PAST TALKS
17/05/2023 at 15h
Speaker: Joaquim Serra
ETH Zurich
Title: Nonlocal approximation of minimal surfaces: optimal estimates from stability.
Abstract: Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren- Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and
Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture.
In a recent paper with Chan, Dipierro, and Valdinoci we set the ground for a new approximation based on
nonlocal minimal surfaces. More precisely, we prove that stable s-minimal surfaces in the unit ball of $\R^3$ satisfy curvature estimates that are robust as s approaches 1 (i.e. as the energy approaches that of classical minimal surfaces).
Moreover, we obtain optimal sheet separation estimates and show that critical interactions are encoded by nontrivial solutions to a (local) "Toda type" system.
As a nontrivial application, we establish that hyperplanes are the only stable s-minimal hypersurfaces in $\R^4$, for $s$ sufficiently close to 1.
19/04/2023 at 15h
Speaker: María Soria-Carro
Rutgers University
Title: Regularity of Elliptic Transmission Problems
Abstract:
Transmission problems describe phenomena in which a physical quantity changes behavior across some fixed surface, known as the interface. The analysis of such problems started in the 1950s with the pioneering work of Picone in elasticity. Nowadays, they have a wide range of applications in different areas, such as electromagnetic processes, composite materials, and climatology. Typically, solutions are not differentiable across the interface, and the primary interest is to study their optimal regularity from each side of this surface.
In this talk, I will discuss two elliptic transmission problems where the interface has minimal regularity. The first problem is for harmonic functions, and I will explain the main ideas and techniques used to prove the optimal regularity of solutions at the boundary. The second problem is for fully nonlinear equations, and I will focus on how to obtain a maximum principle (ABP estimate) for this problem, which plays a key role in the regularity theory of viscosity solutions. These results are part of my Ph.D. dissertation, and they are in collaboration with Luis Caffarelli and Pablo Raúl Stinga.
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22/03/2023 at 15h
Speaker: Marek Fila
Comenius University. Bratislava.
Title: Solutions with moving singularities for nonlinear diffusion equations
Abstract: We give a survey of results on solutions with singularities moving along a prescribed curve for equations of fast diffusion or porous medium type. These results were obtained in collaboration with J.R. King, P. Mackova, J. Takahashi and E. Yanagida.
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22/02/2023 at 15h
Speaker: Sylvia Serfaty
Courant Institute.
Title: Mean-field limits for singular flows
Abstract: We discuss the derivation of PDEs as limits as N tends to infinity of the dynamics of N points for a certain class of Coulomb and Riesz-type singular pair interactions. The method is based on studying the time evolution of a certain "modulated energy" and on proving a functional inequality relating certain "commutators" to the modulated energy. When additive noise is added, some uniform in time convergence results can be obtained. Based on joint works with Hung Nguyen, Matthew Rosenzweig, Antonin Chodron de Courcel.
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18/01/2023 at 15h
Speaker: Alessio Porretta.
Università di Roma Tor Vergata.
Title: Time decay rates of Fokker-Planck equations with confining drift
Abstract: The convergence to equilibrium of Fokker-Planck equations with confining drift is a classical issue, starting with the basic model of the Ornstein-Uhlenbeck process. I will discuss a new approach to obtain estimates on the time decay rate, which applies to both local and nonlocal diffusions. This is based on duality arguments and oscillation estimates for transport-diffusion equations, which are reminiscent of coupling methods used in probabilistic approaches.
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14/12/2022 at 16h.
PAY ATTENTION: Not usual time.
Speaker: Luis Silvestre. University of Chicago.
Title: Holder continuity up to the boundary for kinetic equations
Abstract: We consider a kinetic Fokker-Planck equation with rough coefficients and the spatial variable restricted to a bounded domain. There are recent results concerning interior Holder estimates for this class of equations following techniques by De Giorgi, Nash and Moser. In this talk, we discuss the regularity of the solutions on the boundary.
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16/11/2022
Susanna Terracini, Dipartimento di Matemática "Giuseppe Peano". Università di Torino
Title: Higher order boundary Harnack principle on nodal domains via degenerate equations
Abstract: The ratio v/u of two solutions to a second order elliptic equation in divergence form solves a degenerate elliptic equation if u and v share the zero set; that is, Z(u) ⊆ Z(v). The coefficients of the degenerate equation vanish on the nodal set as u^2. Developing a Schauder theory for such equations, we prove C^{k,α}-regularity of the ratio from one side of the regular part of the nodal set in the spirit of the higher order boundary Harnack principle established by De Silva and Savin in [4]. Then, by a gluing lemma, the estimates extend across the regular part of the nodal set. Eventually, using conformal mapping in dimension n = 2, we provide local gradient estimates for the ratio which hold also across the singular part of the nodal set and depends on the highest value attained by the Almgren frequency function.
References
[1] S. Terracini, G. Tortone and S. Vita, Higher order boundary Harnack prin- ciple on nodal domains via degenerate equations, preprint, 2022.
[2] Y. Sire, S. Terracini, S. Vita. Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions. Comm. Partial Differential Equations, 46-2 (2021), 310-361.
[3] Y. Sire, S. Terracini, S. Vita. Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions. Math- ematics in Engineering, 3-1 (2021), 1-50.
[4] D. De Silva, O. Savin. A note on higher regularity boundary Harnack in- equality. DCDS-A, 35(12), (2015) 6155-6163.
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19/10/2022
Guido de Philippis, Courant Institute of Mathematics NYU, USA
Title: Non-degenerate minimal surfaces as energy concentration sets: a variational approach.
Abstract: I will show that every non-degenerate minimal sub-manifold of codimension 2 can be obtained as the energy concentration set of a family of critical points of the (rescaled) Ginzburg Landau functional. The proof is purely varia-tional, and follows the strategy laid by Jerrard and Sternberg in 2009. The same proof applies also to the Yang-Mills-Higgs and to the Allen-Cahn-Hillard energies. This is a joint work with Alessandro Pigati
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Maria Colombo, EPFL
29/06/2022
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations.
Abstract: 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.
Maria Eugenia Pérez, Universidad de Cantabria
15/06/2022
Title: Spectral boundary homogenization problems with high contrasts
Abstract: We consider a spectral homogenization problem for the elasticity operator posed in a bounded domain of the upper half-space, a part of its boundary being in contact with the plane. We assume that this surface is free outside small regions in which we impose Robin-Winkler boundary conditions linking stresses and displacements by means of a symmetric and positive definite matrix and a reaction parameter. These small regions are periodically placed along the plane while its size is much smaller than the period. We provide all the possible spectral homogenized problems depending on certain asymptotic relations between the period, the size of the regions and the reaction-parameter. We show the convergence of the eigenelements, as the period tends to zero, which deeply involves the corresponding microscopic stationary problemsobtained by means of asymptotic expansions.
Some references
[1] D. Gómez, S.A. Nazarov, ; M.-E. Pérez-Martínez. Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. Journal of Elasticity, 2020, V. 142, p. 89-120.
[2] D. Gómez, S.A. Nazarov ; M.-E. Pérez-Martínez. Spectral homogenization problems in linear elasticity with large reaction terms concentrated in small regions of the boundary. In: Computational and Analytic Methods in Science and Engineering. Birkäuser, Springer, N.Y., 2020, pp. 121-143
[3] M.-E. Pérez-Martínez. Homogenization for alternating boundary conditions with large reaction terms concentrated in small regions. In: Emerging problems in the homogenization of Partial Differential Equations. ICIAM2019 SEMA SIMAI Springer Series 10, 2021, pp. 37-57.
Zineb Hassainia, New York University Abu Dhabi
01/06/2022
Title: Time quasi-periodic vortex patch solutions of the 2D-Euler equations
Abstract:
In this talk I will discuss recent results concerning the emergence of time quasi-periodic vortex patch solutions of the 2D-Euler equations set either in the whole plan or in the unit disc. In the first case, the search of such solutions near Rankine vortices is not clear due to the resonances of the linear frequencies and the absence of an exterior parameter. However, we are able to prove the existence of these structures close to Kirchhoff elliptical vortices when aspect ratios belongs to a Borel set of asymptotically full Lebesgue measure. In the case of Euler equations in the unit disc, we highlight the importance of the boundary effects on the construction of quasi periodic vortex patches solutions close to Rankine vortices with radius belonging to a suitable massive Cantor-like set. Both proofs are based on Nash-Moser implicit function theorem and KAM theory in infinite dimensional spaces. The first result is joint work with Massimiliano Berti and Nader Masmoudi, whereas the second is joint work with Emeric Roulley.
Daniela de Silva, Columbia University
18/05/2022
Title: Inhomoeneous global minimazers to the one-phase free boundary problem.
Abstrat: Given a global 1-homogeneous minimazer U to the Alt-Caffarelli energy functional with sing(F(U))={0}, we rpovide a foliation of the half-space with dilations of graphs of global minimazers U_<U<U^ with analytic free boundaries at distance 1 from the origin. This is a joint work with D. Jerison and H. Dhahgholian.
Jaemin Park, Universidad de Barcelona
04/05/2022
Title: Existence of non-radial stationary solutions to the 2D Euler equation.
Abstract: In this talk, we study stationary solutions to the 2D incompressible Euler equations in the whole plane. It is well-known that any radial vorticity is stationary. For compactly supported vorticity, it is more difficult to see whether a stationary solution has to be radial. In the case where the vorticity is non-negative, it has been shown that any stationary solution has be radial. By allowing the vorticity to change the sign, we prove that there exist non-radial stationary patch-type solutions. We construct patch-type solutions whose kinetic energy is infinite or finite. For the finite energy case, it turns out that a construction of a stationary solution with compactly supported velocity is possible.
Eduardo García-Juarez
20/04/2022
Tilte: Recent results for the Peskin problem
Abstract: The Peskin problem models the dynamics of a closed elastic membrane immersed in an incompressible Stokes fluid. This set of equations was proposed as a simplified model for the motion of red blood cells and it serves as a canonical test problem for numerical methods. Studying the well-posedness of the problem is necessary to perform numerical analysis and to guarantee that numerical methods based upon different formulations of the problem converge to the same solution. Mathematically, the problem can be seen as a generalization of the Stokes two-phase interface with surface tension, and shares the linear structure with the Muskat problem. We will review some of the latest well-posedness problems, and in particular we will focus on the global regularity issue for 2D Peskin with viscosity jump and the local well-posedness for 3D membranes.
David Ruiz, Universidad de Granada
23/03/2022
Title: Symmetry results for compactly supported solutions of the 2D steady Euler equations.
Abstract: In this talk we present some recent results regarding compactly supported solutions of the 2D steady Euler equations. Under some assumptions on the support of the solution, we prove that the streamlines of the flow are circular. The proof uses that the corresponding stream function solves an elliptic semilinear problem. One of the main difficulties in our study is that the nonlinear term can fail to be Lipschitz continuous near the boundary values.
In some cases we can apply a local symmetry result of F. Brock to conclude. Otherwise, we are able to use the moving plane scheme to show symmetry, despite the possible lack of regularity of the nonlinear term. We think that such result is interesting in its own right and will be stated and proved also for higher dimensions. The proof requires the study of maximum principles, Hopf lemma and Serrin corner lemma for elliptic linear operators with singular coefficients.
Wenxian Shen, Auburn University
09/03/2022
Title: Dynamics in Parabolic-Elliptic Chemotaxis Models with Logistic Source
Abstract: Chemotaxis models are used to describe the movements of biological species
or living organisms in response to certain chemicals in their environments.
There are various types of chemotaxis models. The current talk is concerned with the dynamics in parabolic-elliptic chemotaxis models with logistic source. The talk consists of two parts.
In the first part, I will discuss the finite-time blow-up prevention by logistic source in parabolic-elliptic chemotaxis models on bounded domains. In the second part, I will consider front propagation dynamics including spatial spreading speeds and
traveling wave solutions in parabolic-elliptic chemotaxis models on the whole space $R^N$.
Peter Polacik, School of Mathematics, University of Minnesota
23/02/2022
Title: Finding quasiperiodic solutions of elliptic equations on the entire space using center manifold and KAM theorems
Abstract: We consider positive solutions of nonlinear elliptic equations on the (N+1)-dimensional Euclidean space with some predetermined behavior (such as decay and symmetry) in the first N variables. We examine the behavior of these solutions in the remaining variable. Families of solutions periodic in the last variable have been found by several authors; our goal is to prove the existence of quasiperiodic solutions. For this purpose, we examine the spatial dynamics of the equation on a center manifold and apply KAM-type results. In the lecture, I will outline our general techniques and report on recent progress. This is joint work with Dario Valdebenito.
Anibal Rodríguez-Bernal, UCM-ICMAT
09/02/2022
Title: "Nonlinear like behaviour in the linear heat equation in R^{N}”
Abstract: In this talk we show that considering an optimal class of large initial data, the solutions of the linear heat equation display a very rich dynamical behavior, including blow-up in finite time and wild unbounded oscillations.
We characterise blow-up solutions as well as their blow-up set, solutions that exist globally and provide suitable conditions for
asymptotic decay.
This is a joint work with J. Robinson (U. Warwick).
Luis Vega, BCAM UPV/EHU
26/01/2022
Title: Conservation Laws and Energy Cascade for 1d Cubic NLS
Abstract: I’ll present some recent results concerning the IVP of 1d cubic NLS at the critical level of regularity. I’ll also exhibit a cascade of energy for the 1D Schrödinger map which is related to NLS through the so called Hasimoto transformation. For higher regularity these two equations are completely integrable systems and therefore no cascade of energy is possible.
Robin Neumayer, Carnegie Mellon University.
12/01/2022
Title: Quantitative Faber-Krahn Inequalities and the ACF Monotonicity Formula
Abstract: 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.
15/12/2021
Edoardo Bocchi, Universidad de Sevilla
Title: Local well-posedness of an oscillating water column in shallow water with time-dependent air pressure
Abstract: We consider a particular wave energy converter, the so-called oscillating water column. Waves waves governed by the one-dimensional nonlinear shallow water equations arrive from the offshore, encounter a step in the bottom topography and then arrive into a chamber to change the volume of the air to activate the turbine.The system is reformulated as two hyperbolic transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. In this talk we focus on the second transmission problem. First, taking into account a time-dependent air pressure inside the chamber, we derive in a new way a transmission condition that permits to close the system. Secondly, we address the well-posedness of the system, which is obtained by deriving a Kreiss symmetrizer. This is a joint work with J. He and G. Vergara-Hermosilla.
24/11/2021
Nastasia Grubic, ICMAT.
Title: On the free boundary incompressible Euler equations with interface that exhibit cusps and corners.
Abstract: We will construct a class of solutions to the 2d free boundary incompressible Euler equations where the interface self-intersects and the fluid domain has either cusps or corners. Contrary to what happens in all the previously known non-$C^1$ water waves, the angle can change in time. This is joint work with D.Cordoba and A.Enciso.
10/11/2021
Francisco Torres, University of Toronto.
Title: Chaos in the incompressible Euler equation on manifolds of high dimension
Abstract: We will construct finite dimensional families of non-steady, global in time solutions to the Euler equations, that exhibit all kinds of qualitative dynamics in the phase space of divergence-free vector fields, like chaos, invariant manifolds of any topology, and strange attractors.
27/10/2021
Félix del Teso, UCM.
Title: On asymptotic expansions and approximation schemes for the p-Laplacian
Abstract: The aim of this talk is to introduce the topic of asymptotic expansions and approximation schemes for p-Laplacian type operators.
We will present the results in collaboration with J. J. Manfredi and M. Parviainen ([3]). Here, we show a unified framework to prove convergence of approximation schemes for boundary value problems regarding normalized p-Laplacian, which has to be treated in the context of viscosity solutions.
While for the normalized p-Laplacian, asymptotic expansion and finite difference discretizations were very well known, this was not the case for p-Laplapcian. In the second part of the talk, we will present such results. This is a work in collaboration with E. Lindgren ([1, 2]). Here, we introduce new asymptotic expansions and finite difference discretizations and show convergence of approximation schemes for associated problems.
References
[1] del Teso, Felix; Lindgren, Erik; A mean value formula for the variational p-Laplacian. NoDEA Nonlinear Differential Equations Appl., 28 (2021),
no. 3, Paper No. 27, 33 pp.
[2] del Teso, Felix; Lindgren, Erik; A finite difference method for the varia-tional p-Laplacian Preprint, https://arxiv.org/abs/2103.06945. (2021)
[3] del Teso, Felix; Manfredi, Juan J.; Parviainen, Mikko; Convergence of dynamic programming principles for the p-Laplacian Advances in Calculus
of Variations, Ahead of print. (2019).
13/10/2021
Alessandro Audrito, ETH.
Title: 1D symmetry of solutions to a class of semilinear one-phase elliptic PDE.
Abstract: We study minimizers of a family of functionals arising in combustion theory, which converge, for infinitesimal values of the parameter, to minimizers of the \emph{one-phase free boundary} problem.
We prove a $C^{1,\alpha}$ estimate for the ``interfaces'' of critical points (i.e. the level sets separating the burnt and unburnt regions).
As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole $\RR^N$ for $N \le 4$, answering positively a conjecture of Fernández-Real and Ros-Oton.
Our results are to the one-phase free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces.
This is a joint work with J. Serra (ETHZ).
29/09/2021
Rafael de la Llave, Georgia Tech.
Title: Solutions of delay differential equations, including applications to electrodynamics.
Abstract: The study of electrodynamics and other applied problems leads to the study of equations in which the forces are functions of the trajectory at previous times. These delays depend on the trajectory.
For such equations, not even the appropriate phase space is known. Much less a theory of existence and smoothness of evolution.
We show that, nevertheless, there is a systematic theory to construct many solutions. In particular we develop a theory of perturbations for small delays (the 1/c expansions in classical electrodynamics).
The theory is efective and leads to constructive numerical algorithms.
This is based on joint works with J.Gimeno, X. He, J. Yang. and A. Casal, L. Corsi.
This is based on joint works with J.Gimeno, X. He, J. Yang and A. Casal, L. Corsi.
30/06/2021
Daniel Lear (University of Illinois at Chicago)
Title: Grassmannian reduction of Cucker-Smale systems and dynamical opinion games
Abstract: In this talk, we study a new class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long-time dynamics via a new method which allows to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than $\pi$. In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions. Joint work with David N. Reynolds & Roman Shvydkoy
16/06/2021
Valeria Banica (Laboratoire Jacques-Louis Lions)
Title: Title: Microlocal analysis of singular measures.
Title: Title: Microlocal analysis of singular measures.
Abstract: In this talk I shall present a study of scalar and vectorial measures from a microlocal point of view, by introducing a notion of $L1$-regularity wave front set. I shall give several results including a full $L1$-elliptic regularity result, and properties of the singular part of measures constrained by PDEs. This is a joint work with Nicolas Burq.
02/06/2021
Juan José López Velázquez (U. Bonn)
Title: Homoenergetic solutions of the Boltzmann equation.
Abstract: In this talk I will discuss the properties of a class of solutions of the Boltzmann equation, the so-called homoenergetic solutions which were introduced by Galkin and Truesdell in the 1960’s. These solutions are a particular type of non-equilibrium solutions of the Boltzmann equation. One of their most relevant properties as that they do not behave asymptotically as Maxwellian distributions for long times, at least for all the choices of the collision kernels. The current knowledge about the long time behaviour of the homoenergetic solutions as well as some conjectures and open problems will be discussed.
Title: Homoenergetic solutions of the Boltzmann equation.
Abstract: In this talk I will discuss the properties of a class of solutions of the Boltzmann equation, the so-called homoenergetic solutions which were introduced by Galkin and Truesdell in the 1960’s. These solutions are a particular type of non-equilibrium solutions of the Boltzmann equation. One of their most relevant properties as that they do not behave asymptotically as Maxwellian distributions for long times, at least for all the choices of the collision kernels. The current knowledge about the long time behaviour of the homoenergetic solutions as well as some conjectures and open problems will be discussed.
19/05/2021
Philippe Souplet
(Université Sorbonne Paris Nord)
Title: Diffusive Hamilton-Jacobi equations and their singularities
Abstract: We consider the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$ with homogeneous Dirichlet boundary conditions, which plays an important role in stochastic optimal control theory and in certain models of surface growth (KPZ).
(Université Sorbonne Paris Nord)
Title: Diffusive Hamilton-Jacobi equations and their singularities
Abstract: We consider the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$ with homogeneous Dirichlet boundary conditions, which plays an important role in stochastic optimal control theory and in certain models of surface growth (KPZ).
Despite its simplicity, in the superquadratic case p>2 it displays a variety of interesting and surprising behaviors and we will discuss two classes of phenomena:
- Gradient blow-up (GBU): localization of singularities on the boundary, single-point GBU, time rate of GBU, space and time-space profiles, Liouville type theorems and their applications;
- Continuation after GBU as a global viscosity solution: GBU with or without loss of boundary conditions (LBC), recovery of boundary conditions with or without regularization, GBU and LBC at multiple times.
In particular, in one space dimension, we will present the recently obtained, complete classification of all GBU and recovery rates.
This talk is based on a series of joint works in collaboration with A. Attouchi, R. Filippucci, Y. Li, N. Mizoguchi, A. Porretta, P. Pucci, Q. Zhang.
05/05/2021
Jørgen Endal (U. Autónoma de Madrid)
Title: The one-phase fractional Stefan problem
Abstract: We study the existence, properties of solutions, and free boundaries of the one-phase Stefan problem with fractional diffusion posed in $\R^N$. The equation for the enthalpy $h$ reads $\partial_t h+ (-\Delta)^{s}\Phi(h) =0$ where the temperature $u:=\Phi(h):=\max\{h-L,0\}$ is defined for some constant $L>0$ called the latent heat, and $(-\Delta)^{s}$ is the fractional Laplacian with exponent $s\in(0,1)$.
We prove the existence of a continuous and bounded selfsimilar solution of the form $h(x,t)=H(x\,t^{-1/(2s)})$ which exhibits a free boundary at the change-of-phase level $h(x,t)=L$ located at $x(t)=\xi_0 t^{1/(2s)}$ for some $\xi_0>0$. This special solution will be an important tool to obtain that the temperature has finite speed of propagation while the enthalpy has infinite speed, and that the support of the temperature never recedes. Other interesting properties like e.g. $L\to0^+$ and $L\to\infty$ will also be discussed, and the theory itself is illustrated by convergent finite-difference schemes.
21/04/2021
Maria Gualdani
(University of Texas at Austin)
21/04/2021
Maria Gualdani
(University of Texas at Austin)
Title: Hardy inequalities for the Landau equation
Abstract: Kinetic equations are used to describe evolution of interacting particles. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936, Lev Landau derived a new mathematical model for motion of plasma. This latter equation was named the Landau equation. One of the main features of the Landau equation is nonlocality, meaning that particles interact at large, non-infinitesimal length scales. Moreover, the coefficients are singular and degenerate for large velocities. Many important questions, such as whether or not solutions become unbounded after a finite time, are still unanswered due to their mathematical complexity.
Abstract: Kinetic equations are used to describe evolution of interacting particles. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936, Lev Landau derived a new mathematical model for motion of plasma. This latter equation was named the Landau equation. One of the main features of the Landau equation is nonlocality, meaning that particles interact at large, non-infinitesimal length scales. Moreover, the coefficients are singular and degenerate for large velocities. Many important questions, such as whether or not solutions become unbounded after a finite time, are still unanswered due to their mathematical complexity.
In this talk we concentrate on the mathematical results of the homogeneous Landau equation. We will first review existing results and open problems and in the second part of the talk we will focus on recent developments of well-posedness and regularity theory. This is a joint work with Nestor Guillen.
Title: Multiscale decomposition of functions in Metric Random Walk Spaces
Abstract:
Abstract:
Title: On the stability of a point charge for the Vlasov-Poisson system
Abstract: A point charge is a particularly basic and important equilibrium of the Vlasov-Poisson equations, and the study of its stability has inspired several major contributions. In this talk we present some recent work, which brings a fresh perspective on this problem. Our new approach combines a Lagrangian analysis of the linearized problem with an Eulerian PDE framework in the nonlinear analysis, all the while respecting the symplectic structure. As a result, for the case of radial initial data, we see that solutions are global and in fact disperse to infinity via a modified scattering along trajectories of the linearized flow.
Abstract: A point charge is a particularly basic and important equilibrium of the Vlasov-Poisson equations, and the study of its stability has inspired several major contributions. In this talk we present some recent work, which brings a fresh perspective on this problem. Our new approach combines a Lagrangian analysis of the linearized problem with an Eulerian PDE framework in the nonlinear analysis, all the while respecting the symplectic structure. As a result, for the case of radial initial data, we see that solutions are global and in fact disperse to infinity via a modified scattering along trajectories of the linearized flow.
This is joint work with Benoit Pausader (Brown University).
10/03/2021
Julio Rossi
(Universidad de Buenos Aires)
10/03/2021
Julio Rossi
(Universidad de Buenos Aires)
Title: Convexity and quasiconvexity and their associated equations
Abstract: Our goal is to look at the notion of convexity and the equations associated with the convex envelope of a datum in different contexts. In the first part of this talk we deal with PDEs given in terms of eigenvalues of the Hessian and their relation with concave/convex functions. We will also include a fractional version of the involved ideas.In the second part we will describe a notion of convexity for functions defined on a regular tree (a graph in which each node (except one) is connected with a fixed number of successors and one predecessor). Based on joint works with P. Blanc, N. Frevenza, L. Del Pezzo and Alexander Quaas.
24/02/2021
Néstor Guillén
(Texas State University)
Abstract: Our goal is to look at the notion of convexity and the equations associated with the convex envelope of a datum in different contexts. In the first part of this talk we deal with PDEs given in terms of eigenvalues of the Hessian and their relation with concave/convex functions. We will also include a fractional version of the involved ideas.In the second part we will describe a notion of convexity for functions defined on a regular tree (a graph in which each node (except one) is connected with a fixed number of successors and one predecessor). Based on joint works with P. Blanc, N. Frevenza, L. Del Pezzo and Alexander Quaas.
24/02/2021
Néstor Guillén
(Texas State University)
Title: Free boundary problems as Hamilton-Jacobi-Bellman equations
Abstract: Integro-differential equations have been fundamental in the analysis of all kinds of free boundary problems, from the vortex patch equation to interfacial Darcy flows. In work with Chang-Lara and Schwab, we show how certain two-phase free boundary problems such as the quasistatic Stefan problem are equivalent to nonlocal Hamilton-Jacobi-Bellman equations. This equivalence has a number of immediate consequences, such as an existence and uniqueness theory based on viscosity solutions, and propagation of Lipschitz regularity of the initial data.
10/02/2021
SARA MERINO-ACEITUNO (University of Vienna)
Abstract: Integro-differential equations have been fundamental in the analysis of all kinds of free boundary problems, from the vortex patch equation to interfacial Darcy flows. In work with Chang-Lara and Schwab, we show how certain two-phase free boundary problems such as the quasistatic Stefan problem are equivalent to nonlocal Hamilton-Jacobi-Bellman equations. This equivalence has a number of immediate consequences, such as an existence and uniqueness theory based on viscosity solutions, and propagation of Lipschitz regularity of the initial data.
10/02/2021
SARA MERINO-ACEITUNO (University of Vienna)
Title: Applying kinetic theory to the study of collective dynamics
Abstract: The aim of this introductory talk is to illustrate how kinetic theory is used to investigate collective behaviour. Particularly, I will introduce the Vicsek model and Vicsek-related models. In the Vicsek model agents have a constant speed while trying to move in the same direction as their neighbors. The goal is to derive continuum/fluid-like equations from the discrete dynamics of the system. This derivation corresponds to linking microscopic and macroscopic phenomena.
27/01/2021
Juan Dávila (U. of Bath)
Abstract: The aim of this introductory talk is to illustrate how kinetic theory is used to investigate collective behaviour. Particularly, I will introduce the Vicsek model and Vicsek-related models. In the Vicsek model agents have a constant speed while trying to move in the same direction as their neighbors. The goal is to derive continuum/fluid-like equations from the discrete dynamics of the system. This derivation corresponds to linking microscopic and macroscopic phenomena.
27/01/2021
Juan Dávila (U. of Bath)
Title: Solutions of the Euler equation with concentrated vorticity
Abstract: I will discuss solutions to the incompressible Euler equation in two dimensions with vorticity close to a finite sum of Dirac deltas (vortices). The motion of the vortices was known formally for a long time and proved rigorously by Marchioro-Pulvirenti. In collaboration with Manuel del Pino (U. Bath), Monica Musso (U. Bath) and Juncheng Wei (UBC) we have a different point of view, which allows a very precise description of the solution near the vortices. Our construction can be generalized to other situations, such as the construction of leapfrogging vortex rings of the 3D incompressible Euler equations.
13/01/2021
ROBERTA BIANCHINI (Italian National Research Council)
Abstract: I will discuss solutions to the incompressible Euler equation in two dimensions with vorticity close to a finite sum of Dirac deltas (vortices). The motion of the vortices was known formally for a long time and proved rigorously by Marchioro-Pulvirenti. In collaboration with Manuel del Pino (U. Bath), Monica Musso (U. Bath) and Juncheng Wei (UBC) we have a different point of view, which allows a very precise description of the solution near the vortices. Our construction can be generalized to other situations, such as the construction of leapfrogging vortex rings of the 3D incompressible Euler equations.
13/01/2021
ROBERTA BIANCHINI (Italian National Research Council)
Title: Some mathematical aspects of 2D stably stratified fluids
Abstract: In this talk, I will present a couple of recent results on the mathematical description of some phenomena in the dynamics of 2D incompressible stably stratified fluids.
Abstract: In this talk, I will present a couple of recent results on the mathematical description of some phenomena in the dynamics of 2D incompressible stably stratified fluids.
We will consider internal waves as the response to small perturbations of the hydrostatic equilibrium.
The inclination of their group velocity with respect to the vertical (gravity) is completely determined by their time frequency. Therefore the reflection on a sloping boundary cannot follow Descartes’ laws and gives rise to interesting phenomena. In the second part, we will also investigate the stability of a more general class of steady states (shear flows).
9/12/2020
THOMAS ALAZARD (École Normale Supérieure de Paris-Saclay)
9/12/2020
THOMAS ALAZARD (École Normale Supérieure de Paris-Saclay)
Title: The Muskat equation is well-posed on the critical Sobolev space
Abstract: This talk is about a series of papers with Quoc-Hung Nguyen, devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. I will describe our main result, which states that the Cauchy problem is well-posed on the endpoint Sobolev space of L^2 functions with three-half derivative in L^2 (locally in time for large data, and globally for small
Abstract: This talk is about a series of papers with Quoc-Hung Nguyen, devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. I will describe our main result, which states that the Cauchy problem is well-posed on the endpoint Sobolev space of L^2 functions with three-half derivative in L^2 (locally in time for large data, and globally for small
enough data). This result is optimal with respect to the scaling of the equation. For the proof, we introduce weighted fractional laplacians and use these operators to estimate the solutions for a norm which depends on the initial data themselves. Another key ingredient of the proof is a null-type structure, allowing to compensate for the degeneracy of the parabolic behavior for large slopes.
25/11/2020
XAVIER CABRÉ (ICREA and Universitat Politecnica de Catalunya (Barcelona))
XAVIER CABRÉ (ICREA and Universitat Politecnica de Catalunya (Barcelona))
Title: Stable solutions to semilinear elliptic equations are smooth up to dimension 9
Abstract: The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems
Abstract: The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems
11/11/2020
MARÍA MEDINA (Universidad Autónoma de Madrid)
MARÍA MEDINA (Universidad Autónoma de Madrid)
Title: Blow-up analysis of a curvature prescription problem in the disk
Abstract: We will establish necessary conditions on the blow-up points of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures, where a non local restriction will appear. Conversely, given a point satisfying these conditions, we will construct an explicit family of approximating solutions that explode at such a point. These results are contained in several works in collaboration with A. Jevnikar, R. López-Soriano and D. Ruiz, and with L. Battaglia and A. Pistoia
28/10/2020
JEAN DOLBEAULT (Université Paris-Dauphine)
Abstract: We will establish necessary conditions on the blow-up points of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures, where a non local restriction will appear. Conversely, given a point satisfying these conditions, we will construct an explicit family of approximating solutions that explode at such a point. These results are contained in several works in collaboration with A. Jevnikar, R. López-Soriano and D. Ruiz, and with L. Battaglia and A. Pistoia
28/10/2020
JEAN DOLBEAULT (Université Paris-Dauphine)
Title: Stability in Gagliardo-Nirenberg inequalities
Abstract: Optimal constants and optimal functions are known in some functional inequalities. The next question is the stability issue: is the difference of the two terms controlling a distance to the set of optimal functions ? A famous example is provided by Sobolev's inequalities: in 1991, G. Bianchi and H. Egnell proved that the difference of the two terms is bounded from below by a distance to the manifold of the Aubin-Talenti functions. They argued by contradiction and gave a very elegant although not constructive proof. Since then, estimating the stability constant and giving a constructive proof has been a challenge.
Abstract: Optimal constants and optimal functions are known in some functional inequalities. The next question is the stability issue: is the difference of the two terms controlling a distance to the set of optimal functions ? A famous example is provided by Sobolev's inequalities: in 1991, G. Bianchi and H. Egnell proved that the difference of the two terms is bounded from below by a distance to the manifold of the Aubin-Talenti functions. They argued by contradiction and gave a very elegant although not constructive proof. Since then, estimating the stability constant and giving a constructive proof has been a challenge.
This lecture will review some contructive results, mostly on subcritical inequalities, for which explicit constants can be provided. The main tool is based on entropy methods and nonlinear flows. Proving stability amounts to establish, under some constraints, a version of the entropy - entropy production inequality with an improved constant. In simple cases, for instance on the sphere, rather explicit results have been obtained by the « carré du champ » method introduced by D. Bakry and M. Emery. In the Euclidean space, results have been obtained recently using entropy methods and constructive regularity estimates for the solutions of the nonlinear flow.
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14/10/2020
CHRISTOPHE PRANGE (CNRS and Paris Cergy University)
14/10/2020
CHRISTOPHE PRANGE (CNRS and Paris Cergy University)
Title: Quantitative regularity for the Navier-Stokes equations via spatial concentration
Abstract: In this talk I will focus on two related aspects of the regularity theory for the three-dimensional Navier-Stokes equations: quantitative regularity estimates on the one hand and concentration estimates for blow-up solutions on the other hand. This connection enables in particular a quantification of Seregin's 2012 regularity criterion in terms of the critical $L^3$ norm. A counterpart of this is that we are able to give lower bounds on the blow-up rate of certain critical norms near potential singularities in the wake of Tao's work in 2019. This talk is based on recent works in collaboration with Tobias Barker (University of Warwick).
Abstract: In this talk I will focus on two related aspects of the regularity theory for the three-dimensional Navier-Stokes equations: quantitative regularity estimates on the one hand and concentration estimates for blow-up solutions on the other hand. This connection enables in particular a quantification of Seregin's 2012 regularity criterion in terms of the critical $L^3$ norm. A counterpart of this is that we are able to give lower bounds on the blow-up rate of certain critical norms near potential singularities in the wake of Tao's work in 2019. This talk is based on recent works in collaboration with Tobias Barker (University of Warwick).
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30/09/2020
MARÍA ÁNGELES GARCÍA FERRERO (Heidelberg University)
Title: Unique continuation for nonlocal operators
Abstract: Roughly speaking, a unique continuation property states that a solution of certain partial differential equation is determined by its behaviour in a subset. In this talk we will see this kind of properties, including their strong and quantitative versions, for some classes of nonlocal operators like the Hilbert transform, which arise in medical imaging, or the (higher order) fractional Laplacian. The results I will present rely on commonly used tools like Carleman estimates and the Caffareli-Silvestre extension, but also on two alternative mechanisms. As an application we will see Runge approximation results. This is a joint work with Angkana Rüland
MARÍA ÁNGELES GARCÍA FERRERO (Heidelberg University)
Title: Unique continuation for nonlocal operators
Abstract: Roughly speaking, a unique continuation property states that a solution of certain partial differential equation is determined by its behaviour in a subset. In this talk we will see this kind of properties, including their strong and quantitative versions, for some classes of nonlocal operators like the Hilbert transform, which arise in medical imaging, or the (higher order) fractional Laplacian. The results I will present rely on commonly used tools like Carleman estimates and the Caffareli-Silvestre extension, but also on two alternative mechanisms. As an application we will see Runge approximation results. This is a joint work with Angkana Rüland