Schedule for: 23w6003 - Nonlinear Diffusion and nonlocal Interaction Models - Entropies, Complexity, and Multi-Scale Structures

We will review some basic models of nonlinear diffusion driven by nonlocal operators. Then we will present some lines of recent progress in the study of $p$-Laplacian evolution equations both of local or nonlocal type.

We present recent results with L. Chizat, M. Colombo, A. Figalli, on the infinite-width limit of deep linear neural networks initialized with random parameters. We show that, when the number of neurons diverges, the training dynamics converge (in a precise sense) to the dynamics obtained from a gradient descent on an infinitely wide deterministic linear neural network.

It is well known that the stochastic incompleteness of a Riemannian manifold $M$, namely the fact that the trajectories of the Brownian motion on $M$ do not stay locally bounded almost surely, is equivalent to the non-conservation of probability for the heat semigroup. Such a property is in turn equivalent to the non-uniqueness of bounded solutions of the related Cauchy problem. We establish full equivalence between the property of stochastic incompleteness, which is intrinsic to $M$, and non-uniqueness results for a large class of nonlinear diffusion equations ranging from the fast diffusion to the porous medium regime. Closely related issues for semilinear elliptic equations on $M$ will also be discussed.

In this talk we describe some aspects related to the theory of the anisotropic porous medium equation, in the suitable fast diffusion regime. In particular, we show the existence of selfsimilar fundamental solutions, which is uniquely determined by its mass, and the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. If time permits, we will show similar results related to the study of anisotropic parabolic equations of $p$-Laplacian type. The investigation of both models are objects of joint works with F. Feo and J. L. Vázquez.

Entropy methods coupled to nonlinear diffusions are powerfool tools to study some functional inequalities and identify the set of optimal functions. This lecture is devoted to a review of some recent results of stability that can be achieved using well chosen flows and entropies.

Free boundaries arise naturally in many non-linear elliptic (or parabolic) problems: roughly speaking, free boundaries are interfaces at which solutions display some kind of discontinuous behaviour. Such interfaces are “free” in the sense that they are not prescribed. As an example one can take the boundary of the spatial support of a solution to the porous medium equation. The main (open) problem in the field is to determine if free boundaries are in fact smooth hypersurfaces and what is going on if they are not. A model case that has been studied for long is the so-called Obstacle Problem. In this talk - aimed to a general PDE audience - I will give an introduction to the Obstacle Problem and present the current understanding of the free boundary regularity in this case. I will also explain some recent results (obtained with W. Zatoń) concerning the $C^\infty$ regularity of the free boundary outside of a small set.

Vlasov-Poisson type systems are well known kinetic models for plasma. The precise structure of the model differs according to which species of particle it describes, with the `classical’ version of the system describing electrons. The model for ions, however, includes an additional exponential nonlinearity in the equation for the electrostatic potential, which creates several additional mathematical difficulties. The Debye length is a characteristic length scale of a plasma describing the scale of electrostatic interaction. In real plasmas this length is typically very small, and in physics applications frequently assumed to be very close to zero. This motivates the study of the limiting behaviour of Vlasov-Poisson type systems as the Debye length tends to zero relative to the observation scale—known as the ‘quasi-neutral’ limit. In the case of the ionic model, the formal limit is the kinetic isothermal Euler system; however, this limit is highly non-trivial to justify rigorously and known to be false in some cases without very strong regularity conditions and/or structural conditions.

The Lenard-Balescu equation was formally derived in the 1960s as the fundamental description of the collisional process in a spatially homogeneous system of interacting particles. It can be viewed as correcting the standard Landau equation by taking into account collective screening effects. Due to the reputed complexity of the Lenard-Balescu equation in case of Coulomb interactions, its mathematical theory has remained void apart from the linearized setting. In this contribution, we focus on the case of smooth interactions and we show that dynamical screening effects can then be handled perturbatively. Taking inspiration from the Landau theory, we establish global well-posedness close to equilibrium, local well-posedness away from equilibrium, and we discuss the convergence to equilibrium and the validity of the Landau approximation. Joint work with Mitia Duerinckx (Université libre de Bruxelles).

While the case of Wasserstein gradient flows generated by displacement-convex functionals is well understood, in part due to the possibility of constructing the evolution using the JKO/Minimizing Movement method, the general picture of dissipative evolutions (i.e., when the driving probability vector field is not generated by the Wasserstein gradient of some functional) appears more complicated. The theory seems to present two very different situations, depending on whether the probability vector field is dissipative along any coupling of measures (which we will call totally dissipative) or only along the optimal ones, that minimize the Wasserstein distance. In this talk we will focus on the first case and show that it is possible to construct a fully satisfying Lagrangian theory based on classical results in Hilbert spaces; this leads to new interesting results also in the case of gradient flows. Perhaps more surprisingly, we will also show that a dissipative probability vector field defined everywhere and continuous is always totally dissipative. As a consequence we can see that in the Wasserstein setting it is impossible to approximate a dissipative (but not totally dissipative) probability vector field by continuous ones. This fact clarifies why the case of non-totally dissipative probability vector fields needs to be investigated on an ad hoc basis. (In collaboration with Giulia Cavagnari and Giacomo Sodini)

The convergence of gradients systems $(X,E_\varepsilon,R_\varepsilon)$ in the sense of the “Energy-Dissipation Principle” is discussed. Generalized gradient systems on a Banach space $X$ are given by a driving functional $E_\varepsilon$ (e.g. free energy or relative entropy) and a dissipation potential $R_\varepsilon$. In analogy to $\Gamma$ convergence for functionals, one defines EDP-convergence by looking at functionals only, avoiding the usage of the solutions of the gradient-flow equation; however, convergence of solutions is implied by EDP-convergence. The major feature of EDP convergence is that the effective dissipation $R_\mathrm{eff}$ may depend on the microscopic information in the energy encoded in $E_\varepsilon-E_{\mathrm{eff}}$, and quadratic dissipation potentials $R_\mathrm{\varepsilon}$ may converge to non-quadratic $R_\mathrm{eff}$. The theory is applied to slow-fast gradient systems which arise in chemical reaction-diffusion systems where the fast subsystem moves along a family of Non-Equilibrium Steady States determining the effective slow dynamics. For a scalar diffusion equation with a thin membrane region with low mobility we show that the arising transmission is generated by a nonlinear kinetic relation. The research is partially joint work with Th. Frenzel, M. Peletier, and A. Stephan.

In this talk we will consider conservation equations of the form \[ \begin{cases} u_t = \mathrm{div} ( u^\alpha \nabla v) & \\ -\Delta v = u & \end{cases} \] This system can equivalently be written as one equation $u_t = \mathrm{div} ( u^\alpha \nabla W*u)$ where $W$ is the Newtonian potential. For linear mobility, $\alpha = 1$, the equation and some variations have been proposed as a model for superconductivity or superfluidity. In that case the theory leads to uniqueness of bounded weak solutions having the property of compact space support, and in particular there is a special solution in the form of a disk vortex showing a discontinuous leading front. The aim of the talk is to discuss the cases $\alpha \in (0,1)$ and $\alpha > 1$ which, as for the Porous Medium Equation, exhibit very different behaviours. First, we discuss self-similar solutions. Then, we restrict the analysis to radial solutions and construct solutions by the method of characteristics. We introduce the mass function, which solves an unusual variation of Burger's equation, and plays an important role in the analysis. We show well-posedness in the sense of viscosity solutions and construct a convergent finite-difference numerical schemes. For sublinear mobility $0<\alpha<1$ nonnegative solutions recover positivity everywhere, and moreover display a fat tail at infinity. The model acts in many ways as a regularization of the previous one. For superlinear mobility $\alpha > 1$ we show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case $\alpha = 1$. We show a waiting time phenomena. The talk presents joint work with José A. Carrillo (U. Oxford) and Juan Luis Vázquez (U. Autónoma de Madrid).

In this talk we consider the stability of entropy solutions for nonlinear scalar conservation laws with respect to perturbations of the initial datum, the space-time dependent flux and the entropy inequalities. Such general stability theorem has several primary benefits, among which the possibility to study conditional existence and uniqueness of entropy solutions for evolutions whose flux depends non-locally on the solution itself. In addition, the relaxation of the entropy inequality allows to treat approximate solutions arising from various numerical schemes and to derive their corresponding rates of convergence. In particular, we discuss the case of a recent deterministic particle method introduced to study nonlocal transport models with congestion and we obtain for the first time the (sharp) convergence rate. This is a joint work with E. Marconi and F. Stra.

I will present a recent result in collaboration with Sergio Conti and Stefan Muller. We analyse a nonlocal phase transition model in dimension two known, in the literature of crystal defects (dislocations), as the Nabarro Peierls model. The phase field represents the slip of a crystal (i.e. a lattice) along a given slip plane, the multi-well potential penalises slips which are not lattice preserving, while the singular perturbation with the regularised effect is a non local fractional norm representing the elastic distortion due to incompatibilities of the lattice. Under an appropriate scaling we study the Gamma limit of this energy which results in a macroscopic model for elasto-plastic deformations.

Introduced almost ten years ago, the five gradients inequality has been used to provide estimates on sobolev norms of minimizers involving the Wasserstein distance. In conjunction with the JKO scheme, this inequality can grant compactness for the minimizing movement scheme. We investigate the geometric and functional meaning of the five gradients inequality in two generalizations. In the setting of Lie groups the proof naturally suggest that it is a second order optimality condition for the Kantorovich potentials, while in general compact Riemannian manifolds the curvature plays a role. This is a joint work with Simone Murro and Emanuela Radici.

Stochastic gradient descent is one of the most common optimization algorithms in deep learning. In these applications, the noise intensity generally scales with the objective function to be minimized and the covariance matrix of the gradient estimators has low rank. We discuss impacts of low rank and noise degeneracy in separate models. In a toy model capturing only noise degeneracy, we identify the invariant distribution and discuss convergence in a continuous time model both for `ML type noise' and classical homogeneous and isotropic noise. Our methods are associated to fast diffusion equations and related Poincaré inequalities.

In this talk I will introduce the concept of metaplexes and the dynamical systems on them. A metaplex combines the internal structure of the entities of a complex system with the discrete interconnectivity of these entities in a global topology. We focus here on the study of diffusive processes on metaplexes, both model and real-world examples. We provide theoretical and computational evidence pointing out the role of the endoand exo-structure of the metaplexes in their global dynamics, including the role played by the size of the nodes, the location, strength and range of the coupling between nodes. We show that the internal structure of brain regions (corresponding to the nodes of the network) in the macaque visual cortex metaplex dominates almost completely the global dynamics. On the other hand, in the linear metaplex chain and the landscape metaplex the diffusion dynamics display a combination of the endo- and exo-dynamics, which we explain analytically and in numerical results. Metaplexes are expected to facilitate the understanding of complex systems in an integrative way, which combine dynamical processes inside the nodes and between them.

In some functional inequalities, best constants and minimizers are known. The next question is stability: suppose that a function "almost attains the equality", in which sense it is close to one of the minimizers? In this lecture, I will address a recent result on the quantitative stability of a subfamily of Gagliardo-Nirengerg-Sobolev inequalities. The approach is based on the entropy method for the fast diffusion equation and allows us to obtain completely constructive estimates. The results are based on joint work with M. Bonforte, J. Dolbeault, and B. Nazaret.

The evolution of a cloud of charged particles subjected to an external magnetic field can be described by the magnetized Vlasov–Poisson system. We study the existence of classical solutions to this system by using the velocity moment method introduced by Lions and Perthame. In the case of a constant external magnetic field, one can obtain an explicit representation formula for the charge density which allows us to show propagation of velocity moments directly using an Eulerian approach. Conversely, for a time-dependent and position independent magnetic field this method breaks down and we need to use a Lagrangian approach, where the characteristic flow of the system is carefully studied, in order to show our propagation result. A common interesting feature in both works is the use of a recurrence argument depending on the cyclotron frequency. If time permits, we will also discuss uniqueness results for our system.

I will discuss ongoing works related to Cahn-Hilliard equations which can be seen as gradient flows with respect to transport metrics in probability measures. Homogeneity arguments together with the Nagy inequality leads to a classification of the regimes in the scalar case. Systems of these equations can be derived as limits from aggregation-diffusion models that explain the Steinberg cell sorting mechanism in mathematical biology. They also show a similar behavior in terms of stationary states of the corresponding system. We use gradient flow techniques to show the global existence of steepest descent solutions with certain regularity for the scalar and the system cases. This talk is a summary of works in collaboration with R. Baker, C. Elbar, A. Esposito, C. Falcó, A. Fernández, and J. Skrzeczkowski.

Chemotactic blow up in the context of the Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that when the Keller-Segel equation is coupled with passive advection, blow-up can be prevented if the flow possesses mixing or diffusion-enhancing properties, and its amplitude is sufficiently strong. In this talk, we consider the Keller-Segel equation coupled with an active advection, which is an incompressible flow obeying Darcy's law for incompressible porous media equation and driven by buoyancy force. We prove that in contrast with passive advection, this active advection coupling is capable of suppressing chemotactic blow up at arbitrary small coupling strength: namely, the system always has globally regular solutions. (Joint work with Alexander Kiselev and Zhongtian Hu)

In recent years, there has been a spike in interest in multi-phase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law, Brinkman's law or Darcy's law. While each of these velocity-pressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this paper, we want to address this dearth in the literature providing a rigorous argument that bridges the gap between a viscoelastic tumour model (of Brinkman type) and an inviscid tumour model (of Darcy type).

In this talk I will discuss the connection between nonlocal dynamics on graphs and the corresponding local counterparts in the underlying Euclidean space. Equations on graphs have been recently introduced in applications to data science, social dynamics, and synchronisation. Starting from the nonlocal interaction equation on graphs, we can obtain a class of nonlocal interaction equations with the presence of tensor-mobility encoding the information on the localised graph. This talk is based on joint works with G. Heinze (Augsburg), F. Patacchini (Paris), A. Schlichting (Muenster), and D. Slepčev (Pittsburgh).

In the class of nonlinear diffusion PDEs one of the equations which attracted most attention is for sure the porous medium equation $u_t=\Delta(u^m)$, for which many regularity and qualitative properties are now well-understood. For instance, it is well-known that compactly supported initial data stay compactly supported and that the regularity of the solution drastically improves after all possible holes in the initial support have been filled. Many asymptotic regularity estimates which are available only work after this hole-filling time and include constants depending on the initial support. I will address in this talk a work that we are currently completing together with my postdoc Noemi David, where we obtain new estimates with a different approach: we consider the maximum $M(t)$ of a suitable quantity involving u and Du and obtain estimates such as $M' \leq -cM^2$ or $M' \leq -cM$ (the latter if a suitable uniformly convex confining potential is considered). Unfortunately, the conditions to obtain $c>0$ require that the exponent $m$ should be close enough to $1$, but the same analysis can be performed for the fast diffusion case $m<1$ where this limitation is already contained in the standard one $m>1-2/d$. The results that we are able to prove include instantaneous regularization for $t>0$, asymptotic decay of suitable Lipschitz constants as $t\to\infty$ which are sharp but independent of the initial support, and stronger convergence to the Barenblatt solutions. Some computations also work in the presence of general potentials.

Social dinner at Carmen de la Victoria: https://carmendelavictoria.ugr.es/ Link to map: https://goo.gl/maps/kMXuaCcnGCcCy7YS7

This talk reviews recent work on the Landau equation, a substitute of the Boltzmann equation in the case of charged particles interacting via the Coulomb potential. Our main result is that axisymmetric weak solutions of the Landau equation (constructed by Villani in the late 1990s) are smooth outside the axis of symmetry. (joint work with C. Imbert and A. Vasseur)

After a brief review on the state of the art regarding global well-posedness versus blow up in finite time for the homogeneous Landau equation, we present recent results on existence of solutions from the point of view of $L^p$ theory. The talk will also touch upon open problems, future directions and possible approaches.

This is based on a joint work with Angeliki Menegaki. We study the BGK equation on the 1d torus coupled to a spatially inhomogeneous thermostat. This equation has a non-equilibrium steady state. I will mainly talk about how we show linear stability of this steady state and how this might be connected to flows of macroscopic quantities related to the nature of the NESS.

In this talk we will study the problem of long time propagation of chaos for weakly interacting diffusions. We will show that the natural obstruction to obtain such results are phase transitions of the mean field equation, and show sharp results all the way up to the phase transition in specific simple cases. We show how these ideas are applicable to problems in quantum field theory.

We will present recent results developping the De Giorgi method to prove Hölder continuity and Harnack inequalities, in particular for hypoelliptic equations with rough coefficients. These results are based on Poincaré inequalities derived from the construction of suitable trajectories. These are joint works with Guerand and Anceschi-Dietert-Guerand-Loher-Rebucci.

We will discuss two types of objects that can be approximated in high-dimensions. Recent results have established that sliced-Wasserstein (SW) distance can be approximated accurately in high dimensions based on samples of the measures considered. We will discuss the geometry of the SW distance. In particular we will characterize tangent space to the SW space as a certain weighted negative Sobolev space and obtain the local metric. We show that SW space is not a length space and establish properties of the geodesic distance, relevant to gradient flows in the space. To obtain gradient flows that can be approximated in high dimensions we introduce the projected Wasserstein distance where the space of velocities has been restricted to have low complexity. We will show some of the basic properties of the distance and the corresponding gradient flows. Application towards interacting particle methods for sampling will also be discussed The talk is based on joint works with Sangmin Park and Lantian Xu.

Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications are essential. This has motivated researchers to investigate the problem of adversarial training —or how to make models robust to adversarial attacks— but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore two questions: 1) Can we use analytical tools to find lower bounds for adversarial robustness problems?, and 2) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? I will showcase how ideas from optimal transport theory can provide answers to these questions. This talk is based on joint works with Camilo Andrés García Trillos, Matt Jacobs, and Jakwang Kim.

Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations. An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions? In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.

We consider a free energy functional, consisting of an attractive interaction term with a homogeneous kernel $|x-y|^\lambda$ and a repulsive entropy term involving the $q$-th power of the density. Here $\lambda>0$ and $0 (Main Meeting Room - Calle Rector López Argüeta)

We explore in this talk the links between mesoscopic models (often of reaction-diffusion type) and macroscopic models including at least one cross diffusion term. We explain some modeling issues related to fast reaction limits by detailing examples coming out from population dynamics. We also present the interest of those limits in the mathematical analysis of cross diffusion systems.

We consider a population structured with a phenotypic trait, which influences the adaptation of individuals to their environment and the population is subjected to small mutations. The problem involves two time scales: the one in which the population evolves and the one where the small mutations occur. The expected long-time behaviour is that the population concentrates around the dominant traits. The dominant traits evolve in time with the effect of mutations. The concentration phenomenon is studied mathematically by means of a Hopf-Cole transform and in the asymptotic limit, a constrained Hamilton-Jacobi equation is obtained [Barles, Perthame 2008]. The uniqueness of the constrained Hamilton-Jacobi equation is shown recently [Calvez, Lam 2020]. We present a numerical treatment of this model and the main challenge comes from the nonlinearity of the constraint leading to jumps in the solution. More precisely, we propose an asymptotic preserving scheme for the problem transformed with Hopf-Cole and show that the approximations converge to the solution of the PDE and the scheme is asymptotically stable. We also show that the limit scheme is convergent for the constrained Hamilton-Jacobi equation. This is a joint work with V. Calvez and H. Hivert.

The nonlinear integrate-and-fire equation is a mean-field model used for the behaviour of large groups of neurons, and exhibits a wide variety of phenomena: stable and unstable steady states, blow-up, and periodic solutions when a delay is added. By using spectral gap techniques in kinetic theory and tools from delay equations we are able to give a simple criterion for the stability and instability of equilibria, which can be numerically checked in all cases, and can be checked analytically in some of them. This is a collaboration with María José Cáceres and Alejandro Ramos Lora.

We will present a new quantitative approach to the problem of proving hydrodynamic limits from microscopic stochastic particle systems, namely the zero-range and the Ginzburg-Landau process with Kawasaki dynamics, to macroscopic partial differential equations. Our method 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 is simplified as it avoids the use of the block estimates. This is a joint work with Daniel Marahrens and Clément Mouhot (University of Cambridge).

After a quick derivation of the quasilinear diffusion models for electrostatic and highly magnetized plasma systems, We will present their numerical approximations to non-equilibrium statistical states. This is model reduction of Vlasov-Maxwell systems in nonequilibrium regimes showing that perturbation of moving Gaussians exhibit instabilities that derail the approximation of Maxwellian limiting states for long time. In addition, we show that the electrostatic case model, for a one dimensional momenta-spectral space system, admits existence of solutions in Sobolev spaces for long time. The proof is based on strong connections to Porous media flows models with non-linear gradient forms and source terms. Its stationary state may not be a Maxwellian state for the probability distribution in momenta space. This is work in collaboration with Kun Huang, Michael Abdelmalik and Boris Briezman, and just with Kun Huang for the analytical properties of the electrostatic case model.