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Three Nonlinear Days in Coimbra

a CAMGSD―CMUC Workshop in Nonlinear Analysis

Coimbra, 13-15 July 2022

About

The joint meeting aims at gathering international top-level researchers, whose scientific agenda relates to ongoing projects led by members of the research centers CAMGSD (Centre for Mathematics, Geometry and Dynamical Systems) and CMUC (Centre for Mathematics at University of Coimbra). As a consequence, it expects to broaden local scientific interaction and strengthen pre-existing collaborations. As a by-product, the event also exposes young researchers and advanced graduate students to an environment of scientific excellence.

The scope of the meeting covers, though it is not limited to, nonlinear analysis, variational problems and methods, regularity theory for elliptic equations, free boundary problems, and related topics. The scientific program comprises two minicourses, eight plenary lectures and four contributed talks.

Organisers:
Edgard A. Pimentel (CMUC, Universidade de Coimbra)
Hugo Tavares (CAMGSD, Universidade de Lisboa)

Organisation:

Support:

Partially supported by FCT/Portugal projects PTDC/MAT-PUR/1788/2020, PTDC/MAT-PUR/28686/2017, UIDB/MAT/04459/2020 and UIDB/00324/2020.

Mini-courses


José A. Carrillo (University of Oxford)
Title: Nonlocal Aggregation-Diffusion Equations: entropies, gradient flows, phase transitions and applications
Abstract: This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L2 framework, nonlinear diffusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.

Yannick Sire (Johns Hopkins University)
Title: Variations on the theory of harmonic maps
Abstract: In this series of lectures, I will introduce and develop the basic theory for harmonic maps and some variations of them from a PDE point of view. After describing classical results due to Helein and Evans on the regularity of harmonic maps into spheres, I will move to the case of harmonic maps with free boundaries. This latter topic has been instrumental in the works of Fraser and Schoen on extremal Steklov eigenvalues, that I will briefly motivate. In connection with recent works of Da Lio and Riviere on half-harmonic maps, I will present two different proofs of regularity of those maps and explain at the end of the course, how to investigate their heat flow.

Plenary talks


Isabeau Birindelli (Università di Roma "La Sapienza")
Title: TBA

Denis Bonheure (Université Libre de Bruxelles)
Title: The motion of a solid body in a Stokes or Navier-Stokes flow
Abstract: I will consider a fluid-solid model where a solid is immersed in flow constrained in a 2D infinite channel. After presenting some results on the stationary problem and the existence of local or global time-depending solutions, the focus will be put on the various open questions and challenges concerning periodic solutions. The talk is based on an ongoing research program with G. Galdi, F. Gazzola, C. Grandmond, M. Hillairet, C. Patriarca and G. Sperone.

Jean-Baptiste Casteras (Universidade de Lisboa)
Title: Almost sure well-posedness for a class of cubic Schrödinger type equations
Abstract: I will present some results concerning the local well-posedness of a cubic Schrödinger equation in R^d involving a quite general operator of order s>= 2 and with rough initial data belonging to H^S. It is well-known that there exists an explicit regularity threshold S_max(s,d) for the initial data such that if S_max(s,d) is too small then this equation is ill-posed. I will show that by randomising the initial data it is possible to prove the local well-posedness of this equation for some S below this threshold. Based on a joint work with Juraj Földes and Gennady Uraltsev.

Simão Correia (Universidade de Lisboa)
Title: Self-similar solutions and the modified Korteweg-de Vries equation
Abstract: Self-similar solutions to the modified Korteweg-de Vries equation naturally in the evolution of vortex patches, describing either sharp corners or logarithmic spirals. Moreover, they present a natural blow-up behavior at t=0 and describe the asymptotics of small solutions. However, the profile of a self-similar solution has rough decay and regularity assumptions. As such, their existence is fairly nontrivial and the dynamical flow around these solutions has to be constructed in a very delicate manner. We'll discuss the existence of self-similar solutions and their stability at both infinite and blow-up time. The results presented were obtained in collaboration with R. Cöte and L. Vega.

Cristiana De Filippis (Università di Parma)
Title: Schauder estimates for any taste
Abstract: So-called Schauder estimates are a standard tool in the analysis of linear elliptic and parabolic PDE. They have been originally obtained by Hopf (1929, interior case), and by Schauder and Caccioppoli (1934, global estimates). The nonlinear case is a more recent achievement from the '80s (Giaquinta & Giusti, Ivert, Lieberman, Manfredi). All these classical results hold in the uniformly elliptic framework. I will present the solution to the longstanding open problem (~’70s) of proving estimates of such kind in the nonuniformly elliptic setting. I will also cover the case of nondifferentiable functionals and of mixed local and nonlocal problems. From joint work with Giuseppe Mingione (University of Parma).

Manuel del Pino (University of Bath)
Title: Dynamics of concentrated vorticities in 2d and 3d
Abstract: A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on the existence and asymptotic behaviour of these solutions. We describe, with precise asymptotics, interacting vortices, and travelling helices. We rigorously establish the law of motion of "leapfrogging vortex rings", originally conjectured by Helmholtz in 1858.

Jean Dolbeault (Ceremade - Paris Dauphine)
Title: TBA

Fabiana Leoni (Università di Roma "La Sapienza")
Title: New concentration phenomena for sign-changing radial solutions of fully nonlinear elliptic equations
Abstract: We present recent results about radial solutions of a class of fully nonlinear elliptic Dirichlet problems posed in a ball, driven by the extremal Pucci's operators and provided with power zero order terms. In particular, we show that, for the existence of sign-changing solutions, a new critical exponent appears. Furthermore, we analyze the new concentration phenomena occurring as the exponents approach the critical values.

Gianmaria Verzini (Politecnico di Milano)
Title: Singular analysis of the optimizers of the principal eigenvalue of the Laplacian with an indefinite weight
Abstract: When analyzing the survival threshold for a species in population dynamics, one is led to consider the principal eigenvalue of some indefinite weighted problems in a bounded domain. We study the minimization of such eigenvalue, associated with Neumann boundary conditions, performing the analysis of the singular limit in case of arbitrarily small favorable region. We show that, in this regime, the favorable region is connected and it concentrates on the boundary of the domain. Though widely expected, these properties are still unknown in the general case. We also deal with some related problem, with Dirichlet boundary conditions. Joint works with Dario Mazzoleni and Benedetta Pellacci, and with Lorenzo Ferreri.

Contributed talks


Makson Santos (Universidade de Lisboa)
Title: Improved regularity for the parabolic normalized p-Laplace equation
Abstract: We study regularity properties of the parabolic normalized p-Laplace equation, in the case where the exponent p is sufficiently close to 2. We argue by approximation methods relating our model with the heat equation and show that the gradient of the solutions are asymptotically Lipschitz. In addition we also establish improved regularity estimates in Sobolev spaces.

Delia Schiera (Universidade de Lisboa)
Title: Principal spectral curves for Lane-Emden fully nonlinear type systems
Abstract: I will present some recent results, obtained in collaboration with Ederson Moreira dos Santos, Gabrielle Nornberg and Hugo Tavares, about spectral properties of fully nonlinear Lane-Emden type systems with possibly unbounded coefficients and weights. In particular, we prove the existence of two principal spectral curves on the plane, which turn out to be related to the solvability of the corresponding Dirichlet problem, and to the validity of maximum principles. We also construct a possible third spectral curve related to a second eigenvalue and an anti-maximum principle.

Rafayel Teymurazyan (Universidade de Coimbra)
Title: Liouville type theorems for the porous medium equation
Abstract: We prove Liouville type theorems for the porous medium equation, provided it stays universally close to the heat equation. This is a joint work with D.J. Araújo (UFPB, Brazil).

David Jesus (Universidade de Coimbra)
Title: TBA

Schedule

13 July 14 July 15 July
8:50 ― 09:00 Opening


9:00 ― 09:50 Manuel Del Pino
Mini-course Yannick Sire
Mini-course José A. Carrillo
09:55 ― 10:45 Denis Bonheure
Mini-course Yannick Sire
Mini-course José A. Carrillo
10:50 ― 11:20 - Coffee break - - Coffee break - - Coffee break -
11:20 ― 12:10 Cristiana De Filippis Mini-course Yannick Sire Mini-course José A. Carrillo
12:15 ― 12:45 Delia Schiera Rafayel Teymurazyan Closure
12:45 ― 14:45 - Lunch - - Lunch - - Lunch -
14:45 ― 15:35 Simão Correia Gianmaria Verzini
15:40 ― 16:30 Jean-Baptiste Casteras Isabeau Birindelli
16:30 ― 17:00 - Coffee break - - Coffee break -
17:00 ― 17:50 Fabiana Leoni Jean Dolbeault
17:55 ― 18:25 David Jesus Makson S. Santos

Further information


Venue
Department of Mathematics, University of Coimbra


To reach Coimbra, you may fly to Oporto or Lisbon, and then either:
- take an airport shuttle directly to Coimbra;
or
- take a train. For train tickets and schedules, check CP - Comboios de Portugal's website. Choose "Lisboa - Oriente -> Coimbra" if you arrive at Lisbon's airport; "Porto - Campanha -> Coimbra" if you arrive at Oporto's airport. Don't hesitate to contact one of the organisers if you need some information. To go from the airport to the train stations we recommend that you take the metro or a taxi.

Registration

If you wish to participate in this workshop, please fill the following form until July 10. Speakers do not need to register.