#### Advances in Applied Mathematical Analysis and Numerical Methods:

Ten Years of CoLab

### Abstracts

Ricardo Alonso

PUC Rio de Janeiro

*Emergence of exponentially weighted Lp-norms and Sobolev regularity for the Boltzmann equation*

Abstract: We consider the homogeneous Boltzmann equation for Maxwell and hard potentials, without cutoff, and study the appearance and propagation of Lp-norms, including polynomial and exponential weights. Propagation of Sobolev regularity with such weights is also considered. Classical and novel ideas are combined to elaborate an elementary argument that proves the result in the full range of integrability p ∈ [1, ∞] and singularity s ∈ (0, 1). For the case p = ∞, we use an adaptation of the classical level set method by De Giorgi.

Damião Araujo

Joao Pessoa

*Sharp Regularity for the Inhomogenous Porous Medium Equation*

Abstract: In this talk we shall consider the inhomogeneous porous medium equation

$$

\partial u_t – \Delta u^m = f \in L^{q,r} \quad m>1.

$$

Here we show that weak solutions are H\”older continuous, with the following sharp exponent

$$

\min \left\{ \frac{\alpha_0^-}{m} , \frac{[(2q-n)r-2q]}{q[mr-(m-1)]} \right\},

$$

where $\alpha_0$ denotes the optimal H\”older exponent for solutions of the Homogeneous equation. The method relies on an approximation lemma and geometric iteration with the appropriate intrinsic scaling. This talk is based on joint work with J. Miguel Urbano and A.F. Maia – University of Coimbra – Portugal.

Silvia Barbeiro

University of Coimbra

*Numerical solution of time-dependent Maxwell’s equations for modeling light scattering in human eye’s structures*

Abstract: We propose an explicit iterative leap-frog discontinuous Galerkin method for time-domain Maxwell’s equations in anisotropic materials and we focus on deriving stability and convergent estimates of fully discrete schemes. We consider anisotropic permittivity tensors, which arise naturally in our application of interest. An important aspect in computational electromagnetic problems, which will be discussed, is the implementation of the boundary conditions. We illustrate the theoretical results with numerical examples. We also present the results of numerical computations in the context of modeling scattered electromagnetic wave’s propagation through human eye’s structures.

Daniele Boffi

University of Pavia

*A fictitious domain approach for fluid-structure interaction problems*

Abstract: We discuss a distributed Lagrange multiplier formulation of the Finite Element Immersed Boundary Method for the numerical approximation of the interaction between fluids and solids. The discretization of the problem leads to a mixed problem for which a rigorous stability analysis is provided. Optimal convergence estimates are proved for the finite element space discretization. The model, originally introduced for the coupling of incompressible fluids and solids, can be extended to include the simulation of compressible structures.

Sun-Sig Byun

Seoul National University

*Obstacle problems for nonlinear elliptic equations with nonstandard growth*

Abstract: We discuss higher regularity results for nonlinear obstacle problems with nonstandard growth

Filippo Cagnetti

University of Sussex

*Optimal regularity and structure of the free boundary for minimizers in cohesive zone models*

Abstract: We consider minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $C^{1, 1/2}$ on each side of the fracture. Moreover, we prove that near non-degenerate points the fracture set is $C^{1, \alpha}$, for some $\alpha \in (0,1)$. This is joint work with Luis Caffarelli and Alessio Figalli.

Eric Carlen

Rutgers University

*Spectral Gaps for Markov Processes with Chaotic Invariant Measures*

Abstract: This talk presents a method for estimating spectral gaps for Markov jump process for systems of N interacting particles with jump rates that are not bounded from below below, but for which the sequence of invariant measures f_Ng, where _N is the invariant measure for the N particle system, is chaotic in the sense of Mark Kac. The main example that motivates this work is the Kac model for energy and momentum conserving\hard sphere” collisions in three dimensions. However, there are not many techniques available for controlling spectral gaps when the jump rates are not bounded from below, and the method is developed in its wider natural context where it is likely to _nd other applications. This is joint work with Maria Carvalho and Michael Loss.

Maria Carvalho

University of Lisbon

*The Entropy Problem for the Kac Model*

Abstract: The Entropy Problem for the Kac Model has been actively studied in recent years. We present some new results and discuss some of the remaining problems. This is joint work with E. Carlen and A. Einav.

Fabio Chalub

New University of Lisbon

*The Kimura Equation*

Abstract: The Kimura Equation was introduced in the 60’s by the Japanese geneticist Motoo Kimura and is considered one of the most important models in population genetics. It is a degenerated partial differential equation of drift diffusion type modelling the evolution of the probability distribution among different genotypes in a population.

Clint Dawson

The University of Texas at Austin

*Hybrid Discontinuous Galerkin Methods for Nonlinear Dispersive Water Waves*

Abstract: We will describe models used to describe near-shore wave behavior. These models are based on simplifications of the Navier-Stokes equations and are suitable for regimes where shallow water theory typically breaks down. We will discuss one particular model known as the Green-Naghdi equations. We will then discuss numerical approximation of this system of equations using the Hybrid Discontinuous Galerkin method, and discuss the computational science tools used to implement the method in a parallel computing environment. Numerical results will also be discussed and we will outline how this methodology will be used in a newly funded UT-Portugal project.

Guido De Philippis

SISSA

*Boundary Regularity for Mass Minimising currents*

Abstract: In this talk I will present a first boundary regularity result for mass minimising currents in any co-dimension and some of its consequences. In particular I will show that the regular points are dense in the boundary. In this general setting it was unknown even the existence of one regular point. This is a joint work with C. De Lellis, J. Hirsch and A. Massaccesi.

Stanislav Kondratyev

University of Coimbra

*Unbalanced transport: gradient flows and transportation inequalities*

Abstract: We show that a few Fokker–Planck-type equations with reaction can be viewed as gradient flows with respect to metrical structures associated with unbalanced transport. The driving entropies are not supposed to be geodesically convex nor semi-convex. We study isoperimetric-type transportation inequalities naturally associated with the flows and conclude the exponential convergence of the flows to the steady states

Tuomo Kuusi

University of Oulu

*Numerical methods in stochastic homogenization*

Abstract: I will discuss a new method for computing solutions of elliptic equations with random rapidly oscillating coefficients. This is a joint work with S. Armstrong, A. Hannukainen and J.-C. Mourrat.

Diego Marcon

UFRGS

*The calculus of thermodynamical formalism*

Abstract: Given a finite-to-one map T acting on a compact metric space \Omega and an appropriate Banach space of functions X, one classically constructs for each potential A \in X a transfer operator L_A acting on X. Under suitable hypotheses, it is well-known that L_A has a maximal eigenvalue, has a spectral gap and defines a unique Gibbs measure. Moreover there is a unique normalized potential B that acts as a representative of the class of all potentials defining the same Gibbs measure.

We study the geometry of the set of normalized potentials N, of the normalization map, and of the Gibbs map. We give an easy proof of the fact that N is an analytic submanifold of X and that the normalization map is analytic; we compute the derivative of the Gibbs map; last we endow N with a natural weak Riemannian metric (derived from the asymptotic variance) with respect to which we compute the gradient flow induced by the pressure with respect to a given potential, e.g. the metric entropy functional. We also apply these ideas to recover in a wide setting existence and uniqueness of equilibrium states, possibly under constraints. This is a joint work with P. Giulietti, B. Kloeckner, and A. Lopes.

Rui Vilela Mendes

CMAFCIO – University of Lisbon

*Non-commutative tomography and signal processing*

Abstract: Tomograms, a generalization of the Radon transform to arbitrary pairs of non-commuting operators are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of signals, are robust in the presence of noise and avoid the interpretation problems associated to other bilinear transforms Explicit construction of tomogram transforms and applications to component separation and arbitrary feature identification are presented. Also their interpretation as operator symbols provides an adequate tool for nonlinear filtering.

Stefania Patrizi

The University of Texas at Austin

*On some segregation models*

Abstract: Segregation phenomena occurs in many areas of mathematics and science: from equipartition problems in geometry, to social and biological processes (cells, bacteria, ants, mammals) to finance (sellers and buyers). Segregation problems model a situation of high competition for resources and involve a combination of diffusion andannihilation between populations. In this talk we present three different segregations models: in the first one competing species interact at long distance, in the second one species follow a fully nonlinear diffusion, in the third one two competing species follow different propagation equations, one of them involving a local diffusion while the other one involving a non-local diffusion.

This is a joint work with Luis Caffarelli, Veronica Quitalo and Monica Torres.

Adelia Sequeria

University of Lisbon

*Cardiovascular Modeling and Simulations. Applications to some Clinical Studies*

Abstract: Mathematical modeling and simulations of the human circulatory system is a challenging wide-range research field that has seen a tremendous growth in the last few years and is rapidly progressing, motivated by the fact that cardiovascular diseases are a major cause of death in developed countries. The continuous development of surgical techniques such as angioplasty, stents placements, etc. has given a great impulse to the investigation of blood flow in vessels. The acquisition of medical data and the understanding of the local hemodynamics and its relation with global phenomena, in both healthy and pathological conditions, using appropriate and accurate numerical methods, play an important role in the medical research. This helps, for instance, in predicting the consequences of surgical interventions, or identifying regions of the vascular systems prone to the formation and growth of atherosclerotic plaques or aneurysms.

In this talk we consider some mathematical models of the cardiovascular system and comment on their significance to yield realistic and accurate numerical results, using stable, reliable and efficient computational methods. They include fluid-structure interaction (FSI) models to account for blood flow in compliant vessels, analysis of absorbing boundary conditions to deal with the numerical spurious reflections due to the truncation of the computational domain and a quick overview of the so-called geometrical multiscale approach to simulate the reciprocal interactions between local and systemic hemodynamics. Data Assimilation techniques in Cardiovascular Mathematics, with particular emphasis to the velocity tracking approach based on the solution of an optimal control problem will also be addressed.

Results on the simulation of some image-based patient-specific clinical cases will be presented.

Ana Jacinta Soares

University of Minho

*On the derivation of reaction-diffusion equations from the kinetic theory of reactive mixtures*

Abstract: We consider a chemically reactive mixture of polyatomic gases described in the frame of the Boltzmann equation and study the reaction-diffusion limit of the kinetic system of equations. Under certain assumptions, we formally derive a reaction-diffusion system of Maxwell-Stefan type. This is based on joint work with B. Anwasia, M. Bisi and F. Salvarani.

Rolf Stenberg

Aalto University

*Stabilised Finite Element Methods for Variational Inequalities*

Abstract: We survey our recent and ongoing work [1,2] on finite element methods for contact problems. Our approach is to first write the problem in mixed form, in which the contact pressure act as a Lagrange multiplier. In order to avoid the problems related to a direct mixed finite element discretisation, we use a stabilised formulation, in which appropriately weighted residual terms are added to the discrete variational forms. We prove that the formulation is uniformly stable, which implies an optimal a priori error estimate. Using the stability of the continuous problem, we also prove a posteriori estimates, the optimality of which is ensured by local lower bounds. In the implementation of the methods, the discrete Lagrange multiplier is locally eliminated, leading to a Nitsche-type method [3].

For the problems of a membrane and plate subject to solid obstacles, we present numerical results.

Joint work with Tom Gustafsson (Aalto) and Juha Videman (Lisbon).

References:

[1] T. Gustafsson, R. Stenberg, J. Videman. Mixed and stabilized finite element methods for the obstacle problem. SIAM Journal of Numerical Analysis 55 (2017) 2718–2744

[2] T. Gustafsson, R. Stenberg, J. Videman. Stabilized methods for the plate obstacle problem. BIT– Numerical Mathematics (2018) DOI: 10.1007/s10543-018-0728-7

[3] E. Burman, P. Hansbo, M.G. Larson, R. Stenberg. Galerkin least squares finite element method for the obstacle problem. Computer Methods in Applied Mechanics and Engineering 313 (2017) 362–374

Eduardo Teixeira

University of Central Florida

*On a new class of variable diffusibility operators*

Abstract: We introduce a new class of non-divergence form elliptic operators whose degree of degeneracy/singularity varies accordantly to a prescribed power law. Such an endeavor parallels the by now well established minimization theory of functionals satisfying p-growth condition, which in particular encompasses the theory of p(x)-laplacian. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are differentiable; sharp estimates are also derived.

Rafayel Teymurazyan

University of Coimbra

*A free boundary optimization problem for the infinity Laplacian*

Abstract: We study an optimization problem for the infinity Laplacian as a limit of quasilinear optimization problems in heat conduction. We prove uniform estimates which then allow to obtain regularity results and derive geometric properties for the solution and the free boundaries.

Alexis Vasseur

The University of Texas at Austin

*The 3D Quasi-geostrophic equation: existence of solutions, lateral boundary conditions and regularity.*

Abstract: The 3D Quasi-geostropic equation is a model used in climatology to model the evolution of the atmosphere for small Rossby numbers. It can be derived from the primitive equation. The surface quasi-geostrophic equation (SQG) is a special case where the atmosphere above the earth is at rest. The evolution then depends only on the boundary condition, and can be reduced to a 2D model.

In this talk, we will show how we can derive the physical lateral boundary conditions for the inviscid 3D QG, and construct global in time weak solutions. Finally, we will discuss the global regularity of solutions to the QG equation with Ekman pumping.

Dmitry Vorotnikov

University of Coimbra

*Some gradient flows of spatial curves*

Abstract: We study gradient flows on certain spaces of spatial curves parametrized by the arc length. The spaces under consideration can be viewed as submanifolds of the Wasserstein space of probability measures endowed with Otto’s Riemannian structure. In particular, the gradient flow of the potential energy models overdamped motion of a falling inextensible string. We establish existence of generalized solutions to the corresponding nonlinear evolutionary PDE and their exponential decay to the equilibrium. The main difficulty comes from the fact that the diffusivity is an unknown Lagrange multiplier, so parabolicity may fail. The system admits solutions backwards in time (which is of course impossible for a conventional parabolic problem like the heat equation), which leads to funny non-uniqueness of trajectories. Another flow under consideration is a new geometric flow which we call uniformly compressing mean curvature flow. This flow (similarly to MCF) collapses in finite time, but after time/space rescaling one gets another gradient flow which has a mechanical interpretation related to the one above. For the corresponding PDE, we show the wellposedness and study some asymptotics. This flow does not behave in the the bizarre branching way mentioned above. We will also touch upon similar evolution of higher-dimensional objects (membranes). Based on joint works with W. Shi.

Mary Wheeler

The University of Texas at Austin

*Modeling Flow, Reactive Transport and Wave Propagation in Porous Media*

Abstract: In this presentation, we discuss enriched Galerkin (EG) algorithms for modeling Darcy flow, reactive transport, and elastic wave propagation. This approach involves enriching the continuous Galerkin finite element method with discontinuous elements. For transport EG is coupled with entropy residual stabilization for transport. The method provides locally and globally conservative fluxes, which are crucial for coupled flow and transport problems. In particular, numerical sim-ulations of viscous fingering instabilities in heterogeneous porous media and Hele-Shaw cells are illustrated as well as results for two phase flow. Here dynamic adaptive mesh refinement is applied in order to save computational cost for large-scale three-dimensional applications. In addition, entropy residual based stabilization for high order EG transport systems prevents any spurious oscillations. Recently EG was applied to simulated elastic wave propagation in a fractured media. Here Linear Slip theory was used for incorporating fractures and faults. Specifically, this new approach has the advantage of DG with a computational cost comparable to that of the Spectral Element Method. Computational results demonstrating the effectiveness of EG for these flow and reactive transport and wave propagation are provided. The work on flow and transport was done in collaboration with Sanghyun Lee and the work on elastic wave propagation with Mrinal Sen and Janaki Vamaraju.

Jingang Xiong

Beijing Normal University

*Local analysis for singular solutions of conformally invariant elliptic equations*

Abstract: I will talk about a higher order extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation in 1989. The approach is based on the Green’s representation and is unified for different orders of semilinear elliptic equations with Sobolev critical exponents.

This is joint work with Tianling Jin.