Analysis Seminar

Whether the Landau equation can develop a finite time singularity is an important open problem in kinetic equations. In this talk, we will discuss the slightly perturbed homogeneous Landau equation with very soft potentials, where we increase the nonlinearity from $ c(f) f$ in the Landau equation to $\alpha c(f) f$ with $\alpha>1$. For $\alpha > 1 $ and close to $1$, we establish finite time nearly self-similar blowup from some smooth nonnegative initial data, which can be radially symmetric or non-radially symmetric. The blowup results are sharp as the homogeneous Landau equation $(\alpha=1)$ is globally well-posed, which was recently established by Guillen and Silvestre. The proof builds on our previous framework on sharp blowup results of the De Gregorio model with nearly self-similar singularity to overcome the diffusion. Our results shed light on potential singularity formation in the inhomogeneous setting.

The Boltzmann hierarchy is an infinite system of coupled equations that are instrumental for the rigorous derivation of the Boltzmann equation from many particles. In this talk we will first show uniqueness of the mild solutions to the Boltzmann hierarchy by combining, for the first time, a combinatorial technique known as the Klainerman-Machedon board game argument together with an $L^\infty$-based estimate. Then we will show existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data. The proof of existence is constructive, and employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.

In this talk we will focus on the derivation and the analysis of the inhomogeneous kinetic wave equation (IKWE). Although there is an extended amount of literature for the space homogeneous KWE, the inhomogeneous problem has barely been addressed. In the first part we examine the validity of the IKWE for quadratic nonlinearities and dispersion relations close to Laplacian, based on joint work with Collot, Germain. In the second part of the talk, we investigate the global well-posedness and stability of the $4$-wave inhomogeneous kinetic wave equation near vacuum, employing techniques from classical kinetic theory, such as the Kaniel-Shinbrot iteration.

Far-from-equilibrium behavior in physical systems is widespread. A statistical description of these events is provided by macroscopic fluctuation theory, a framework for non-equilibrium statistical mechanics that postulates a formula for the probability of a space-time fluctuation based on the constitutive equations of the system. This formula is formally obtained via a zero noise large deviations principle for the associated fluctuating hydrodynamics, which postulates a conservative, singular stochastic PDE to describe the system out-of-equilibrium. In this talk, we will focus particularly on the fluctuations of certain interacting particle processes about their hydrodynamic limits. We will show how the associated MFT and fluctuating hydrodynamics lead to a class of conservative SPDEs with irregular coefficients, and how the study of large deviations principles for the particles processes and SPDEs leads to the analysis of parabolic-hyperbolic PDEs in energy critical spaces. The analysis makes rigorous the connection between MFT and fluctuating hydrodynamics in this setting, and provides a positive answer to a long-standing open problem for the large deviations of the zero range process.

We consider axisymmetric incompressible inviscid flows without swirl. When the axial vorticity is non-positive in the upper half space and odd in the last coordinate, we call the flow anti-parallel and we may expect a head-on collision of anti-parallel vortex rings. By establishing monotonicity and infinite growth of the vorticity impulse on the upper half-space, we obtain infinite growth of vorticity maximum at infinite time for certain classical vorticity. On the other hand, a finite but faster growth for some smooth vorticity is obtained thanks to the stability of Hill's vortex. This talk is based on joint work with In-Jee Jeong(SNU).

The breakthrough of Mouhot and Villani (2011) on the so called Landau Damping for the collision-less Vlasov equation \begin{eqnarray} & \partial_t F +v\cdot \nabla_x F \pm \nabla_x V\cdot\nabla_v F=0 \; \hbox{with} \; (x,v) \in (\Omega \subset (\mathbf R_v)^d\times \mathbf R^d)\\ & -\Delta V =G+f = F- \langle F\rangle, \; \langle F \rangle =\int_{\Omega} F(x,v,t) dx \end{eqnarray} has generated in the mathematical community many activities around the qualitative behavior (in particular for large time) of their solutions and for the behavior of the space independent average: \begin{equation} G(v,t) = \int_{\Omega} G(t,v,x)dx\,. \end{equation} As I intend to show, starting from physical motivation, such behavior depends on the spectra of the linearized operator and on the size of the perturbation. Existing mathematical results are in line with a very rich and diverse behavior.

In this presentation, I will establish the global existence of entropy weak solutions for scalar balance laws with nonlocal singular sources, along with a partial uniqueness result. A detailed description of the solution is provided for a general class of initial data in a neighborhood where two shocks interact.

In this talk, we give an overview of several recent results where quantitative estimates play a key role. In the first part of the talk, we discuss convex integration constructions for fluid systems with external forcing. We will then discuss a novel application of these ideas to the surface quasi-geostrophic (SQG) equation. Moving forward with the theme of quantitative estimates, in the second part of the talk we will describe new bounds for the defocusing energy-supercritical Nonlinear Schrödinger equation (NLS) and use these to give a universal blow-up criteria which goes below the scaling invariant threshold. These results are in line with a recent breakthrough construction of finite-time blow-up solutions, and in particular give the first generic result distinguishing potential defocusing blow-up phenomena from many of the known examples of blow-up in the focusing setting. At the end of the talk, we will briefly describe applications to related models.

We consider the long-time behavior of a passive scalar advected by an incompressible velocity field. If the velocity field is autonomous or time periodic, long-time behavior follows by studying the spectral properties of the transfer operator associated with the finite time flow map. When the flow is uniformly hyperbolic, it is well known that it is possible to construct anisotropic Sobolev spaces where the transfer operator becomes quasi-compact with a spectral gap, yielding exponential decay in these spaces. In the non-autonomous and non-uniformly hyperbolic case this approach breaks down. In this talk, I will discuss how in the stochastic ergodic velocity setting one can recover analogous results under the expectation using pseudo differential operators to obtain exponential decay of solutions to the transport equation from $H^{-\delta}$ to $H^{-\delta}$ -- a property we call annealed mixing. As a result, we show that the Markov process obtained by considering advection with a source term has a unique stationary measure. This is a joint work with Jacob Bedrossian and Patrick Flynn.

The dynamics for the interface in many free boundary problems is driven by the normal derivative of a corresponding pressure function, and often the free boundary is the boundary of the positivity set of this unknown pressure, which evolves in time. One well-known example of these types of models is called the Hele-Shaw problem. In this talk we will describe the set-up of some of these free boundaries and show how in many situations, their solution becomes equivalent to solving a nonlinear fractional heat equation (in one fewer space variables). These fractional heat equations fall into the general scope of what are called Hamilton-Jacobi-Bellman equations, which have enoyed extensive study in the past 20 years or so (at least for the fractional setting, and much longer for the first and second order settings). Furthermore, many of the well established properties about existence, uniqueness, and regularity for Hamilton-Jacobi-Bellman equations can then be transferred back to the original free boundary problem. We will discuss various recent results in this direction.

We will present the conjecture of vortices concentrated around a filament for the 3D Euler equations. While this question is open in general, we will give several geometric configurations where results have been obtained: 2D Euler, axisymmetric 3D Euler without swirl, lake equations and helical 3D Euler without swirl. I will show the link between the kernels of the Biot-Savart law for these equations and for the 2D Euler equations. I will present the main arguments to show the persistence of the concentration towards filaments that move according to the "binormal curvature flow". This work is in collaboration with Martin Donati, Lars Eric Hientzsch and Evelyne Miot.

The kinetic theory of plasma features a surprising variety of equations such as the Landau equation, the Vlasov-Poisson equation and the Balescu-Lenard equation. This wealth of different dynamics is due to the onset of collective effects and the scale-invariance of the Coulomb potential. In this talk, we discuss the theory in the framework of scaling limits of interacting particle systems and present recent advances.

The talk is concerned with semilinear elliptic inequalities with a potential on infinite weighted graphs. Given a distance on the graph, we assume an upper bound on its Laplacian, and a growth condition on a suitable weighted volume of balls. Under such hypotheses, we discuss the nonexistence of nonnegative nontrivial solutions. We also see the optimality of our conditions.

Such results have been recently obtained jointly with D.D. Monticelli and J. Somaglia.

Introduction to nonlinear models, to anomalous diffusion and to nonlinear fractional diffusion. The focus will be on two main models: the Gagliardo norms and the evolution sp-Laplacian, and the fractional Stefan problem.

In a recent preprint with Luis Silvestre we show that smooth solutions to the homogeneous Landau equation do not break down in finite time for a broad range of potentials — including Coulomb. This result, which relies on showing the monotonicity for the Fisher information, is made possible by three ingredients: 1) a “lifting” of the Landau equation to a linear degenerate parabolic PDE in double the number of variables, 2) a decomposition of this linear PDE in terms of rotations and 3) a functional inequality on the sphere that is closely related to the log-Sobolev inequality. In this talk I will describe these ingredients starting by motivating some of them for the simpler case of the heat equation.

The Landau equation is an important subject in kinetic equations. Recently, Guillen and Silvestre showed that the solutions of the Landau equation remain bounded for very soft potentials, including Coulombic potentials. Their breakthrough was achieved by proving that the Fisher information is monotone decreasing. An isotropic modification of the Landau equation was introduced by Krieger and Strain and has been studied over the last decade. Based on the method of Guillen-Silvestre, we prove that the Fisher information of the isotropic Landau equation (the Krieger-Strain equation) is also monotone decreasing. As a consequence, we establish the global existence of smooth solutions for a reasonable class of initial data. This talk is based on joint work with David Bowman.

We will review several recent results on global dynamics of radial self-gravitating compressible Euler flows, which arise in the mathematical description of stars. We will discuss classes of smooth initial data that lead to the formation of imploding finite-time singularities. Our main focus is on the role of scaling invariances and their interaction with the nonlinearities.

In this talk, I will present a result obtained in a recent paper about the study of the stability in time of a family $(\underline{U}_\epsilon)_{0 < \epsilon < \epsilon_0}$ of traveling waves solutions to \begin{align*} \partial_t u+\partial_x (f(u))=g(u)+\epsilon \partial_x^2u \end{align*} that approximate a given Riemann shock, and we aim at showing some uniform asymptotic orbital stability result of these waves under some conditions that guarantee the asympotic orbital stability of the corresponding Riemann shock, as proved in a previous work of V. Duchêne and L. M. Rodrigues. Even at the linear level, to ensure uniformity in $\epsilon$, the decomposition of the Green function associated with the (fast-variable) linearization about $\underline{U}_\epsilon$ of the above equation into a decreasing part and a phase modulation is carried out in a highly non-standard way. Furthermore, we introduce a multi-scale norm depending in $\epsilon$ that is the usual $W^{1,\infty}$ norm when restricted to functions supported away from the shock location. To avoid the use of arguments based on parabolic regularization that would preclude a result uniform in $\epsilon$, we close nonlinear estimates on this norm through some suitable maximum principle.

G-equation is a well known level set model in turbulent combustion, and becomes an advective mean curvature type evolution equation when the curvature effect is considered: $$ G_t + \left(1-d\, \mathrm{div}{\frac{DG}{|DG|}}\right)_+|DG|+V(x)\cdot DG=0. $$ In this talk, I will show the existence of effective burning velocity under the above curvature G-equation model when $V$ is a two dimensional cellular flow, which can be extended to more general two dimensional incompressible periodic flows. Our proof combines PDE methods with a dynamical analysis of the Kohn-Serfaty deterministic game characterization of the curvature G-equation based on the two dimensional structures. In three dimensions, the effective burning velocity will cease to exist even for simple periodic shear flows when the flow intensity surpasses a bifurcation value.

The dynamics for the interface in many free boundary problems is driven by the normal derivative of a corresponding pressure function, and often the free boundary is the boundary of the positivity set of this unknown pressure, which evolves in time. One well-known example of these types of models is called the Hele-Shaw problem. In this talk we will describe the set-up of some of these free boundaries and show how in many situations, their solution becomes equivalent to solving a nonlinear fractional heat equation (in one fewer space variables). These fractional heat equations fall into the general scope of what are called Hamilton-Jacobi-Bellman equations, which have enjoyed extensive study in the past 20 years or so (at least for the fractional setting, and much longer for the first and second order settings). Furthermore, many of the well established properties about existence, uniqueness, and regularity for Hamilton-Jacobi-Bellman equations can then be transferred back to the original free boundary problem. We will discuss various recent results in this direction.

Even though homogenization has a long history, this talk revisits this problem by providing a completely variational approach in obtaining a homogenization result for Fokker Planck equation (FKE). FKE is interpreted as a gradient flow in Wasserstein space in connection with optimal transport. We make use of Sandier-Serfaty's characterization of gradient flows. The spatial inhomogeneity is introduced through a Benamou-Brenier formulation. The limiting procedure is achieved by means of gamma convergence. The talk is based on joint work with Yuan Gao.

When a mean field game has a unique mean field equilibrium, the value function on the unique equilibrium would satisfy a master equation, which is an infinitely dimensional PDE on the Wasserstein space of probability measure. This equation is wellposed when the time duration is small. However, for large time duration, it is well known that mean field games typically have multiple equilibria, and the uniqueness requires certain structural conditions, mostly known as monotonicity conditions. In this talk we will go over various types of monotonicity conditions and explain how they lead to the global (in time) wellposedness of the master equations.

For $ k \in \{2, \ldots , n \} $, the $ k $-th mean curvature of an embedded hypersurface in the $ (n+1) $-dimensional Euclidean space is given by the $ k $-th elementary symmetric polynomial of its principal curvatures. In this talk I discuss some recent results on the asymptotic behaviour of sequences of closed hypersurfaces in the Euclidean space whose $ k $-th mean curvature converges in $ L^1 $ to a constant.

We present a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the current distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems up to degrees of freedom given by rotations: one for the evolution of mean and covariance of the interpolating curve, and one for its shape. Similarly, on the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target.

The derivation of the quantum Boltzmann equation is an open problem in mathematical physics, with at least fifty years of history. On the other hand, the derivation of mean-field equations from quantum systems has seen important developments in the last two decades, and is now regarded as a well-established field. In particular, this progress has led to the rigorous implementation of key physical ideas. Among them, is the “bosonization” of Fermi gases at low temperatures. In this talk, I present new results in which the emergence of quantum Boltzmann dynamics is linked to this phenomenon. This talk is based on joint work with Thomas Chen.

We consider a class of nonlocal conservation laws in which the flux function depends on the solution $\rho$ through the convolution with an exponential kernel. We show that quantities involving the nonlocal impact, $W:={1}_{(-\infty,0]}(\cdot)\exp(\cdot) \ast \rho(t,\cdot)$, satisfy an Oleinik-type entropy condition. As a byproduct, we deduce that, when the nonlocality is shrunk to a local evaluation (i.e., when the kernel tends to a Dirac delta distribution), the (unique) weak solution of the nonlocal problem converges to the (unique) entropy-admissible solution of the corresponding local conservation law, under the assumption that the initial datum is essentially bounded and not necessarily of bounded variation. Furthermore, the Oleinik-type inequality—combined with a suitable scaling argument—allows us to show the long-time convergence of a nonlocal regularization of the Burgers equation towards the local N-wave solution.

Graph-based semi-supervised learning is a field within machine learning that uses both labeled and unlabeled data with an underlying graph structure for classification and regression tasks. Recent work has established connections between graph-based semi-supervised learning and boundary value PDE problems involving the p-Laplacian (including p=\infty) and various types of singularly weighted Laplace and Poisson equations in the big data continuum limit. In this talk, we will overview some of this work and highlight where PDE analysis can play a role in machine learning. In the latter part of the talk, we will focus on a specific recent work on the continuum limit of Poisson learning, which involves the discrete graph to continuum PDE scaling limit with measure-valued data.

The seminal result of Benamou and Brenier provides a dynamical description of the Wasserstein distance as the path of the minimal action in the space of probability measures, where paths are solutions of the continuity equation and the action is the kinetic energy. We consider a fundamental modification of the framework where the paths are solutions of nonlocal continuity equations and the action is a nonlocal kinetic energy. The resulting nonlocal Wasserstein distances are relevant to fractional diffusions and Wasserstein distances on graphs. We will present the basic properties of the distance and conditions on the (jump) kernel that determine whether the topology metrized is the weak or the strong topology. We will also discuss quantitative comparisons between the nonlocal and local Wasserstein distance. Finally we will discuss gradient flows with respect to the nonlocal Wasserstein metric.

Motivated by nonlocal models in continuum mechanics, we present several nonlocal analogues of Sobolev rigidity relations for vector fields. In each setting, we show that a class of vector field spaces whose semi-norm involves the magnitude of ``directional'' difference quotients is in fact equivalent to the class of vector fields characterized by a semi-norm involving the full difference quotient. This equivalence can be considered a Korn-type characterization of nonlocal Sobolev spaces. We apply these inequalities to obtain quantitative statements for solutions to variational problems arising in peridynamics and dislocation models. We additionally demonstrate nonlocal analogues of a Poincare-Korn inequality for a nonlinear peridynamic-type model characterized by a nonconvex strain energy.

Solitons are particle-like solutions to dispersive evolution equations whose shapes persist as time goes by. In some situations, these solitons appear due to the balance between nonlinear effects and dispersion, in other situations their existence is related to topological properties of the model. Broadly speaking, they form the building blocks for the long-time dynamics of dispersive equations.

In this talk I will present forthcoming joint work with W. Schlag on long-time decay estimates for perturbations of the soliton for the 1D focusing cubic Klein-Gordon equation (up to exponential time scales), and I will discuss our previous work on the asymptotic stability of the sine-Gordon kink under odd perturbations. While these two problems are quite similar at first sight, we will see that they differ by a subtle cancellation property, which has significant consequences for the long-time dynamics of the perturbations of the respective solitons.

Mean-field games (MFG) is a framework for modeling and analysis of vast populations of agents that play differential games. It is an actively growing field with numerous applications in economics, finance, industrial engineering, crowd motion, swarm control, and recently machine learning and data science. In this talk, I will discuss several computational, applied, and theoretical aspects of MFG. In particular, I will focus on nonlocal models, applications in machine learning, and MFG PDE.

First introduced in the context of potential theory a century ago, and later expanded to its full generality by Ishii in the 1980's, the Perron method is an essential theoretical tool used to construct viscosity solutions to nonlinear PDE. In work with Stan Osher and Alex Tong Ling (UCLA), we show the Perron method provides us with a simple principle for the numerical computation of solutions to Hamilton-Jacobi (HJ) equations -- as well as any problem admitting a comparison principle, such as obstacle problems or Hele-Shaw flows. Applications include grid-less methods for HJ equations using artificial neural networks or linear bases such as multidimensional Fourier series.

Applications of Optimal Transport (OT) theory have gained popularity in several fields such as machine learning and signal processing. In this seminar, we will address this point by introducing embeddings or transforms based on OT. First, we will present the Cumulative Distribution Transform (CDT), its version for signed signals/measures, and the Linear Optimal Transport Embedding (LOT). Here, the underlying conservation of mass law will be a benefit. Since these tools are new signal representations based on OT, they have suitable properties for decoding information related to certain signal displacements. We will demonstrate this by describing a Wasserstein-type metric in the embedding space and showing applications in classifying (detecting) signals under random displacements, parameter estimation problems for certain types of generative models, and interpolation. Moreover, these techniques allow faster computation of the classical Wasserstein between pairs of probability measures. However, even though the balanced mass requirement from classical OT is crucial, it also limits the performance of these transforms/embeddings. Therefore, we will finally move to Optimal Partial Transport (OPT) theory and propose a new (linear) embedding.

Surfaces whose curvature minimizes the Dirichlet energy are central in applications such as surface design in industry and architecture and are generally constructed by using computer-aided design (CAD). We present the system of equations and prove the first results on existence of classical and weak solutions. This is joint work with Luis A. Caffarelli (UT Austin) and Hern\’an Vivas (Universidad Nacional de Mar del Plata, Argentina).

I will present a general framework to establish the stability of inequalities of the form $\langle Lu, u \rangle \ge F(u)$; where $L$ is a positive linear operator and $F$ is a $2$-homogeneous nonlinear functional. We will then see how this framework can be employed to obtain some stability results concerning the fractional Caffarelli-Kohn-Nirenberg inequality. This is joint work with N. De Nitti and T. Konig.

Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations by using a variational approach based on the gradient theory, the potential and the wells may have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered.

In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. The supercritical case for fixed wells is also addressed, now leading to an isotropic interfacial energy. In the subcritical case with moving wells, where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains.

This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), USA), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA).

We introduces a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension $2$ together with a partial result in dimension $3$. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted $L^p$ norms of the marginals or observables of the system, uniformly in the number of particles. This allows to qualitatively derive the mean-field limit for very singular interaction kernels between the particles, including repulsive Poisson interactions, together with quantitative estimates for a general kernel in $L^2$.

We will discuss recent results about a class of Hamilton-Jacobi-Bellman (HJB) equations in spaces of probability measures that arise in the study of stochastic optimal control problems for systems of $n$ particles with common noise, interacting through their empirical measures. We will present a procedure to show that the value functions $u_n$ of $n$ particle problems, when converted to functions of the empirical measures, converge as $n\to\infty$ uniformly on bounded sets in the Wasserstein space of probability measures to a function $V$, which is the unique viscosity solution of the limiting HJB equation in the Wasserstein space. The limiting HJB equation is interpreted in its "lifted" form in a Hilbert space, a technique introduced by P.L. Lions. The proofs of the convergence of $u_n$ to $V$ use PDE viscosity solution techniques. An advantage of this approach is that the lifted function $U$ of $V$ is the value function of a stochastic optimal control problem in the Hilbert space. We will discuss how, using Hilbert space and classical stochastic optimal control techniques, one can show that $U$ is regular and there exists an optimal feedback control. We then characterize $V$ as the value function of a stochastic optimal control problem in the Wasserstein space. The talk will also contain an overview of existing works and various approaches to partial differential equations in abstract spaces, including spaces of probability measures and Hilbert spaces.

We consider general two-dimensional autonomous velocity fields and prove that their mixing and dissipation features are limited to algebraic rates. As an application, we consider a standard cellular flow on a periodic box, and explore potential consequences for the long-time dynamics in the two-dimensional Euler equations.

A classical model to describe the dynamics of Newtonian stars is the gravitational Euler-Poisson system. The Euler-Poisson system admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In this talk, I will discuss some recent progress on those star solutions with focus on expansion and collapse. If time permits, I will also discuss the non-radial stability of self-similarly expanding Goldreich-Weber star solutions.

The Boussinesq equations are a member of a family of models of incompressible fluid equations, including the 3D Euler equations, for which the problem of global existence of solutions is open. The Boussinesq equations arise in fluid mechanics, in connection to thermal convection and they are extensively studied in that context. Formation of finite time singularities from smooth initial data in ideal (conservative) 2D Boussinesq equations is an important open problem, related to the blow up of solutions in 3D Euler equations. The Voigt Boussinesq equations are a conservative approximation of the Boussinesq equations which have certain attractive features, including sharing the same steady solutions with the Boussinesq equations. In this talk, after giving a brief description of issues of local and global existence, well-posedness and approximation in the incompressible fluids equations, I will present a global regularity result for critical Voigt Boussinesq equations. Some of the work is joint with Jingyang Shu.

Plasmas in a nonequilibrium state experience complex behavior at the large time due to their collective meanfield interaction, including phase mixing, Landau damping, dispersion, and oscillations also known as Langmuir's oscillatory waves. Part of the talk is to overview a complete linear theory of the dynamics of charged particles near spatially homogeneous equilibria, focusing on the classical collision-less models including Vlasov-Poisson and Vlasov-Maxwell systems, while the other part surveys some recent nonlinear results on the subject. Amusingly, a threshold of the wavenumber is provided for the survival of oscillations: namely, below the threshold pure oscillatory waves that obey a Klein-Gordon’s type dispersion relation are found, at the threshold waves are damped by the classical Landau damping (i.e. the faster electrons decay or vanish, the weaker Landau damping is), and above the threshold waves decay polynomially or exponentially fast via the phase mixing mechanism, the full picture of which is deeply linked to the resonant interaction between waves and particles.

We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernández, R. Frank, D. Gómez-Castro, F. Hoffmann, M. Lewin, and J. L. Vázquez.

Orlicz-Sobolev spaces are the natural setting for the study of variational problems with nonstandard growth, meaning that the energy under consideration is given by a potential whose behavior is different from a power. Such problems are typical, for instance, of statistical physics, where the exponential and entropic functions play a crucial role. Integro-differential equations, on the other hand, appear in the study of Lévy processes with jumps in which the infinitesimal generator of a stable pure jump process is given, through the Lévy-Khintchine formula, by an integro-differential operator. These have proven to be accurate models to describe phenomena in physics, biology, meteorology, and finance among many other fields.

In this talk we will discuss some recent results for integro-differential equations posed in fractional Orlicz-Sobolev spaces, ranging from eigenvalue problems to regularity and qualitative issues, and present some open problems and questions which we consider of interest. These are joint works with Julián Fernández Bonder and Ariel Salort.

The deep linear network (DLN) is a simplified model of training by gradient descent. It was popularized a few years by computer scientists (Arora, Cohen and Hazan, especially).

I will present several results on the gradient flow. The main insights all reduce to an elegant in trinsic Riemannian geometry of the DLN.

This is joint work with Nadav Cohen (Tel Aviv) and Zsolt Veraszto (Brown).

In this talk I will briefly discuss the Monge-Kantorovich optimal transportation problem. This problem involves optimally mapping one mass distribution onto another, where optimality is measured against a given cost function $c(x, y)$. I will review some important results, such as Brenier’s theorem, and then discuss a new transportation problem involving path-dependence and interaction effects.

In this talk, I will first give an overview of stochastic homogenization for nondivergence form elliptic equations, from both the PDE perspective and the probability perspective. I will then present new quantitative homogenization results on the parabolic Green Function and large-scale averages of the unique, ergodic, mutually absolutely continuous, invariant measure. This invariant measure is a solution of the adjoint equation in doubly divergence form satisfying certain integrability conditions. Time permitting, I will present a large-scale $C^{0,1}$-regularity result for the invariant measure. This talk is based on joint work with Scott Armstrong and Benjamin Fehrman.

In this talk, we will discuss a global existence result on the Poiseuille flow of nematic liquid crystals via full Ericksen-Leslie model. The existing results on the Ericksen-Leslie model for the liquid crystals mainly focused on the parabolic and elliptic type models by omitting the kinetic energy term. We established a new method to study the full model. A singularity formation result will also be discussed, together with the global existence result showing that the solution will in general live in the Holder continuous space. The earlier related result on the stability of variational wave equation using the optimal transport method, and the recent result on singularity formation due to geometric effect will also be discussed. The talk is on the joint work with Tao Huang & Weishi Liu.

We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.

The analysis of many-body interacting systems is a very challenging task due to the high complexity caused by the interaction, that makes an analytical and numerical treatment difficult and often out of reach. However, in my situations it is possible to approximate the many-body dynamics with simplified PDEs, called effective equations, that accurately reflect the empirically relevant features of the system in certain regimes. In this talk I will focus on the mean-field and semiclassical regime to study the approximation of a system of many interacting fermions by the Vlasov-Poisson equation. Furthermore, in presence of a magnetic field governed by the Maxwell equations, we will prove the semiclassical approximation of the dynamics by the Vlasov-Maxwell system. This is more relevant for applications in plasma physics than Vlasov-Poisson, as it provides a more accurate prediction of the dynamics in a tokamak. Based on joint works with J. Chong, L. Lafleche, N. Leopold.

I will review some recent results on the transport of Gaussian and Gibbs measure along the flow of Hamiltonian PDEs, such as nonlinear Schroedinger equations and Benjamin-Ono type equations underlining the main open problems in the field.

I will discuss a vector valued version of the Phi4 model with $N$ components, known as the linear sigma model in quantum field theory. I will review the well posedness theory for the Langevin dynamic and present some new uniform in $N$ estimates. These are applied to describe the large $N$ behavior of the model in both equilibrium and non-equilibrium settings, the corresponding mean-field limit being a singular SPDE of Mckean-Vlasov type. Time permitting, I will also discuss a related continuum random matrix model, where the large $N$ limit of correlation functions leads to a hierarchy of elliptic PDEs. Based on joint work with Hao Shen, Rongchan Zhu, and Xiangchan Zhu.

Liquid crystal polymeric networks are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy. We characterize the zero energy deformations and prove that the energy lacks rank-1 convexity. We propose a finite element method to discretize the problem. To address the lack of convexity of the membrane energy, we regularize with a term that mimics a higher order bending energy. We prove that minimizers of the discrete energy converge to minimizers of the continuous energy using techniques inspired by Frieseke, James, and Müller (2002). For minimizing the discrete problem, we employ a nonlinear gradient flow scheme, which is energy stable. Additionally, we present computations showing the geometric effects that arise from liquid crystal defects. Computations of configurations from nonisometric origami are also presented.

I will discuss how to use mean curvature flow to give nearly optimal lower bounds on the density of topologically nontrivial minimal hypercones in low dimensions. This compliments work of Ilmanen-White who gave analogous bounds for topologically nontrvial area-minimizing hypercones. This is joint work with L. Wang.

We study the well/ill-posedness of the Boltzmann equation with dispersive methods. We take the constant collision kernel case as the first example. We construct a family of special solutions, which are neither near equilibrium nor self-similar, and prove the ill-posedness in $H^s$ Sobolev space for $s<1$, despite the fact that the equation is scale invariant at $s=1/2$. Combining with the previous Chen-Denlinger-Pavlovic result regarding well-posedness, we have found the exact well/ill-posedness threshold.

In this talk, I will present $C^\infty$ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to point-wise bounds on their mass, energy and entropy densities. Joint work with Luis Silvestre.

We plan to discuss parts of the construction of incompressible flows of an ideal fluid with vorticity in Lebesgue classes in dimension two that imply nonuniqueness in the Cauchy problem.

The notion of pathwise viscosity solutions was developed by Lions and Souganidis to study fully nonlinear stochastic partial differential equations set on the entire space. I will explain how this notion can be generalized to treat equations set on a convex sub-domain with Neumann boundary conditions. A comparison principle is proved for sub and supersolutions, which yields existence and uniqueness of solutions as well as continuity with respect to the driving noise. As an application, I show how the well-posedness theory is used to study the long-time behavior of a stochastically perturbed mean-curvature flow in a cylinder-like domain with right angle contact boundary condition. This is joint work with Paul Gassiat.

The topic of this talk will be the validity of a kinetic description for wave turbulence of a model quadratic dispersive equation. We focus on deriving an inhomogeneous (transport) effective equation for the Wigner transform of the microscopic model solution, up to a small polynomial loss of the kinetic time. We will present nonlinearities where the kinetic description holds and cases where it might fail. We achieve that by examining the convergence of the Dyson series under random data in the weakly nonlinear regime.

In his seminal work, Leray demonstrated the existence of global weak solutions, with nonincreasing energy, to the Navier-Stokes equations in three dimensions. In this talk we exhibit two distinct Leray solutions with zero initial velocity and identical body force. The starting point of our construction is Vishik's answer to another long-standing problem in fluid dynamics, namely whether the Yudovich uniqueness result for the 2D Euler system can be extended to the class of L^p-integrable vorticity. Building on Vishik's work, we construct a `background' solution which is unstable for the 3D Navier-Stokes dynamics in similarity variables; the second solution from the same initial datum is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Sverak.

The binary-ternary Boltzmann equation was recently rigorously derived by Ampatzoglou and Pavlovic for a dense hard-spheres gas in which particles interact via either binary or ternary interactions. We study analytic properties of the spatially homogeneous version of this equation for a wider range of potentials. In particular, we study polynomial and exponential moment estimates and show that the presence of the ternary part can improve moment estimates of the classical (binary) Boltzmann equation. We will also present a result on global in time existence and uniqueness of solutions which is based on moment estimates and an abstract ODE theory in Banach spaces. This is a joint work with Ioakeim Ampatzoglou, Irene Gamba and Natasa Pavlovic.

In this talk we discuss a full resolution result for minimizers in the exterior isoperimetric problem with respect to a compact obstacle in the large volume regime. This is achieved by the study of a Plateau-type problem with obstacle and boundary datum at infinity, which identifies the first obstacle-dependent term in energy expansion for large volumes of the exterior isoperimetric problem, and is therefore called the isoperimetric residue of the obstacle. A crucial tool in the analysis is a new “mesoscale flatness criterion” for hypersurfaces with bounded mean curvature, which we obtain as a development of ideas originating in the theory of minimal surfaces with isolated singularities. This is a joint work with Francesco Maggi.

It is well known that the prescribed Gauss-Kronecker curvature problem for a graph on a convex domain can be described via optimal transport. In this talk, I will discuss an alternate interpretation, again as an optimal transport problem but in a setting different from the traditional approach. This talk is based on joint work with N. Guillen.

Among all drum heads of a fixed area, a circular drum head produces the vibration of lowest frequency. The general dimensional analogue of this fact is the Faber-Krahn inequality, which states that balls have the smallest principal Dirichlet eigenvalue among subsets of Euclidean space with a fixed volume. I will discuss new quantitative stability results for the Faber-Krahn inequality on Euclidean space, the round sphere, and hyperbolic space, as well as an application to the Alt-Caffarelli-Friedman monotonicity formula used in free boundary problems. This is based on joint work with Mark Allen and Dennis Kriventsov.

I will discuss a probabilistic approach to the Nash embedding theorems along with several applications. The initial motivation for this work were the results of De Lellis and Szekelyhidi linking Nash embedding with the Euler equations. The main idea in my work is to replace ad hoc construction schemes with a principled choice of stochastic flows, in order that one may answer the question “what does a typical embedding look like?”. More precisely, we ask how one may construct natural probability measures supported on solutions to nonlinear PDE. This idea will be illustrated with examples. This program is still some ways away from completely new proofs of these theorems, however it yields many new models and new insights into Nash-Moser and KAM theorems. Most of the effort at this point is in sharpening the fundamental insight through numerics. In particular, there is a stong interplay with interior point methods for semidefinite programming.

Building upon the pioneering work of Merle-Rodnianski-Szeftel, we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents. For the particular exponent $7/5$, corresponding to air and akin to the result of Merle-Raphael-Rodnianski-Szeftel, we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability and non-linear stability which allows us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the adiabatic exponent $7/5$ and density bounded from below.

In this talk, we will define fractional powers of nondivergence form elliptic operators in bounded domains under minimal regularity assumptions and highlight several applications. We will characterize a Poisson problem driven by such operators with a degenerate/singular extension problem. We then develop the method of sliding paraboloids in the Monge–Ampère geometry to prove Harnack inequality for classical solutions to the extension equation. This in turn implies Harnack inequality for solutions to the fractional Poisson problem. This work is joint with Pablo Raúl Stinga (Iowa State University).

The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory of turbulence.