## Ohio State University Partial Differential Equations Seminar## Year 2018-2019Time/Location: Wednesdays 11:00 am- noon / MW 154 (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 “free boundary.” 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

Robert L Jerrard (Apr 10, 2019)

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