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 founding 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, he has 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 he has written 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 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 has served as referee for 20 research journals, and he
is a member of the editorial boards of ten journals; he has supervised
20 Ph.D. dissertations.
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Multiscale Research: DOE Project