on an evolving surface with prescribed evolution. Secondly, we outline a new, constructive approach to a fluid-rigid body interaction problem with no slip boundary conditions. This approach is based on an iteration of problems posed on prescribed evolving domains.

This talk is based on joint works with Charles Elliott.

4th March - Iwona Chlebicka (University of Warsaw) -- Asymptotics for the Cauchy problem for the fast p-Laplace evolution equation

Title: Asymptotics for the Cauchy problem for the fast p-Laplace evolution equation

We focus on the fast diffusion equation involving p-Laplacian for p<2. The properties of the solutions to the p-Laplace Cauchy problem change in several special values of the parameter p. In the range of p when mass is conserved, non-negative solutions behave like the Barenblatt (or fundamental) solutions for large times. The convergence was established in the literature for p close to 2, but no rates were available.

We show the polynomial rates of the convergence in the uniform relative error for a natural class of initial data.

In particular, we allow for the values of p, for which the entropy is not displacement convex, placing our study outside of the reach of the optimal transportation tools.

Joint project with Matteo Bonforte and Nikita Simonov, arXiv:2405.05405.

11th March - Yohance Osborne (Durham University) - Analysis and Numerical Approximation of Mean Field Game Partial Differential Inclusions

The Mean Field Game (MFG) system of Partial Differential Equations (PDE), introduced by Lasry & Lions in 2006, models Nash equilibria of large population stochastic differential games of optimal control where the players of the game have unique optimal controls, and the convex Hamiltonian of the underlying optimal control problem is differentiable. In this talk, we introduce a new class of model problems called Mean Field Game Partial Differential Inclusions (MFG PDI), which extend the MFG system of Lasry and Lions to situations where players may have possibly nonunique optimal controls, and the resulting Hamiltonian of the underlying optimal control problem is not required to be differentiable.

We prove the existence of unique weak solutions to MFG PDI for a broad class of Hamiltonians that are convex, Lipschitz, but possibly nondifferentiable, under a monotonicity condition similar to one considered previously by Lasry & Lions. Moreover, we introduce a class of monotone finite element discretizations of the weak formulation of MFG PDI and present theorems on the strong convergence of the approximations to the value function in the $L^2(H_0^1)$ -norm and the strong convergence of the approximations to the density function in $L^p(L^2)$ -norms. We conclude the presentation with discussion of a numerical experiment involving a non-smooth solution.

Term 1 24-25

1st October - Alpar Meszaros (Durham)

Title

From the porous medium equation to the Hele-Shaw flow: an optimal transport perspective

Abstract

In this talk we will revisit the classical problem on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. We will demonstrate that the theory of optimal transport via gradient flows can bring new perspectives, when it comes to consider confining potentials or nonlocal drift terms within the problem. In particular, we provide quantitative convergence rates in the 2-Wasserstein distance for the singular limit, which are global in time thanks to the contractive property arising from the external potentials. The talk will be based on a recent joint work with Noemi David and Filippo Santambrogio.

7th October - Clement Mouhot (Cambridge) - D1.07 MONDAY 15th October - Jakub Woznicki (Warsaw)

Title: Strong convergence of the vorticities in the 2D viscosity limit on a bounded domain

Abstract: In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and,

consequently, the possible formation of a boundary layer. Within a purely interior framework,
Constantin and Vicol showed that the two-dimensional viscosity limit is justified for any arbitrary
but finite time under the assumption that on each compactly contained subset of the domain, the
enstrophies are bounded uniformly along the viscosity sequence. Within this framework, we upgrade to local strong convergence of the vorticities under a similar assumption on the p-enstrophies, p > 2. The talk is based on a recent publication with Christian Seis and Emil Wiedemann.

22nd October - Richard Medina (Paris Dauphine)

THE BOLTZMANN EQUATION ON C^1 AND CYLINDRICAL DOMAINS NEAR THE HYDRODYNAMIC LIMIT.

We present the well-posedness for the Boltzmann equation near its hydrodynamic limit on a bounded domain. We consider two types of domains, namely $C^1$ domains with Maxwell boundary conditions where the accommodation coefficient is a continuous space dependent function $\iota(x) \in [\iota_0,1]$ for any $\iota_0 \in (0,1]$ , or cylindrical domains with diffusive reflection on the basis of the cylinder and specular reflection on the rest of the boundary.

Furthermore we construct an explicit rate of the decay for solutions near the equilibrium of the equation in a weighted framework for either polynomial, stretched exponential and gaussian weights.

29th October - Esther Bou-Dagher (Paris Dauphine)

Title: Exponential inequalities in probability spaces revisited

Abstract: In this talk, we revisit several results on exponential integrability in probability spaces and derive some new ones. In particular, we give a quantitative form of recent results by Cianchi, Musil, and Pick in the framework of Moser-Trudinger-type inequalities, and recover Ivanisvili-Russell’s inequality for the Gaussian measure. One key ingredient is the use of a dual argument, which is new in this context, that we also implement in the discrete setting of the Poisson measure on integers. This is a joint work with Ali Barki, Sergey Bobkov, and Cyril Roberto.

5th November - Lorenzo Pareschi (Heriot Watt)

A PDEs perspective on metaheuristics

Metaheuristic optimization methods, widely used in applications ranging from machine learning to optimal control, often lack a rigorous mathematical foundation. Many of these approaches are driven by stochastic particle systems and rely on heuristic techniques that are challenging to analyse formally. Recently, tools from statistical physics have offered a new perspective on metaheuristic algorithms through partial differential equations (PDEs). This approach provides a foundation for developing a robust mathematical theory of convergence to the global minimum and opens up opportunities for systematically enhancing algorithm performance. In this talk, we will explore popular metaheuristic algorithms, such as simulated annealing, genetic algorithms, and particle swarm optimization, by showing how, in the limit of large particle numbers, these algorithms can be described by kinetic and mean-field PDEs. We will also discuss connections to other optimization approaches, including Langevin dynamics and consensus-based optimization.

12th November 19th November - Jakub Skrzeczkowski (Oxford)

Title: The Stein-log-Sobolev inequality and the exponential rate of convergence for the continuous Stein variational gradient descent method.

Abstract: The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information as a duality pairing between

$H^{-1}$ and

$H^{1}$ , which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions.

This is a joint work with J. A. Carrillo and J. Warnett.

26th November - Noemi David (Rouen) 3rd December - Amirali Hannani (KU Leuven)

Title: On the effect of the Coriolis force on the enstrophy cascade

Abstract:
We study the enstrophy cascade at small spatial scales in statistically stationary forced-dissipated 2D Navier-Stokes equations subjected to the Coriolis force. We provide physically reasonable sufficient conditions to prove that on small scales, in the presence of the Coriolis force, the so-called third-order structure function's asymptotics follow the third-order universal law of 2D turbulence without the Coriolis force.

Our result indicates that on small scales, the enstrophy flux from larger to smaller scales is not affected by the Coriolis force, confirming experimental and numerical observations. To the best of our knowledge, this is the first mathematically rigorous study of the above equations. We also proved well-posedness and certain regularity properties to obtain the mentioned results.
This is a joint work with Yuri Cacchio (GSSI) and Gigliola Staffilani (MIT).