Ovidiu Savin, Columbia University

Title: The Alt-Phillips free boundary problems

Abstract: We discuss a family of free boundary problems that interpolate between the more classical models of the obstacle problem, the Alt-Caffarelli problem and minimal surfaces.

Yao Yao, National University of Singapore

Title: Symmetry and uniqueness via a variational approach

Abstract: For some nonlocal PDEs, their steady states can be seen as critical points of some associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progress in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan). I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

Benjamin Fehrman, Louisiana State University

Title: Conservative stochastic PDEs and non-equilibrium fluctuations

Abstract:

Stochastic PDEs of fluctuating hydrodynamics type, like the Dean–Kawasaki equation and related models, have for over three decades attracted a large amount of attention in both mathematics and statistical physics due to their success describing non-equilibrium phenomenon in physical systems [3,8,11].

However, their application has until recently lacked a precise mathematical meaning for several reasons. The stochastic PDEs are formally supercritical in the language of regularity structures and singular SPDEs and are therefore not renormalizable [7], they exhibit non-Lipschitz noise coefficients including the square root, and attempts at establishing certain weak solution theories have been shown to be either ill-posed or trivial [9]. And at first glance, this rather negative evidence might suggest that the SPDEs are merely formal rewritings of the underlying microscopic dynamics, and that they themselves offer little in the way of understanding and simulation.

One purpose of these lectures will be to show that this is not the case. After introducing a suitable spatial regularization of the noise—a step motivated, for example, by the grid-length of the particle system, coarse-graining, and numerical approximations—we will establish a robust well-posedness theory for a general class of SPDEs of fluctuating hydrodynamics type and show that, along appropriate scaling limits, the SPDEs accurately describe the physical system in terms of a law of large numbers, central limit theorem, and large deviations principle [5,6]. Furthermore, we will explain how the study of large deviations principles describing the far-from-equilibrium behavior of the system leads to the analysis of certain parabolic-hyperbolic PDEs in energy critical spaces. The resulting well-posedness and stability theory developed for the stochastic PDEs and skeleton equation extend to the nonlinear setting concepts of renormalized solutions, introduced by DiPerna and Lions [4] and Ambrosio [1], and combine these with the concept of kinetic solutions introduced by Lions, Perthame, and Tadmor [10] and Chen and Perthame [2].

A second purpose will be to introduce the audience to the diverse analytic and probabilistic techniques that have led to the above results. In particular, the students will be introduced to the kinetic formulation of skeleton equation and SPDEs of fluctuating hydrodynamics type. The kinetic formulation generalizes the notion of entropy solutions, and it is a particularly useful framework for studying elliptic and parabolic PDEs with degenerate and potentially singular coefficients. Furthermore, the theory relies crucially on optimal entropy-based estimates of the solutions, which will be derived in detail, as well as \( L^\infty \)-based estimates that are based on a novel parabolic Moser iteration extended to the stochastic setting. The lectures will also introduce the students to fundamental notions in probability, including central limit fluctuations and large deviations principles for SPDEs, and demonstrate a rich interplay between analytic methods in elliptic and parabolic equations, stochastic PDEs, and interacting particle systems.

References:

[1] L. Ambrosio. Transport equation and Cauchy problem for BV vector fields. Invent. Math., 158(2):227–260, 2004.

[2] G.-Q. Chen and B. Perthame. Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 20(4):645–668, 2003.

[3] D. Dean. Langevin equation for the density of a system of interacting Langevin processes. J. Phys. A: Math. Gen., 29(24):L613, 1996.

[4] R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98(3):511–547, 1989.

[5] B. Fehrman and B. Gess. Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift. Invent. Math., 234:573–636, 2023.

[6] B. Fehrman and B. Gess. Well-posedness of the Dean–Kawasaki and the nonlinear Dawson–Watanabe equation with correlated noise. Arch. Ration. Mech. Anal., 248(2):Paper No. 20, 60, 2024.

[7] M. Hairer. A theory of regularity structures. Invent. Math., 198(2):269–504, 2014.

[8] K. Kawasaki. Stochastic model of slow dynamics in supercooled liquids and dense colloidal suspensions. Physica A, 208(1):35–64, 1994.

[9] V. Konarovskyi, T. Lehmann, and M.-K. von Renesse. Dean-Kawasaki dynamics: ill-posedness vs. triviality. Electron. Commun. Probab., 24:8, 9, 2019.

[10] P.-L. Lions, B. Perthame, and E. Tadmor. A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc., 7(1):169–191, 1994.

[11] H. Spohn. Large Scale Dynamics of Interacting Particles. Springer, 2012.