a CAMGSD―CMUC Workshop in Nonlinear Analysis
Coimbra, 1315 July 2022
José A. Carrillo (University of Oxford)
Title: Nonlocal AggregationDiffusion 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 FokkerPlanck 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 CuckerSmale orientation dynamics, McKeanVlasov equations, and nonlinear diffusion KellerSegel 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 CrandallRabinowitz 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 NonlocalCPD 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 halfharmonic 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.
Isabeau Birindelli (Università di Roma "La Sapienza")
Title: Propagation of minima: local and non local case
Abstract: I will give a geometrical characterization of the zero level set for non negative supersolutions for operators that don't satisfy the strong maximum principle. The point of view is quite different from the classical "sharp maximum principle" of Bony. We will treat both the case of local operators and non local operators.
These are joint work with Giulio Galise and Hitoshi Ishii.
Denis Bonheure (Université Libre de Bruxelles)
Title: The motion of a solid body in a Stokes or NavierStokes flow
Abstract: I will consider a fluidsolid 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 timedepending 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.
JeanBaptiste Casteras (Universidade de Lisboa)
Title: Almost sure wellposedness for a class of cubic Schrödinger type equations
Abstract: I will present some results concerning the local wellposedness 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 wellknown 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 illposed. I will show that by randomising the initial data it is possible to prove the local wellposedness 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: Selfsimilar solutions and the modified Kortewegde Vries equation
Abstract: Selfsimilar solutions to the modified Kortewegde Vries equation naturally in the evolution of vortex patches, describing either sharp corners or logarithmic spirals. Moreover, they present a natural blowup behavior at t=0 and describe the asymptotics of small solutions. However, the profile of a selfsimilar 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 selfsimilar solutions and their stability at both infinite and blowup 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: Socalled 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: Selfsimilar solutions, relative entropy and applications
Abstract: Selfsimilar Barenblatt solutions are well known for their role in the study of large time asymptotics of the solutions of fast diffusion equations. In the framework of entropy methods, they play a key role in applications to rigidity results, symmetry or stability in interpolation inequalities. This lecture will review some results on these issues.
Fabiana Leoni (Università di Roma "La Sapienza")
Title: New concentration phenomena for signchanging 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 signchanging 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.
Makson Santos (Universidade de Lisboa)
Title: Improved regularity for the parabolic normalized pLaplace equation
Abstract: We study regularity properties of the parabolic normalized pLaplace 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 LaneEmden 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 LaneEmden 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 antimaximum 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: A degenerate fully nonlinear free transmission problem with variable exponents
Abstract: We discuss optimal regularity for equations with a degeneracy rate which varies in a discontinuous fashion over the domain. Moreover, this discontinuity depends in an implicit way on the solution itself. We introduce the notion of pointwise sharp regularity which requires an alternative characterization of Holder spaces. These ideas are combined with a geometric tangential analysis argument to obtain the optimal regularity.
13 July  14 July  15 July  

8:50 ― 09:00  Opening 


9:00 ― 09:50  Manuel Del Pino 
Minicourse Yannick Sire 
Minicourse José A. Carrillo 
09:55 ― 10:45  Denis Bonheure 
Minicourse Yannick Sire 
Minicourse José A. Carrillo 
10:50 ― 11:20   Coffee break    Coffee break    Coffee break  
11:20 ― 12:10  Cristiana De Filippis  Minicourse Yannick Sire  Minicourse 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  JeanBaptiste 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 
Venue
Room Pedro Nunes, Department of Mathematics, 1st (ground) floor, 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.
Choose "Porto  Campanha > Coimbra" if you arrive at Oporto's airport.
Both from Oporto or Lisbon you will arrive by train first at "CoimbraB". With the same ticket you can then take a local train to "Coimbra" which will leave you at the city center (from which you can walk up to the hotel). Another (cheap) option is to simply take a taxi from Coimbra B to the hotel.
To go from the airport to the train stations we recommend that you take the metro or a taxi.
Don't hesitate to contact one of the organisers if you need some information.