## Ohio State University Partial Differential Equations Seminar## Year 2018-2019Time/Location: Thursdays 11:00 am- 11:55am / CH 240 (unless otherwise noted) |
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Abstract: Fully nonlinear partial differential equations play important roles in geo- metric problems, such as curvature equations in classical geometry and the Yamabe problems on manifolds. A key to understand these equations is to establish a priori estimates for these equations. The Dirichlet problems have received much attention. In this talk we report some recent results joint with Professor Guan Bo for the Neu- mann problem of fully nonlinear elliptic equations on Riemannian manifolds. We try to delete the structural condition for the Neumann boundary in deriving estimates for second derivatives. And we use a priori estimates and blow-up methods to derive the gradient estimates.

Chris Klausmeier (Oct 3, 2018)

Title: Theoretical Approaches to Phytoplankton EcologyAbstract: Phytoplankton, the microscopic primary producers in lakes and oceans, are an ideal system for doing theoretical ecology. In this talk I will talk about three dimensions in which phytoplankton show intriguing patterns: in space, in time, and in the space of functional traits. In space, we focus on the vertical distribution of phytoplankton in the water column. Phytoplankton require light and nutrients to grow, but these essential resources often form contrasting gradients with depth. We use reaction-diffusion-advection models along with game theoretical approaches to figure out how phytoplankton resolve this problem. In time, plankton communities are regularly driven away from equilibrium by the changing of the seasons. We use forced differential equation models and analytical approximations to study the dynamics of the seasonal succession of species. In trait-space, we use trait-based modeling techniques adapted from evolutionary game theory to understand the emergence and maintenance of biodiversity in ecological communities. This is joint work with E. Litchman.

Title: The role of nonlocal information in the dispersal of animals in spatiotemporally varying environments

Abstract: Recent research on reaction-advection-diffusion models and related integro-differential models for animal movement has shown that in spatially varying but temporally constant environments, animals can achieve an evolutionarily stable spatial distribution on the basis of purely local information about the environment. However, there is empirical evidence that in some situations animals use nonlocal information to inform their movements. Numerical computations give evidence that by using nonlocal information on how to advect and diffuse, animals can improve their success at foraging in some spatiotemporally varying environments. Furthermore, in time periodic environments, it is sometimes possible for animals to achieve an evolutionarily stable spatial distribution by means of diffusion and advection, but to do so requires the use of nonlocal information. This talk will give some background and present some recent results on these topics. The modeling and analysis will be done in the mathematical framework of reaction-advection-diffusion equations.

Title: Dynamics in Chemotaxis Models with Logistic Source

Abstract: The current talk is concerned with the asymptotic dynamics in chemotaxis models with logistic source. In particular, I will present some of my recent joint works on the asymptotic dynamics of the following three types of chemotaxis models: chemotaxis models with time and space dependent logistic source on bounded domains; chenotaxis models with logistic source on bounded moving domains with a free boundary; and chemotaxis models with logistic source on the whole space.

Title: Rheumatoid arthritis: a mathematical model

Abstract: A joint is a structure that connects two parts of the skeleton; in particular, the synovial joint is a joint where two bones are connected. This joint consists of cartilage (as cushion) at each bone-end, synovial fluid (as shock absorber when bones are rotated) and synovial membranes between the cartilages and the fluid. Rheumatoid arthritis (RA) is an autoimmune inflammatory degenerative disease of the synovial joints. The inflammations begins in the synovial membrane by immune cells, and it leads to the destruction of the cartilage. There are two million Americans with RA. In this talk, I will present a novel mathematical model of RA. The model is presented as a system of PDEs in the three compartments of the synovial joint. As the cartilage layer degrades it becomes thinner, and its boundary that is in contact with the synovial membrane is moving in time as a âfree boundary.â There is no cure to RA, but drugs are used to try slow the progression of the disease. I shall use the model to evaluate the efficacy of several approved drugs, combination of drugs, and experimental drugs. Finally, I will briefly present open mathematical problem in PDE with free boundary that are associated with the model. This a joint work with Nicola Moise from the medical school in Bucharest, Romania

Title: Spreading speeds and Traveling waves of chemotaxis models

Abstract: Chemotaxis described the oriented movements of mobile species in response to chemical signals by moving along gradient of the chemical substance, either toward higher concentration or away from it. Mathematical models describing chemotaxis have gained more interests over the last few decades. It is well known that such mathematical models exhibit very colorful dynamics. In this talk, we study the spreading speeds of solutions with compactly supported initials, and the existence/non-existence of Traveling wave solutions of the classical Keller-Segel chemotaxis models with logistic sources on R^N. We shall also discuss about some interesting problems.

Title: Dirac concentration in an integro-PDE model from adaptation

Abstract: We consider a mutation-selection model of a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We establish the existence and asymptotic profile of a steady state solution. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. Similar result was independently obtained by B. Perthame and T. Souganidis via an elegant method. I will present a sketch of proof blending the arguments of both papers. This is joint work with Yuan Lou and Wenrui Hao (Penn State).

Emeric Bouin (Univ. Dauphine, Paris) (Feb 5, 2019)

Title: About propagation in integro-dfferential equationsAbstract: In this talk, I will review and discuss various results about finite and accelerated propagation in integro differential equations of KPP type. This will include a Bramson logarithmic delay when the jump kernel is thin, and a rate of acceleration for fat tailed kernels.

Judith Miller (Georgetown) (Mar 28, 2019)

Title: Spatial population dynamics with adaptation to a heterogeneous environmentAbstract: We model the joint evolution of a population density and the mean, and sometimes variance, of a quantitative trait (that is, a continuous random variable such as flowering time in plants) subject to selection toward an optimum value that varies in space. To do so, we study a family of deterministic models originating from the Kirkpatrick-Barton (1997) reaction-diffusion system. We use analysis and numerics to identify conditions under which the models predict range pinning due to an influx of locally maladapted individuals from the center of a species' range to its borders (“genetic swamping”) versus invasions represented as travelling waves. We highlight issues of solution existence, as well as differences between the predictions of the Kirkpatrick-Barton model and those of related models incorporating features, such as non-Gaussian dispersal kernels and patchy habitat, that are often represented in nongenetic invasion models.

Peiyong Wang (Wayne State) (Apr 4, 2019)

Title: One-Phase Bifurcation of a Free Boundary Problem Associated with p-LaplacianAbstract: This talk will cover the existence of a third solution through the Mountain Pass Lemma when the boundary data of the problem decreases below a threshold value. Meanwhile, an evolutionary perspective of the stability-instability of a solution of the stationary problem is gained through a parabolic comparison principle. This is necessary in examining the stability due to the lack of a comparison principle for the elliptic equation. Technically, the proof for the p-Laplacian is much more involved than for the case of Laplacian.

Robert L Jerrard (Toronto) (Apr 11, 2019)

Title: Concentrated vorticity in the Gross-Pitaevskii equationsAbstract: We study the motion of thin, nearly parallel vortex filaments in 3d solutions of the Gross-Pitaevskii equations. Our main result shows that in a certain scaling limit, these filaments are governed by a system of nonlinear Schroedinger equations formally derived by Klein, Majda, and Damodaran in the mid '90s in the context of the Euler equations. This is the first rigorous justification of the Klein-Majda-Damodaran model in any setting. This is joint work with Didier Smets.

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