# Abstracts

#### September 18-22, 2017Institute for Computational Engineering and Sciences (ICES)Peter O’Donnell Jr Building – 201 East 24th StreetThe Blanton Museum of Art – Capitol RoomThe University of Texas at Austin

Sona AkopianBrown University, USA

On $L^p$ approximations of Landau equation solutions in the Coulomb case

Abstract: We examine a class of Boltzmann equations with an abstract collision kernel in the form of a singular mass concentrated at very low collision angles and relative velocities between interacting particles. Similarly to the classical Boltzmann operator, this particular collision operator also converges to the collision term in the Landau equation as the characterizing parameter $\epsilon$ tends to zero. We will address the existence of $L^p$ solutions to this family of Boltzmann equations and discuss their approximations of solutions to the Landau equation as $\epsilon$ vanishes.

_____________________

Ricardo AlonsoPUC Rio, Brazil

Electromagnetic wave propagation in random waveguides

Abstract: A long range propagation of electromagnetic waves in random waveguides with rectangular cross-section and perfectly conducting boundaries is presented. The waveguide is filled with an isotropic linear dielectric material, with randomly fluctuating electric permittivity. The fluctuations are weak, but they cause significant cumulative scattering over long distances of propagation of the waves. We analyze Maxwell’s equations in this configuration with the diffusion approximation theory. The result is a detailed characterization of the transport of energy in the waveguide, the loss of coherence of the modes and the depolarization of the waves due to cumulative scattering.

_____________________

Kazuo AokiNational Taiwan U, Taiwan

Shock wave structure for polyatomic gases with large bulk viscosities

Abstract: The structure of a standing plane shock wave in a polyatomic gas is investigated on the basis of kinetic theory, with special interest in gases with large bulk viscosities, such as the CO2 gas. The polyatomic version of the ellipsoidal statistical model is employed, and the shock structure is obtained numerically for different upstream Mach numbers and for different (large) values of the ratio of the bulk viscosity to the shear viscosity. The double-layer structure consisting of a thin upstream layer with a steep change and a much thicker downstream layer with a mild change is obtained. An analytical solution, consisting of a jump condition and a slowly varying solution, that can approximate the double-layer structure well is also presented.

This work is a collaboration with Shingo Kosuge (Kyoto University).

_____________________

Dieter ArmbrusterArizona State U, USA

Kinetic Models of Need-based Transfers

Abstract: Kinetic exchange models of markets utilize microscopic binary descriptions of wealth transfers to derive a Boltzmann-like equation describing the evolution of the corresponding wealth distribution. We develop such a model to describe a binary form of welfare called need-based transfer (NBT), motivated by the reciprocal gift-giving of cattle practiced among the Maasai of East Africa. Variants of such welfare schemes can be attributed to other human and animal communities. Specifically, we consider NBTs relative to a given welfare threshold such that individuals with surplus give to individuals with need in order to preserve the recipient’s continued viable participation in the economy. Our NBT kinetic model considers redistribution rules parameterized to vary between regressive and progressive redistribution.

_____________________

Claude BardosUniversity of Paris VII, France

Onsager Conjecture, the Kolmogorv 1=3 law and the 1984 Kato Criteria in bounded domains with boundaries: Joint work in progress with Edriss Titi and E. Wiedemann

Abstract: Several of my recent contributions, with Edriss Titi, Emile Wiedemann and others were motivated by the following issues:
The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence.

As a consequence.

I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.

Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.

Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is equivalent to the persistence of regularity in the zero viscosity limit.

_____________________

Alexandre BobylevKeldysh Institute of Applied Mathematics, RAN. Moscow

On hydrodynamics beyond Navier-Stokes

Abstract: We consider in this talk  the  problem of derivation and regularization of higher (in Knudsen number) equations of hydrodynamics. The author’s approach based on successive changes of hydrodynamic variables is presented in  more detail for the Burnett level. The complete theory is briefly discussed for the linearized Boltzmann equation.

_____________________

Mihai Bostan – Aix Marseille U, France

Gyro-kinetic theory for the Vlasov-Poisson equations

Abstract : The subject matter of this talk concerns the asymptotic regimes of the Vlasov-Poisson system for strongly magnetized plasmas. We appeal to gyro-average methods and perform the multi-scale analysis with respect to the fast cyclotronic motion. We also investigate collisional models, and derive fluid limits when the collision frequency becomes very large.

_____________________

Luis Caffarelli – The University of Texas at Austin, USA

A problem of segregation involving optimal control

Abstract: We consider a segregation model where two competing species optimize the diffusion process along the media in the region they control, through a “fully non linear” process.

Existence and regularity of the configuration was develop by Veronica Quitalo.

In collaboration with Stefania Patrizi, Veronica and Monica Torres we describe the geometry and regularity of the interphase between both species.

_____________________

Jose Cañizo – University of Granada, Spain

N-particle ground states of the interaction energy approximate continuous ground states

We study lowest-energy distributions (ground states) of N classical particles interacting through a pair potential and show that in the limit of N large these ground states converge to lowest-energy distributions of the continuum interaction energy functional. The shape of these N-particle minimisers is of interest both in statistical mechanics and in more recent collective behaviour models. In particular we show the following: if the potential is H-stable (in the sense of statistical mechanics), these N-particle ground states spread without limit as N grows; while if the potential is not H-stable, the N-particle ground states stay within a ball of a fixed radius, independently of N. The singularity of the potential is
one of the main difficulties in showing the convergence of discrete minimisers to continuum minimisers.

This is a joint work with Francesco Patacchini.

_____________________

Jose A. Carrillo – Imperial College, London, U.K.

Swarming models with local alignment effects: phase transition &  hydrodynamics

Abstract: I will make a review of swarming models with repulsive-attractive effects focusing on two new aspects: phase transitions for the local Cucker-Smale type model and self-organized hydrodynamics of the Vicsek model with fixed speed from asymptotic speed Cucker-Smale models by penalization. In short, we will show that the asymptotic speed Cucker-Smale model behaves in terms of hydrodynamics and phase transitions as the Vicsek model with fixed speed for large friction parameter.

_____________________

Thomas Chen – The University of Texas at Austin, USA

On the dynamics of quantum fluctuations around Bose-Einstein condensates

Abstract: In this talk, we discuss an extension to the Hartree equation, which describes thermal fluctuations around the Bose-Einstein condensate. Using quasifree reduction, we derive the Hartree-Fock-Bogoliubov (HFB) equations, and discuss the well-posedness of the corresponding Cauchy problem. In particular, the emergence of Bose-Einstein condensates at positive temperature via a self-consistent Gibbs state is addressed.

This is based on joint work with V. Bach, S. Breteaux, J. Froehlich, and I.M. Sigal.

_____________________

Yingda Cheng – Michigan State U, USA

A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations

In this talk, we present sparse grid discontinuous Galerkin schemes for solving high-dimensional PDEs. We will discuss the construction of the scheme based on hierarchical tensor product finite element spaces, its properties and applications in kinetic transport equations.

_____________________

Pierre Degond – Imperial College, London, U.K.

On the interplay between kinetic theory and game theory

Abstract: We propose a mean field kinetic model for systems of rational agents interacting in a game theoretical framework. This model is inspired from non-cooperative anonymous games with a continuum of players and Mean-Field Games. The large time behavior of the system is given by a macroscopic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. Applications of the presented theory to social and economical models will be given.

_____________________

Bjorn Engquist – The University of Texas at Austin, USA

Coupling particle, kinetic and fluid models by multiscale simulation

We will discuss the heterogeneous multiscale method (HMM), which is a framework for coupling of macro and micro-scale models in order to have efficient and accurate computations of multiscale problems. Local micro-scale simulations give missing data to a macro-scale model. We
will present one example where local kinetic Monte Carlo computations give data to a continuum model for macro-scale epitaxial growth. Another example is parareal simulations of molecular dynamics where the coarse solver is a phase plane map of kinetic type.

_____________________

François Golse – Ecole Polytechnique, France

Linear Boltzmann Equation and Fractional Diffusion F. Golse (work in collaboration with C. Bardos and I. Moyano)

Abstract: Consider the linear Boltzmann equation of radiative transfer in a halfspace, with constant scattering coefficient σ. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient α. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where σ → +∞ and 1 − α ∼ C/σ, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of a kinetic model which is based on the harmonic extension definition of √ −∆. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions (“heavy tails”) or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273–280].

_____________________

Thierry Goudon – INRIA Research Centre Sophia Antipolis, France

From Vlasov-Poisson equations to the Incompressible Euler system, the case with finite mass

Abstract: Magneto-Optical Traps (MOT) are experimental devices used to trap cold atoms. The mathematical modeling of such devices involves the Vlasov-Poisson(-Fokker-Planck) system with an external potential. We can identify physically relevant regimes which allow us to replace this equation by a model of macroscopic nature. The analysis relies on the derivation of the Incompressible Euler system from the Vlasov equation in the quasi-neutral regime by Y. Brenier and N. Masmoudi. However, by contrast to these cases studied on the torus or with infinite charge, here, the strong external field governs the shape of the domain on which the limit equation is posed. The discussion of these phenomena has unexpected connections with the analysis of the obstacle problem.

This is a joint work with J. Barr\’e (Univ. Orl\’eans), D. Chiron (Univ.
C\^ote d’Azur) and N. Masmoudi (CIMS-NYU).

___________________

Maria Gualdani – George Washington U, USA

On Ap-weights and the Landau equation

Abstract: In this talk we present results on regularization of solutions for the spatially homogeneous Landau equation.

For moderately soft potentials, we show that weak solutions become smooth instantaneously
and stay so over all times, the estimates depending merely on the initial mass, energy, and entropy. For very soft potentials, we present a conditional regularity result,
hinging on what can be described as a nonlinear Morrey space bound assumed to hold uniformly over time.

This bound always holds in the case of moderately soft potentials, and nearly holds for general potentials, including Coulomb. This is a joint work with N. Guillen.

_____________________

Yan Guo – Brown University, USA

L^6 Estimate for the Boltzmann Equation

Abstract: We will discuss a recent L^6 estimates for the Boltzmann equation and its applications.

___________________

Jeff Haack – Los Alamos National Laboratory, USA

Interfacial mixing in high energy-density matter with a multiphysics kinetic model

Abstract: We have extended a recently developed multispecies, multitemperature BGK model [Haack {\it et al.}, J. Stat. Phys., 168, (2017)] to include multiphysics capabilities that enable modeling of a wider range of physical conditions. In terms of geometry, we have extended from the spatially homogeneous setting to one spatial dimension. In terms of the physics, we have included an atomic ionization model, accurate collision physics across coupling regimes, self-consistent electric fields, and degeneracy in the electronic screening. We apply the model to a warm dense matter scenario in which the ablator-fuel interface of an inertial confinement fusion target is heated, similar to the recent molecular dynamics study in [Stanton {\it et al.}, submitted to PRX], but for larger length and time scales and for much higher temperatures. Relative to this study, the kinetic model greatly extends the temperature regime and the spatio-temporal scales over which we are able to model. In our numerical results we observe hydrogen from the ablator material jetting into the fuel during the early stages of the implosion and compare the relative size of various diffusion components (Fickean diffusion, electrodiffusion, and barodiffusion) that drive this process. We also examine kinetic effects, such as anisotropic distributions and velocity separation, in order to determine when this problem can be described with a hydrodynamic model.

___________________

Jingwei Hu – Purdue University, USA

Asymptotic-preserving and positivity-preserving implicit-explicit schemes for the stiff BGK equation

Abstract: We develop a family of second-order implicit-explicit (IMEX) schemes for the stiff BGK kinetic equation. The method is asymptotic-preserving (can capture the Euler limit without numerically resolving the small Knudsen number) as well as positivity-preserving — a feature that is not possessed by any of the existing second or high order IMEX schemes. The method is based on the usual IMEX Runge-Kutta framework plus a key correction step utilizing the special structure of the BGK operator. Formal analysis is presented to demonstrate the property of the method and is supported by various numerical results. Moreover, we show that the method satisfies an entropy-decay property when coupled with suitable spatial discretizations. Additionally, we discuss the generalization of the method to some hyperbolic relaxation system and provide a strategy to extend the method to third order. This is joint work with Ruiwen Shu and Xiangxiong Zhang.

___________________

Semiclassical computational methods for quantum dynamics with band-crossing and uncertainty

Band-crossing is a quantum dynamical behavior that contributes to important physics and chemistry phenomena such as quantum tunneling, Berry connection, chemical reaction etc. In this talk, we will discuss some recent works in developing semiclassical methods for band-crossing in surface hopping. For such systems we will also introduce an “asymptotic-preserving” method that is accurate uniformly for all wave numbers, including the problem with random uncertain band gaps.

_____________________

Armando Majorana – Università di Catania, Italy

A semi-discrete Boltzmann equation based on a finite volume scheme

Abstract
This work deals with a semi-discrete model of the classical nonlinear Boltzmann equation. The new unknowns are integrals, with respect to the velocity variables, of the distribution function. The partial derivatives with respect to the spatial and time coordinates are not approximated; so the model consists in a set of partial differential equations in time and space. The cells, used in the finite volume scheme, are truncated octahedra. This reduces both the computing complexity and the number of the constant coefficients arising from the collision operator. These parameters are obtained by means of numerical quadrature or by using the NanbuBabovsky algorithm for variable hard sphere molecules in the framework of the Direct Simulation Monte Carlo methods.

_____________________

Peter Markowich – King Abdullah University, Saudi Arabia

First principle modeling of biological transportation networks

Abstract: We present a PDE modeling framework for biological transportation networks.  Typical examples are leaf venation, neuronal networks and blood vessel systems in mammals.

_____________________

Jose Morales Escalante – TU Vienna, Austria

Boundary Layers in Boltzmann-Poisson: Homogenization, Reflection Boundary Conditions, and Discontinuous Galerkin Schemes

Abstract: The Boltzmann-Poisson system is homogenized with a background medium that has a periodic permittivity and a periodic charge concentration.  The domain is the half-space with a periodic charge concentration on the boundary, so the domain consists of cells periodically repeated in two dimensions and unbounded in the third dimension.

We perform as well numerical simulations of this physical setting by means of Discontinuous Galerkin schemes for Boltzmann – Poisson,  where we consider also the possible influence of reflective boundary conditions such as specular, diffusive and mixed reflection on the boundary of the device.

This talk is related to joint work with Clemens Heitzinger, Pierre Degond, and Fanny Delebecque.

_____________________

Natasa Pavlovic – The University of Texas at Austin, USA

On well-posedness for Boltzmann equation via dispersive tools

Abstract: Boltzmann equation is an evolutionary partial differential equation which describes the behavior of a dilute gas of identical particles in a specific scaling limit. In this talk we will focus on local theory of well-posedness. Our main intention is not to investigate optimal regularity spaces for solving Boltzmann equation. Rather, we demonstrate the close connection between Boltzmann equation and nonlinear Schr{\” o}dinger equations in the density matrix formulation; this connection has been recognized implicitly for some time, but we wish to make it quite explicit and to the best of our knowledge this is the first time such an explicit connection has been established.

The talk is based on joint works with Thomas Chen and Ryan Denlinger.

_____________________

Kui Ren –The University of Texas at Austin, USA

Inverse problems to system of diffusion equations with internal data

Abstract: We will consider some inverse coefficient problems to system of linear and semilinear diffusion equations where the aim is to recover unknown parameters in the equations from partial information on the solutions to the systems. We present some recent theoretical and numerical results, and point out possible applications of such problems in imaging.

__________________

Christian Ringhofer – Arizona State University, USA

Asymptotically preserving numerical methods for kinetic equations on networks with applications to solar cell design
Abstract: The evolution of charged defects in poly – crystalline solar cells leads to a large system of reaction diffusion equations exhibiting largely different time and spatial scales. The basic structure of the system is essentially equivalent to kinetic equations, where the attribute vector evolves on a network, as opposed  to continuous space or a lattice. The arising system shares many features with systems arising in biological applications. We discuss asymptotically preserving computational methods based on an operator splitting approach.

_____________________

Chi-Wang Shu – Brown U, USA

IMEX time marching for discontinuous Galerkin methods

Abstract: For discontinuous Galerkin methods approximating convection diffusion equations, explicit time marching is expensive since the time step is restricted by the square of the spatial mesh size. Implicit methods, however, would require the solution of non-symmetric, non-positive definite and nonlinear systems, which could be difficult. The high order accurate implicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats the diffusion term implicitly (often linear, resulting in a linear positive-definite solver) and the convection term (often nonlinear) explicitly, can greatly improve computational efficiency. We prove that certain IMEX time discretizations, up to third order accuracy, coupled with local discontinuous Galerkin method for the diffusion term treated implicitly, and regular discontinuous Galerkin method for the convection term treated explicitly, are unconditionally stable (the time step is upper bounded only by a constant depending on the diffusion coefficient but not on the spatial mesh size) and optimally convergent. The results have been generalized to multi-dimensional unstructured meshes, to other types of DG methods such as the embedded DG methods, to fourth order PDEs, and to incompressible fluid flow. The method has been applied to the drift-diffusion model in semiconductor device simulations, where a convection diffusion equation is coupled with an electrical potential equation. Numerical experiments confirm the good performance of such schemes. This is a joint work with Haijin Wang, Qiang Zhang, Yunxian Liu, Shiping Wang and Guosheng Fu.

_____________________

Panagiotis Souganidis – University of Chicago, USA

A regularizing result for stochastic Hamilton Jacobi equations and scalar conservation laws

Abstract: I will discuss a regularizing by noise type result for Hamilton-Jacobi equations and scalar conservation laws with convex nonlinearity.

_____________________

Eitan Tadmor – University of Maryland, USA

On the role of spectral gap in two-dimensional hydrodynamic flocking

Abstract: We discuss the question of global regularity for a general class of equations which arise as the hydrodynamic description for agent-based models driven by alignment.

Smooth solutions of such systems must flock, and 2D global regularity follows for sub-critical initial data such that the initial spectral gap of the velocity gradient matrix is “not too large”.

A similar role of the spectral gap appears in the study of two-dimensional pressure-less equations. Here, we develop a new L^1 framework to prove the existence of weak dual solutions for the 2D pressure-less Euler equations based on new BV estimates associated with the spectral gap.

__________________

Minh Binh Tran – University of Wisconsin-Madison, USA

Local existence and uniqueness theory for a kinetic equation of quantum Boltzmann type in wave turbulence theory

Abstract: In the talk, I will present our local existence and uniqueness results for the 4-wave kinetic equation in wave turbulence theory. The dispersion relation is considered to be of the general abstract form. This is my joint work with Pierre Germain and Alexandru D. Ionescu.

__________________

Alexis Vasseur – The University of Texas at Austin, USA

Holder regularity for hypoelliptic kinetic equations with rough diffusion coefficients

Abstract: In this talk, we will present an application of the De Giorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport operator has a mixing effect in the position/velocity phase space. The mixing effect is incorporated in the classical De Giorgi method via the averaging lemmas. The result can be seen as a Holder regularity version of the classical averaging lemmas.

It is a joint work with  F. Golse,C. Impert, and C. Mouhot.

_____________________

Lexing Ying – Stanford University, USA

Entropy monotonic spectral methods for Boltzmann equation

Abstract: In this talk, we propose a spectral method for discretizing the Boltzmann collision operator that satisfies a discrete version of the H-theorem. The method is obtained by modifying the existing Fourier spectral method in order to match a classical form of the discrete velocity method. It preserves the positivity of the solution on the Fourier collocation points and as a result satisfies the H-theorem. The fast algorithms appeared previously in the literature can be readily applied to this method to speed up the computation.

__________________

Thaleia Zariphopoulou – The University of Texas at Austin, USA

Mean-field and n-agent games for optimal investment under relative performance criteria

Abstract: I will discuss the optimal behavior of a population of fund managers who trade, in a common horizon [0,T] and log-normal markets, aiming at maximizing their expected utility but are also concerned about their relative performance. I will present the n-agent and mean field game for both CARA and CRRA risk preferences, construct the equilibria explicitly and provide conditions for their existence and uniqueness.

(Joint work with D. Lacker)

_____________________