## Year 2020-2021

Time/Location: Thursdays 1:50-2:50pm Virtual via Zoom (unless otherwise noted)

## Abstracts

Chun Liu, Illinois Institute of Technology (Aug 3, 2020)

Title: Generalized law of mass action (LMA) with energetic variational approaches (EnVarA) with applications
Abstract: In this talk, we will present a derivation to generalize the mass action kinetics of chemical reactions using an energetic variational approach. Our general framework involves the energy dissipation law for a chemical reaction system, which carries all the information of the dynamics. The dynamics of the system is determined through the choice of the free energy, the dissipation (the entropy production), as well as the kimenatics (conservation of species). The method enables us to capture the coupling and competition of various mechanisms, including mechanical effects such as diffusion, viscoelasticity in polymerical fluids and muscle contraction, as well as the thermal effects. We will also discuss several applications under this approach, in particular, the modeling of wormlike micellar solutions. This is a joint work with Bob Eisenberg, Pei Liu, Yiwei Wang and Tengfei Zhang.

Jiaxin Jin, Univ. of Wisconsin, Madison (Dec 9, 2020)

Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.
Abstract: In this talk, I will discuss a three-species system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. This is a joint work with Gheorghe Craciun, Casian Pantea and Adrian Tudorascu.

Xinyue (Evelyn) Zhao, Notre Dame University (Dec 28, 2020)

Title: A free boundary tumor growth model with a time delay in cell proliferation.
Abstract: Being a leading cause of death, tumor is one of the most important health problems facing the whole world. While there is a lot of work on the tumor growth models, only a few of them included time delay; and nearly in all the literature, only the radially symmetric case was considered with a time delay. In this talk, I will present a non-radially symmetric tumor growth model with a time delay in cell proliferation. The time delay represents the time taken for cells to undergo cell replication (approximately 24 hours). The model is a coupled system of an elliptic equation, a parabolic equation and a backward or- dinary differential equation. It incorporates the cell location under the presence of time delay, with the tumor boundary as a free boundary. The inclusion of a small time delay makes the system non-local, which produces technical difficul- ties for the PDE estimates. I will discuss the stability and bifurcation results we obtained concerning this model. Through stability analysis, the result indicates that tumor with large aggressiveness parameter would trigger instability, which is biologically reasonable.

Daniel Cooney, University of Pennsylvania (Jan 28, 2020)

Title: Long-Time Behavior of a PDE Replicator Equation for Multilevel Selection in Group-Structured Populations.
Abstract: In many biological systems, there is an evolutionary conflict between the incentive of individuals to cheat and the collective incentive to establish cooperation within groups of individuals. In this talk, we consider a hyperbolic PDE describing the evolutionary dynamics of a two-level birth-death process in which individual-level replication favors cheaters and between-group competition favors groups featuring positive levels of cooperation. We derive a threshold level of the relative strength of between-group competition such that defectors take over the population below the threshold while cooperation weakly persists in the long-time population above the threshold. Under stronger assumptions on the initial distribution of group compositions, we further prove that the population converges to a steady state density supporting cooperation for between-group selection strength above the threshold. When the group replication rate is maximized by an intermediate level of cooperation, we additionally see that the average payoff at steady state is limited by the average payoff a full-cooperator group, and that the steady state density concentrates to a delta-function supporting a suboptimal level of cooperation in the limit of infinite strength of between-group competition. In these cases, individual-level selections casts a long shadow on the dynamics of multilevel selection: no level of between-group competition can erase the effects of the individual incentive to defect. This is joint work with Yoichiro Mori and Joshua Plotkin.

Tao Zhou, University of Science and Technology of China (Feb 25, 2021)

Title: Spreading speeds of nonlocal KPP equations in heterogeneous media.
Abstract: This talk mainly concerns the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media. The kernel K is assumed to depend on the media. First, we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues. Second, we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.

Linhan Li, University of Minnesota (Mar 11, 2021)

Title: Carleson measure estimates for the Green function
Abstract: We are interested in the relations between an elliptic operator on a domain, the geometry of the domain, and the boundary behavior of the Green function. In a joint work with Guy David and Svitlana Mayboroda, we show that if the coefficients of the operator satisfy a quadratic Carleson condition, then the Green function on the half-space is almost affine, in the sense that the normalized difference between the Green function with a sufficiently far away pole and a suitable affine function at every scale satisfies a Carleson measure estimate. We demonstrate with counterexamples that our results are optimal, in the sense that the class of the operators considered is essentially the best possible.

Turanova, Olga, Michigan State University (Mar 18, 2021)

Title: Some tumor growth models and connections between them
Abstract: This talk concerns PDEs modeling tumor growth. We show that a novel free boundary problem arises via the stiff-pressure limit of a certain model. We take a viscosity solutions approach; however, since the system lacks maximum principle, there are interesting challenges to overcome. We also discuss connections between these problems and other PDEs arising in tumor growth modeling. This is joint work with Inwon Kim.

Yao Yao, Georgia Tech (Mar 25, 2021)

Title: Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states
Abstract: The aggregation-diffusion equation is a nonlocal PDE driven by two competing effects: local repulsion modeled by nonlinear diffusion, and long-range attraction modeled by nonlocal interaction. In this talk, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint work with Carrillo, Hittmeir and Volzone). In a recent work, we further investigate whether they are unique within the radial class, and show that for general attractive potentials, uniqueness/non-uniqueness of steady states are determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint work with Delgadino and Yan.)

James Nolen, Duke (April 1, 2021)

Title: A free boundary problem from Brownian bees in the infinite swarm limit in R^d
Abstract: I will explain analysis of a free boundary problem for a parabolic PDE in which the solution is coupled to the moving boundary through an integral constraint. The problem arises as the hydrodynamic limit of a stochastic interacting particle system involving branching Brownian motion with selection, the so-called Brownian bees model. At each branch event in the branching Brownian motion, a particle is removed from the system according to a fitness function, so that the total number of particles, N, is preserved. The free boundary PDE arises from the limit as N tends to infinity. In the large time limit, the PDE solution approaches a certain eigenfunction. We also prove that the so-called strong selection principle holds: the large N and large t limits commute for the particle system. This is joint work with Julien Berestycki, Éric Brunet, Sarah Penington.

Eirik Endeve, ORNL (April 8, 2021)

Title: A DG Method for the Micro-Macro Formulation of the Vlasov-Poisson-Lenard-Bernstein System
Abstract: The Vlasov-Poisson-Lenard-Bernstein (VPLB) system can serve as an approximate model to study transport processes in fusion and space plasmas. These systems are often characterized by multiple spatial and temporal scales, depending on the degree of collisionality. On the one hand, when inter-particle collisions are frequent, the plasma can be described quite well by fluid models. On the other hand, when collisions are infrequent, a fully kinetic description is needed to capture non-equilibrium effects. The micro-Macro (mM) formulation decomposes the particle distribution function into fluid and kinetic components, which are evolved with separate equations that couple through upscaling and downscaling terms. One motivation for adopting the mM model is to gain computational efficiency in collision-dominated regimes. However, maintaining structural properties (e.g., particle, momentum, and energy conservation) is challenging. In this talk, we present a numerical method for the mM formulation of the VPLB system. This method is based on the discontinuous Galerkin (DG) method for phase-space discretization and implicit-explicit time stepping. We focus on the design of consistent discretization of the micro and macro components, which is needed in order to recover conservation properties with the mM method. Numerical results, demonstrating conservation properties and accuracy with coarse phase-space resolution, are also presented.

Wujun Zhang, Rutgers (April 15, 2021)

Title: A rate of convergence of numerical optimal transport problem with quadratic cost
Abstract: In recent years, optimal transport has many applications in evolutionary dynamics, statistics, and machine learning. The goal in optimal transportation is to transport a measure $\mu(x)$ into a measure $\nu(y)$ with minimal total effort with respect to a given cost function $c(x,y)$. On way to approximate the optimal transport solution is to approximate the measure $\mu$ by the convex combination of Dirac measure $\mu_h$ on equally spaced nodal set and solve the discrete optimal transport between $\mu_h$ and $\nu$. If the cost function is quadratic, i.e. $c(x,y) = |x-y|^2$, the optimal transport mapping is related to an important concept from computational geometry, namely Laguerre cells. In this talk, we study the rate of convergence of the discrete optimal mapping by introducing tools in computational geometry, such as Brun-Minkowski inequality. We show that the rate of convergence of the discrete mapping measured in $W^1_1$ norm is of order $O(h^2)$ under suitable assumptions on the regularity of the optimal mapping.

Prerona Dutta, North Carolina State (April 22, 2021)

Title: Metric entropy and nonlinear PDEs
Abstract: Inspired by a question posed by Lax in 2002, the study of metric entropy for nonlinear partial differential equations has received increasing attention in recent years. This talk demonstrates methods to obtain sharp upper and lower bounds on the metric entropy for a class of real-valued bounded total variation functions and then for a class of bounded total generalized variation functions taking values in a general totally bounded metric space. Thereafter we use each of these results to establish metric entropy estimates for the set of viscosity solutions to the Hamilton-Jacobi equation with uniformly directionally convex Hamiltonian and the set of entropy admissible weak solutions to a scalar conservation law with weak genuinely nonlinear flux, respectively. Estimates of this type could provide a measure of the order of resolution of a numerical method required to solve the corresponding equation.