Schedule for: 21w5127 - New Trends in Nonlinear Diffusion: a Bridge between PDEs, Analysis and Geometry (Online)

The Stefan problem describes phase transitions such as ice melting to water, and it is among the most classical free boundary problems. It is well known that the free boundary consists of a smooth part (the regular part) and singular points. In this talk, I will describe a recent result with Ros-Oton and Serra, where we analyze the singular set in the Stefan problem and prove a series of fine results on its structure.

We study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions $n\geq3$ is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons’ cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open. The goal of this talk is to present some new results in this direction. On the one hand we find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of "large solutions" for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions \(n<6\) is one-dimensional, independently of the parameter $s\in(0,1)$.

In electrostatic Born-Infeld theory, the electrostatic potential $u_\rho$ generated by a charge distribution $\rho$ on $\mathbb{R}^m$ (typically, a Radon measure) is required to minimize the action \[ \int_{\mathbb{R}^m} \Big( 1 - \sqrt{1-|D\psi|^2} \Big) d x - \langle \rho, \psi \rangle \] among functions with a suitable decay at infinity and satisfying $|D\psi| \le 1$. Formally, the Euler-Lagrange equation $(\mathcal{BI})$ prescribes $\rho$ as being the Lorentzian mean curvature of the graph of $u_\rho$ in Minkowski spacetime $\mathbb{L}^{m+1}$; for instance, if $\rho$ is a finite sum of Dirac deltas, then the graph of $u_\rho$ is a maximal spacelike hypersurface with singularities in $\mathbb{L}^{m+1}$. While the existence/uniqueness of $u_\rho$ follows from standard variational arguments, finding sharp conditions on $\rho$ to guarantee that $u_\rho$ solves $(\mathcal{BI})$ is an open problem that has been addressed only in a few special cases. In this talk, I will report on a recent joint work with J. Byeon, N. Ikoma and A. Malchiodi, where we study the solvability of $(\mathcal{BI})$ and the regularity of $u_\rho$ under mild conditions on $\rho$. One of the main sources of difficulties is the possible presence of light rays in the graph of $u_\rho$, which will be discussed in detail.

It is joint work with Boris Vertman (Oldenburg) and Jørgen Olsen Lye (Oldenburg). We study the convergence of the normalized Yamabe flow with positive Yamabe constant on a class of pseudo-manifolds that includes stratified spaces with iterated cone-edge metrics. We establish convergence under a low-energy condition. We also prove a concentration-compactness dichotomy, and investigate what the alternatives to convergence is.

We study the regularization effect of the inhomogeneous Boltzmann equation without cutoff. We obtain a priori estimates for all derivatives of the solution depending only on bounds of its hydrodynamic quantities: mass density, energy density and entropy density. We use methods that originated in the study of nonlocal elliptic and parabolic equations: a weak Harnack inequality in the style of De Giorgi, and a Schauder-type estimate.

I will review some results for the construction of invariant tori in infinite dimensional systems modeled on lattices and (some) PDEs, with an emphasis on ill-posed PDEs arising in fluids. I will in particular work out the details for the Boussinesq equation and some other long-wave approximations of the water wave system.

We are concerned with finding Fokker-Planck equations in whole space with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrarily close to its infimum. This infimum is $1$, corresponding to the high-rotational limit in the Fokker-Planck drift. Such an "optimal" Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. The proof is based on the recent result that the $L^2$-propagator norms of the Fokker-Planck equation and of its drift-ODE coincide. This talk is based on joint work with Beatrice Signorello.

We consider the spectrum of non-homogeneous partially hinged plates having structural engineering applications. A possible way to prevent instability phenomena is to optimize the frequencies of certain oscillating modes with respect to the density function of the plate; we prove existence of optimal densities and we investigate their qualitative properties. The analysis is carried out by showing fine properties of the involved fourth order operator, such as the validity of the positivity preserving property. Based on a joint work with Alessio Falocchi.

A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid, when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on the existence and asymptotic behaviour of these solutions. We describe, with precise asymptotics, interacting vortices, and travelling helices. We rigorously establish the law of motion of "leapfrogging vortex rings", originally conjectured by Helmholtz in 1858. This is joint work with Juan Davila, Monica Musso, and Juncheng Wei.

We study the heat equation in a half-space or the exterior of the unit ball with a dynamical boundary condition. In this talk, we construct a global-in-time solution of this problem and show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition. Furthermore, we give the optimal rate of convergence.

The Vlasov-Poisson system with massless electrons (VPME) is widely used in plasma physics to model the evolution of ions in a plasma. It differs from the classical Vlasov-Poisson system in that the Poisson coupling has an exponential nonlinearity that creates several mathematical difficulties. In this talk, we will discuss the well-posedness of VPME, the stability of solutions, and its behaviour under singular limits. Then, we will introduce a new class of Wasserstein-type distances specifically designed to tackle stability questions for kinetic equations. As we shall see, these distances allow us to improve classical stability estimates by Loeper and Dobrushin and to obtain, as a consequence, improved rates in quasi-neutral limits.

Aggregation-diffusion models describe the motion of interacting agents towards states of overall balance between diffusion effects and mutual attraction. The Newtonian and the Riesz interaction potentials provide relevant examples of aggregation modeling with long range effects. They give rise to local and nonlocal PDEs for the characterization of stationary states: we will focus on existence, uniqueness and regularity properties of radial entire solutions to the equilibrium equations. This is a joint work with H. Chan, M.D.M. González, Y. Huang and B. Volzone.

I present some results concerning the size of magnetic fields that support zero modes for the three dimensional Dirac equation and related problems for spinor equations. The critical quantity, is the $3/2$ norm of the magnetic field $B$. The point is that the spinor structure enters the analysis in a crucial way. This is joint work with Rupert Frank at Caltech.

We present regularity estimate for the isotropic analogue of the homogeneous Landau equation. This is done for interaction in the range of very soft potentials. The main observation is that the classical weighted Hardy inequality leads to a weighted Poincaré inequality, which in turn implies the propagation of high $ L^p $ norms of solutions. From here, the boundedness follows from certain weighted Sobolev inequalities and De Giorgi-Nash-Moser theory. This is a joint work with N. Guillen.

The competition between directional dispersal caused by Brownian forcing and tendency towards concentration caused by alignment is studied. Main results are the stability of uniform states for dominating Brownian forcing (proven by hypocoercivity methods) as well as the existence of nontrivial steady states (shown by a bifurcation approach).

Given a desired target distribution and an initial guess of that distribution, composed of finitely many samples, what is the best way to evolve the locations of the samples so that they accurately represent the desired distribution? A classical solution to this problem is to allow the samples to evolve according to Langevin dynamics, the stochastic particle method corresponding to the Fokker-Planck equation. In today’s talk, I will contrast this classical approach with a deterministic particle method corresponding to the porous medium equation. This method corresponds exactly to the mean-field dynamics of training a two layer neural network for a radial basis function activation function. We prove that, as the number of samples increases and the variance of the radial basis function goes to zero, the particle method converges to a bounded entropy solution of the porous medium equation. As a consequence, we obtain both a novel method for sampling probability distributions as well as insight into the training dynamics of two layer neural networks in the mean field regime.

We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in $N$-dimensional convex domains, where $N\ge 2$. Jointly with what we already know, i.e. that log-concavity is the strongest power concavity preserved by the Dirichlet heat flow, we see that log-concavity is indeed the only power concavity preserved by the Dirichlet heat flow. This talk is based on a joint work with Paolo Salani (Univ. of Florence, Italy) and Asuka Takatsu (Tokyo Metropolitan Univ., Japan).

This talk is concerned with a quantitative analysis of asymptotic behavior of solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles will be discussed via an energy method. The rate of convergence for positive profiles was recently discussed based on an entropy method by Bonforte and Figalli (2021, CPAM). An alternative proof will also be provided to their result.

We discuss existence of global-in-time solutions to the porous medium equation with a reaction term on Riemannian manifolds, where Sobolev and Poincaré inequalities are assumed to hold. Smoothing estimates are also established. The results have been recently obtained jointly with Gabriele Grillo and Giulia Meglioli (Politecnico di Milano).

In this talk we will review several recent results, in collaboration with Carmen Cortázar (PUC, Chile) and Noemí Wolanski (IMAS-UBA-CONICET, Argentina), on the large-time behaviour of solutions to fully nonlocal heat equations involving a Caputo time derivative and a power of the Laplacian. The Caputo time derivative introduces memory effects that yield new phenomena that are not present in classical diffusion equations.

In this first lecture of a series of three, we discuss stability results in Gagliardo-Nirenberg-Sobolev inequalities, from a joint project with M. Bonforte, J. Dolbeault and N. Simonov. The core of this approach is the use of a non scaling invariant form of the inequalities, equivalent to entropy-entropy production inequalities arising in the study of large time asymptotics for solutions to fast diffusion equations. We only use variational arguments, leading to non constructive estimates, but this paves the way for the constructive results given in the next two lectures.

In this talk, I will discuss the asymptotic behavior of solutions to the fast diffusion equation when the tails of the initial datum have a certain decay. In this setting, I will provide a fully constructive estimate of the threshold time after which the solution enters a neighborhood of the Barenblatt profile in a uniform relative norm. This estimate plays a fundamental role in obtaining a constructive stability result in Gagliardo-Nirenberg-Sobolev inequalities. The results are based on a joint work with Matteo Bonforte, Jean Dolbeault, and Bruno Nazaret.

This lecture is the third lecture on stability issues in Gagliardo-Nirenberg-Sobolev inequalities, a joint project with M. Bonforte, N. Simonov and B. Nazaret. The results are based on entropy methods and the use of the fast diffusion equation (FDE) for studying refined versions of the Gagliardo-Nirenberg-Sobolev inequalities. Using the quantitative regularity estimates, we go beyond the variational results of the first lecture and provide fully constructive estimates, to the price of a small restriction of the functional space which is inherent to the method.

We review developments in geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. We especially discuss a nonlinear analogue of the Gamma-calculus and its applications to isoperimetric and functional inequalities.

In this talk, I will discuss a nonlocal aggregation equation with degenerate diffusion, which describes the mean-field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all stationary solutions must be radially decreasing up to a translation, but uniqueness (for a given mass) within this class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/non-uniqueness criteria are determined by the power of the degenerate diffusion, with the critical power being $m=2$. Namely, for $m \geq 2$, we show the stationary solution for any given mass is unique for any attractive potential, by tracking the associated energy functional along a novel interpolation curve. And for $1< m < 2 $, we construct examples of smooth attractive potentials, such that there are infinitely many radially decreasing stationary solutions of the same mass. This is a joint work with Matias Delgadino and Xukai Yan.

When one looks for radial solutions of an equation with fractional Laplacian, it is not generally possible to use standard ODE methods. If such equation has some conformal invariances, then one may rewrite it in Emden-Fowler (cylindrical) coordinates and use the properties of the conformal fractional Laplacian on the cylinder, which involves some complex analysis techniques. After giving the necessary background, we will briefly consider two particular applications of this technique: 1. Symmetry breaking, non-degeneracy and uniqueness for the fractional Caffarelli-Kohn-Nirenberg inequality (joint work with W. Ao and A. DelaTorre). 2. Existence and regularity for fractional Laplacian equations with drift and a critical Hardy potential (joint with H. Chan, M. Fontelos and J. Wei).

Under the validity of the positive mass theorem, the Yamabe flow on a smooth compact Riemannian manifold of dimension greater or equal to $3$ is known to exist for all time and converges to a solution to the Yamabe problem at infinity. In this talk I will present a result, obtained in collaboration with Seunghyeok Kim, in which we prove that if a suitable perturbation, which may be smooth and arbitrarily small, is imposed on the Yamabe flow on any given Riemannian manifold M of dimension bigger or equal to $5$, the resulting flow may blow up at multiple points on M in the infinite time. We construct such a flow by using solutions of the Yamabe problem on the unit sphere as blow-up profiles. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

This talk addresses the spectral stability of monotone traveling front solutions for reaction-diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusivities which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). Spectral stability is understood as the property that the spectrum of the linearized operator around the wave, acting on an exponentially weighted space, is contained in the complex half plane with non-positive real part. The degenerate fronts under consideration travel with positive speed above a threshold value and connect the (diffusion-degenerate) zero state with the unstable equilibrium point of the reaction function. In this case, the degeneracy of the diffusion coefficient is responsible of the loss of hyperbolicity of the asymptotic coefficient matrices of the spectral problem at one of the end points, precluding the application of standard techniques to locate the essential spectrum (cf. Kapitula, Promislow, 2013). This difficulty is overcome with a suitable partition of the spectrum, a generalized convergence of operators technique, the analysis of singular (or Weyl) sequences and the use of energy estimates. The monotonicity of the fronts, as well as detailed descriptions of the decay structure of eigenfunctions on a case by case basis, are key ingredients to show that all traveling fronts under consideration are spectrally stable in a suitably chosen exponentially weighted $L^2$ energy space. This is joint work with J. F. Leyva (Benemérita Universidad Autónoma de Puebla) y L. F. López Ríos (IIMAS-UNAM).

I will describe results on global existence, stability and interior electroneutrality for Nernst-Planck equations coupled with Navier-Stokes and related equations.

We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow. This has surprising consequences in the context of inviscid damping in 2D Euler and enhanced dissipation in Navier-Stokes.

This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the $L^2$ framework, nonlinear diffusion needs the stronger $L^\infty$ topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.

A manifold is said to be stochastically complete if the free heat semigroup preserves probability. It is well known that this property is equivalent to nonexistence of nonnegative, bounded solutions to certain (linear) elliptic problems, and to uniqueness of solutions to the heat equation corresponding to bounded initial data. We prove that stochastic completeness is also equivalent to similar properties for certain nonlinear elliptic and parabolic problems. This fact, and the known analytic-geometric characterizations of stochastic completeness, allow to give new explicit criteria for existence/nonexistence of solutions to certain nonlinear elliptic equations on manifolds, and for uniqueness/nonuniqueness of solutions to certain nonlinear diffusions on manifolds.

It is known that the projection of a uniform probability measure on the $N$-dimensional sphere to the first $n$ coordinates approximates the $n$-dimensional Gaussian measure. In this talk, I will present that the spectral structure on the $N$-dimensional sphere compatible with the projection to the first $n$ coordinates approximates the spectral structure on the $n$-dimensional Gaussian space.

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation on the whole Euclidean space, in the so-called "good fast diffusion range". It is well known that non-negative solutions behave for large times as B, the Barenblatt (or fundamental) solution, which has an explicit expression. We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of B. This can be considered the nonlinear counterpart of the celebrated Gaussian estimates for the linear heat equation. To the best of our knowledge, analogous issues for the linear heat equation, do not possess such clear answers, only partial results are known. Also, we characterize the maximal (hence optimal) class of initial data such that the GHP holds, by means of an integral tail condition, easy to check. Finally, we derive sharp global quantitative upper bounds of the modulus of the gradient of the solution, and, when data are radially decreasing, we show uniform convergence in relative error for the gradients. This is joint work with Matteo Bonforte (UAM-ICMAT, Madrid, Spain) and Nikita Simonov (Ceremade-Univ. Paris-Dauphine, Paris, France).

We provide new direct methods to establish symmetrization results in the form of mass concentration (i.e. integral) comparison for fractional elliptic equations of the type $ (-\Delta)^s u =f \ $ ($ 0 < s < 1 $) in a bounded domain $ \Omega $, equipped with homogeneous Dirichlet boundary conditions. The classical pointwise Talenti rearrangement inequality is recovered in the limit $ s\rightarrow1 $. Finally, explicit counterexamples constructed for all $ s\in(0,1) $ highlight that the same pointwise estimate cannot hold in a nonlocal setting, thus showing the optimality of our results. The results are contained in a joint paper with Bruno Volzone [Ferone, V.; Volzone, B., Symmetrization for fractional elliptic problems: a direct approach. Arch. Ration. Mech. Anal. 239 (2021), 1733-1770].

We show the stability of viscous shocks of the 1D compressible Navier-Stokes equation. This stability holds uniformly with respect to the viscosity, up to the inviscid limit. Stability results for shocks of the Euler equation are then inherited at the inviscid limit. These stability results hold in the class of wild perturbations of inviscid limits, without any regularity restriction. This shows that the class of inviscid limits of Navier-Stokes equations is better behaved than the larger class of weak entropic solutions to the Euler equation. The result is based on the theory of a-contraction with shifts. This is a joint work with Moon-Jin Kang.

In this talk we will focus on the the existence, multiplicity and local behavior of singular solutions of the fractional Lane–Emden equation with Serrin’s critical exponent and homogeneous Dirichlet exterior condition. These will provide the profile to construct singular metrics with constant (non-local) curvature. We will show radial symmetry close to the origin, a Liouville-type result without any assumption on its asymptotic behavior (showing the necessity of imposing the Dirichlet condition) and the existence of multiple solutions in a bounded domain with any prescribed closed singular set. Moreover, we will show that the singular behavior of the profile is unique, presenting new methods based on the connection between the non-local equation and its associated first order ODE in one dimension. This is a joint work with H. Chan.

The classical Liouville Theorem states that bounded harmonic functions are constant. The talk will revisit this result for the most general class of linear operators with constant coefficients satisfying the maximum principle (characterized by Courrège in [P. Courrège, Générateur infinitésimal d’un semi-groupe de convolution sur $R^n$ , et formule de Lévy-Khinchine. Bull. Sci. Math. (2), 88:3–30, 1964]). The class includes local and nonlocal and not necessarily symmetric operators among which you can find the fractional Laplacian, Relativistic Schrödinger operators, convolution operators, CGMY, as well as discretizations of them. We give a full characterization of the operators in this class satisfying the Liouville property. When the Liouville property does not hold, we also establish precise periodicity sets of the solutions. The techniques and proofs of [N. Alibaud, F. del Teso, J. Endal, and E. R. Jakobsen, The Liouville theorem and linear operators satisfying the maximum principle. Journal de Mathématiques Pures et Appliquées, 142:229–242, 2020] combine arguments from PDEs, group the- ory, number theory and numerical analysis (and still, they are simple, short, and very intuitive).

We consider a number of equations involving nonlinear fractional Laplacian operators where progress has been obtained in recent years. Examples include fractional $p$-Laplacian operators appearing in elliptic and parabolic equations and a number of variants. Numerical analysis is performed.