Research Description

Ralph Showalter joined the Mathematics Department at Oregon State University in the Fall of 2003. Previously he held the Blumberg Centennial Professorship in Mathematics at the University of Texas at Austin and was a member of the Texas Institute for Computational and Applied Mathematics, now ICES. Since receiving the Ph.D. in Mathematics at the University of Illinois as an NSF Fellow, he has published over a hundred research articles, one research monograph co-authored with R.W. Carroll, Singular and Degenerate Cauchy Problems, one graduate text, Hilbert Space Methods in Partial Differential Equations, edited volumes with J.T. Oden, Workshop on Existence Theory in Nonlinear Elasticity, and with M. Peszynska, A. Spagnuolo and N. Walkington, Modeling, Analysis and Simulations of Multiscale Nonlinear Systems, and a volume in the Mathematical Surveys and Monographs of the American Mathematical Society, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. He contributed the chapter ``Micro-structure models of porous Media'' in the book Homogenization and Porous Media edited by Ulrich Hornung. His research interests include singular or degenerate nonlinear evolution equations and partial differential equations, related variational inequalities and free-boundary problems, and applications to initial-boundary-value problems of mechanics and diffusion. Among his technical contributions are the development of existence-uniqueness-regularity theory for pseudo-parabolic and Sobolev-type partial differential equations, existence theory of degenerate evolution equations, particularly the doubly-nonlinear cases and nonlinear systems in mixed form. More applied contributions include the formulation and existence theory for Stefan free-boundary problems for a parabolic system, for the pseudo-parabolic equation and for the hyperbolic telegraphers' equation, the phase-change problem of advection of methane in the hydrate zone, the quasi-static Biot system of poroelasticity, and the coupled Biot-Stokes system. He introduced the fissured medium equation and the layered medium equation as models for diffusion in heterogeneous media and contributed to the development of distributed systems with microstructure, and those with hysteresis. His current research interests are focused on the development of multi-scale models of coupled fluid-solid dynamics and flow in deformable porous media. A member of the American Mathematical Society and the Society for Industrial and Applied Mathematics, he has organized or co-organized the Texas Differential Equations Conference series, originally as the Texas PDE Seminar, a sectional SIAM meeting, an AMS special session, an NSF Workshop on partial differential equations and applications, and a DOE-NSF Workshop on "Modeling, Analysis and Simulation of Multiscale Nonlinear Systems". He regularly serves as referee for 20 research journals, and he is a member of the editorial boards of ten journals; he has supervised 19 Ph.D. dissertations.


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Multiscale Research: DOE Project