# Titles and Abstracts

Title: Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains
Abstract: We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L} F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$\,, with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we can use a quite general class of linear operators that includes the three most common versions of the fractional Laplacian $(-\Delta)^s$, with $s\in (0,1]$\,. We will shortly present some recent results about existence, uniqueness and a priori estimates for a quite large class of very weak solutions, that we call weak dual solutions. We will devote special attention to the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our techniques cover also the local case s = 1 and provide new results even in this setting. A surprising instance of this problem is the possible presence of nonmatching powers for the boundary behavior: for instance, when $\mathcal{L}=(-\Delta)^s$ is a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that, whenever $2s \ge 1 - 1/m$, solutions behave as $dist^{1/m}$ near the boundary; on the other hand, when $2s < 1 - 1/m$, different solutions may exhibit different boundary behaviors even for large times. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the elliptic case. The above results are contained on a series of recent papers in collaboration with A. Figalli, X. Ros-Otón, Y. Sire and J. L. Vázquez.

• Jan Burczak (Oxford)

Title: Fractional Keller-Segel system
Abstract: I will discuss certain regularity results for the parabolic-elliptic Patlak-Keller-Segel system with fractional diffusion. The main part of my talk contains a disproof of the finite-time blowup conjecture in the critical, one dimensional case and certain smoothness/steady-state asymptotics results for the logistically damped case, but with both diffusions and damping weak.

• Charlie Elliott (Warwick)

Title: Coupled bulk surface systems and non-local surface equations
Abstract: In the context of modelling receptor ligand dynamics in cell biology I will discuss the coupling of reaction diffusion equations holding on a bulk domain and its boundary leading to non-local surface free boundary problems of Stefan and Here-Shaw type.

• Marco Fontelos (ICMAT - Madrid)

Title: Similarity solutions to coagulation and fragmentation processes
Abstract: Coagulation and fragmentation processes are inherently nonlocal, with the probability of forming framents of a certain size at a given time depending on the distribution of fragments of any other size at a previous time. An interesting question is what will be the asymptotic distribution of fragment's size or whether such distribution may have a moment blowing up in finite time (gelation). Concerning the equation for coagulation with multiplicative kernel, we propose the problem of self-similar solutions of the second kind for kernels that do produce gelation in finite time. The results consist in the construction of global solutions matching asymptotic expansions and intermediate behaviors. The exponents of self-similarity are computed by solving a suitable nonlinear eigenvalue problem. We then consider the problem of self-similar solutions to the linear equation of irreversible fragmentation. We rigorously determine an explicit formula in the form of infinite product for the self-similar solutions through a Wiener-Hopf method for the calculation of residues in the complex plane; this is done for both the case of fragmentation which allows infinitesimal fragments and for fragmentation which bounds from below the m ́ınimum fragments. This study leads us to accurately determine the asymptotic behaviors of the solutions in both cases.

• María del Mar González (UAM -Madrid)

Title: Gluing methods for the fractional Yamabe problem with isolated singularities
Abstract: We construct solutions for the fractional Yamabe problem with isolated singularities, $(-\Delta)^s u =u^{s*}$. When the singular set is just one point, a Delaunay type solution may be constructed by solving a non-local ODE. In the case of a finite number of singularities, we succesfully use a gluing method despite the nonlocality of the problem. This is joint work with Weiwei Ao, Azahara DelaTorre and Jucheng Wei.

• Gabrielle Grillo (Milano)

Title: Existence and uniqueness of solutions to the porous media equation with measure data in the local and nonlocal setting
Abstract: We consider a weighted version of the fractional porous media equation, and develop existence and uniqueness results for solutions corresponding to measure data, uniqueness being not immediate even in the unweighted case. We also show how methods of the same type, together with delicate potential theoretic results and geometric methods, allow to treat similar problems also for local problems posed on negatively curved Riemannian manifolds.

• Grzegorz Karch (Uniwersytet Wroc\l awski)

Title: Blowup phenomena in conservation laws with fractional Laplacian and nonlocal fluxes
Abstract: Recent results on a nonlinear nonlocal evolution equation involving fractional Laplacian will be presented. This model contains, as a particular example, the celebrated parabolic-elliptic model of chemotaxis. For such a model, global-in-time solutions are constructed under (nearly) optimal assumptions on the size of radial initial data. Moreover, criteria for blowup of radial solutions in terms of suitable Morrey spaces norms are derived.

This is a joint work with Piotr Biler and Jacek Zienkiewicz.

• Moritz Kassmann (Bielefeld)

Title: Nonlocal energy forms and function spaces
Abstract: We discuss function spaces that are defined with the help of nonlocal quadratic forms. One aim of the talk is to investigate conditions under which such a form is comparable with the quadratic form that defines the seminorm of a fractional Sobolev space. We provide several examples including those with singular measures and one example related to the study of the Boltzmann equation. We outline the proof of comparability in the latter case, which involves delicate chaining and renormalization arguments. A second aim of the talk is to explain the significance of the aforementioned comparability results with regard to Dirichlet forms and regularity of solutions to integrodifferential equations. Last, we mention recent results on function spaces that appear in the study of nonlocal Dirichlet problems. The talk is based on joint works with Bartek Dyda resp. with Kai-Uwe Bux and Tim Schulze.

References:

Kai-Uwe Bux, Moritz Kassmann, Tim Schulze, "Quadratic forms and Sobolev spaces of fractional order", work in progress

Bartek Dyda, Moritz Kassmann, "Regularity estimates for elliptic nonlocal operators", https://arxiv.org/pdf/1509.08320

Bartek Dyda, Moritz Kassmann, "Function spaces and extension results for nonlocal Dirichlet problems", https://arxiv.org/pdf/1612.01628

• John King (Nottingham)

Title: Non-local problems in plant systems biology and thin-film flows
Abstract: Formal analyses will be outlined of two distinct problems which have in common that non-local effects arise from nonlinearly diffusive processes. One of these arises in capillary driven spreading and the other in plant-root patterning.

• Philippe Laurencot (Toulouse)

Title: Stationary solutions to a free boundary modeling MEMS
Abstract: An idealized electrostatically actuated microelectromechanical system (MEMS) consists of an elastic plate suspended above a fixed horizontal ground plate and held clamped along its boundary. Applying a voltage difference across the device generates a Coulomb force which in turn induces a deformation of the elastic plate. The state of the device is then described by the electrostatic potential between the two plates and the (vertical) deflection of the elastic plate. Stationary solutions of the corresponding moving boundary problem are constructed by a constrained variational approach and their behaviour in the vanishing aspect ratio limit is analysed. Joint works with Christoph Walker, Hannover, Germany.

• Dong Li (Hong Kong - HKUST)

Title: On refined Kato-Ponce and fractional Leibniz
Abstract: We will discuss a group of new refined Kato-Ponce estimates which is deeply connected with the Kenig-Ponce-Vega estimate for fractional Laplician operators. We will mention some interesting applications to fluid models and related equations.

• Stefano Lisini (Pavia)

Title: Optimal control problem for interaction equations: mean-field limit and Gamma convergence
Abstract: In this talk I will describe a mean-field limit approximation of an optimal control problem for a family of interaction equations. By means of a suitable Gamma-convergence I will show that the the mean-field limit for the equation commutes with the optimization. Techniques of optimal transportation of measures are used.

• Edoardo Mainini (Genova)

Title: Ground states in the diffusion dominated regime
Abstract: We consider a Keller-Segel type model with nonlinear power law diffusion and aggregation driven by Riesz potential interaction. All stationary states of the system are shown to be radially symmetric decreasing. In particular, we show that global minimizers of the associated free energy are compactly supported, uniformly bounded, Hölder regular, and smooth inside their support. This is a joint work with J. A. Carrillo, F. Hoffmann and B. Volzone.

• Xavier Ros-Otón (Zurich)

Title: Regularity of free boundaries in obstacle problems for integro-differential operators
Abstract: We present a brief overview of the regularity theory for free boundaries in different obstacle problems. We describe how a monotonicity formula of Almgren plays a central role in the study of the regularity of the free boundary in some of these problems. Finally, we explain new strategies which we have recently developed to deal with cases in which monotonicity formulas are not available.

• Filippo Santambrogio (Orsay)

Title: BV and Sobolev estimates on the JKo scheme for aggregation-diffusion equations
Abstract: I will consider a large class of evolution PDEs with linear diffusion and a non-local advection term, in gradient form, which includes for instance the parabolic-elliptic Keller-Segel model, and prove decay or bounds on first-order quantities by using fine estimates on the minimizers of the Jordan-Kinderlehrer-Otto scheme which can be obtained via a new inequality in optimal transport theory. Most of the work is in collaboration with Simone Di Marino (SNS Pisa).

• Lucia Scardia (Bath)

Title: Equilibrium measure for a nonlocal dislocation energy
Abstract: In this talk I will present a recent result on the characterisation of the equilibrium measure for a nonlocal and anisotropic energy arising as the Gamma-limit of discrete interacting dislocations, and an extension to more general anisotropies.

This is joint work with J.A. Carrillo, J. Mateu, M.G. Mora, L. Rondi and J. Verdera.

• Christian Schmeiser (Viena)

Title: Fractional diffusion as macroscopic limit of kinetic models
Abstract: > Fat-tailed equilibrium distributions, which do not possess finite second order moments with respect to velocity, lead to macroscopic limits of fractional diffusion type. Recent results on the derivation of fractional diffusion-advection equations and on fractional diffusion on bounded domains will be reviewed. (joint work with P. Aceves Sanchez)

• Maria Schonbek (California - Santa Cruz)

Title: Time decay for solutions to the Stokes equations with drift
Abstract: I will discuss the long time behaviour of the Lebesgue norms $\|v(\cdot,t)\|_p$ of solutions $v$ to the Cauchy problem for the Stokes system with drift $u$, which is supposed to be a divergence free smooth vector valued function satisfying a scale invariant condition. This is joint work with Gregory Seregin.

• Enrico Valdinoci (Berlin)

Title: Recent trends in nonlocal geometric problems
Abstract: We consider some nonlocal problems with geometric flavor and discuss topics such as asymptotics, regularity, rigidity and planelike minimizers.

Some of these problems are also related to models for phase coexistence.

• Bruno Volzone (Naples)

Title: Recent advances in symmetrization techniques for nonlocal equations
Abstract: A detailed abstract can be found here.

• Marie-Therese Wolfram (Warwick)

Title: Mean-field aggregation models in the life and social sciences
Abstract: In this talk we start by presenting different mean-field models for biological aggregation, which were inspired by the behavior of the pre-social German cockroach Blatella germanica. We show that the random diffusive motion of individuals, which respond to the local population density observed in their neighborhood by increasing or decreasing the amplitude of their random motion, can lead to the formation of aggregates. We formally derive the corresponding mean-field limits, which lead to nonlocal degenerate diffusion. For the corresponding first order model we prove existence of weak solutions and present its stability analysis.
Next we discuss how this approach can be generalized to multiple interacting species, as considered in Schelling's segregation model. The economist Thomas Schelling suggested that a slight individual preference to stay within the own group, could lead to total segregation. We present a mean-field version of the Schelling's model and give first insights into its dynamics. (joint work with M. Burger, J. Haskovec, J.-F. Pietschmann and H. Ranetbauer)