Partial Differential Equations and their Applications Seminar
Term 1 24-25
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.
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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.
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.
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.
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.
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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.
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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).
