Marshall Slemrod - US grants

This research project will examine continuous and discrete models of phase transition. Bifurcation theory will be a basic tool in the continuous case. In the discrete case, the discrete-fragmentation equations will be considered with conditions which allow for a breakdown of density conservation in finite time. Phase transitions are an important phenomena in both fluids and solids. This research will extend understanding of the dynamics of phase transitions.

This award will support a Joint US-France Seminar on Partial Differential Equations and Continuum Models of Phase Transitions, to be jointly organized by Prof. Marshall Slemrod, University of Wisconsin-Madison, and Prof. Michel Rascle, University of Nice, France. The seminar, to be held in Nice in January, 1988, will bring together researchers from France and the US to discuss the formulation and solution of nonlinear partial differential equations in the fields of mechanics, metallurgy, and solid state physics. The main subjects to be addressed include crystal defects, van der Waals fluids, liquid crystals, numerical simulation, and current problems in both solid state physics and industrial technology. Particular emphasis will be placed on problems which evolve in time. It is especially appropriate to hold this seminar in France at this time, because 1987-88 has been denoted the Special Year of Nonlinear Phenomena by the French National Center for Scientific Research (CNRS). This seminar will focus on solutions to differential equations that arise in the study of phase transitions, which have a nonstandard mathematical structure. Building on recent advances in theoretical, applied, and numerical mathematics, this will be the first such seminar to emphasize dynamical problems, and should produce results of use in the solution of problems in solid-state and semiconductor electronics and in fluid mechanics.

This is a continuing program of research and cooperative training by four professors specializing in applied mathematics. Their work concentrates on the application and development of functional analysis, asymptotics and scientific computing; it emphasizes the integral connection between these three approaches to scientific discovery. Professor Slemrod plans to investigations on phase transitions and coagulation. Professor M.C. Shen proposes studies of magnetic fluid dynamics, liquid transport in physiology and nonlinear gravity waves. Professor R.E. Meyer intends to improve the approximation of observables in wave processes. Professor J.M. Vanden-Broeck plans analytical and numerical studies on nonlinear surface and internal waves of ship hydrodynamics and geophysics.

This award will support collaborative research between Dr. Marshall Slemrod, University of Wisconsin and two French applied mathe- maticians: Professor Michel Rascle, University of Nice and Professor Denis Serre, Ecole Normale Superieure de Lyon. The objective of the project is to study the existance of solutions to systems of conservation laws arising in continuum mechanics. In particular, the investigators will study the decay, propagation and creation of oscillations using the tools of compensated compactness and the Young measure. One of the most important branches of partial differential equations is the area of conservation laws. In fact most mechanical systems are described by such equations, e.g. motion of elastic bodies, gases fluids, plasmas, etc. Hence their study both analytically and numerically is exceptionally important. In the proposed project, the investigators will focus on the following: 1) dynamics of the Young measure associated with a sequence of approximate solutions 2) systems of conservation laws describing phase transitions. Dr. Slemrod and his French colleagues are leading researchers in a small group of mathematicians working on the analysis of physically reasonable problems in conservation laws. Solution of the proposed problems would be a significant step toward the understanding of one of the central problems of modern applied mathematics: the dynamics of discontinuous, oscillatory, measure-valued physical systems.

The principal investigator will continue his research on several topics in the dynamics of continua. In particular, he will study continuum and kinetic theories of material motion undergoing phase transitions, initial value problems for the Riemann problem for systems of hyperbolic conservation laws, and distributed-parameter control systems. The tools used will be various techniques from the qualitative theory of partial differential equations and from nonlinear analysis. Many natural phenomena involve the transition of a material from one phase to another. As an example, think of what happens to the ice cubes in your drink. The principal investigator will apply modern methods of nonlinear analysis to study the dynamical behavior of materials undergoing phase transitions and also ways to control the instabilities that often result when materials are stressed.

9406295 Slemrod This proposal outlines proposed research in three areas: (i) kinetic modeling of gases exhibiting change of phase and metastable states, (ii) resolution of Riemann problems of hyperbolic systems of conservation laws via hydrodynamic and viscous limits, (iii) analysis of the structure of shock waves for spherically symmetric three dimensional fluid flow. The main goal is the understanding of the dynamics of these highly nonlinear phenomena and applying this insight to interpretation of the motivating physical problems. The technical approach to the above research will be to (i) examine cluster Boltzmann like dynamics and try to simulate van de Waals intermolecular attractions to develop discrete velocity phase transition models: (ii) apply self similar scaling methods to recover hydrodynamic and viscous limits for systems of conservation laws, (iii) impliment the viscous self similarity method to the analysis of spherically symmetric gas dynamics (conservation laws of mass, momentum, energy).

The scientific tools used by Slemrod in his research on finding
multi-scale models for constitutive relations for gases, fluids, and
granular materials are kinetic theories for these materials
and singular perturbation limits. These singular perturbation limits
allow him to recover models on large space and time scales. However
the linking of the large and small scales in unified models will be
accomplished by a new technique developed by Slemrod and motivated
by the procedure used by physicists in the study of critical phenomena.
Slemrod calls the method" generalized rational approximation".
It provides a stable way of extrapolating from small scale to
large scale models of materials.

In the study of the dynamics of liquids, gases, and granular materials
it is important to have good mathematical models to aid in the prediction
of physical phenomena. It is also important to have these models to be
valid on a whole range of length and time scales. For example the motion
of the atmosphere (a rarefied gas) far away from a plane or a space shuttle
re-entering the earth's atmosphere uses a large scale model different
than that used to model the small scale dynamics of a shock layer near
the plane or shuttle. However for computational efficiency one unified
model valid on a large range of length and time scales is desirable.
Slemrod in his research will formulate a method for finding such models
for fluids, gases, and granular materials.

This three-year award provides support for US-France cooperative research on mathematical problems in continuum modeling and analysis for physical systems. The collaboration involves the US research team led by Gui-Qiang Chen at the Northwestern University and the French team led by Michel Rascle at the Mathematics Laboratory of the University of Nice. The investigators will study compressible Euler equations, entropy flux splittings, kinetic formulations, microstructure, phase transition and metastability. The underlying mathematical theme is that of singular limits; the unifying physical theme is that of continuum mechanics. The project takes advantage of the complementary expertise of the US and French investigators and will advance understanding of mathematical underpinnings of problems in physical systems.

9731084
Slemrod

This project will support the participation of researchers from
the United States at the Fifth International Workshop on
Mathematical Aspects of Fluid and Plasma Dynamics to be
held in Wailea, Hawaii from June 28 - July 3, 1998. The
objective of the workshop is to give researchers in the
mathematics of fluids, plasmas, stellar systems, and rarefied
gas dynamics the opportunity to present recent progress and
learn about new directions and tools. The conference will
focus on the following main areas of research:
magnetohydrodynamics, kinetic theory of gases and granular
flow, compressible and incompressible fluid mechanics, semi-
conductor dynamics, and numerical methods for resolution of
these problems. The workshop will bring together researchers
from the U.S., Japan, Canada, Australia, India, and Europe.

9819788
Brualdi
The Department of Mathematics of the University of Wisconsin-Madison
has a tradition of excellence in graduate education that goes back
more than 100 years, successfully training more than 900 PhDs in
Mathematics who now occupy important positions in universities and
industries throughout the USA and indeed the world. For more than 25
years it has had a successful postdoctoral program (the Van Vleck
program), with more than 80 Mathematics PhDs continuing their
training in research and instruction under the mentorship of
established researchers and educators. Our proposal seeks to
sustain, strengthen, enhance, revise, and integrate the five areas of
mathematics education at UW-Madison: (1) Mathematics Undergraduate
Education, (2) Applied Mathematics, Engineering, and Physics
Undergraduate Program (AMEP), (3) Mathematics PhD Program within the
Department of Mathematics, (4) Applied Mathematical Sciences
Interdisciplinary Program within the Center for Mathematical Sciences
(GPAMS), and (5) Postdoctoral Training in Pure and Applied
Mathematics.

Our primary goals are (1) to broaden the mathematics education of
undergraduate and graduate students, and postdoctoral fellows so that
they will be able to (a) interact and communicate effectively with
scientists and engineers (b) carry out mathematical research at a
very high level, and (c) appreciate the role of computing in the
mathematical sciences and, when appropriate, perform computations;
(2) strengthen our traditional undergraduate, graduate, and
postdoctoral programs in core mathematics, and (3) decrease the time
to PhD degree to an average of 5 years or less. The components of
our proposal integrate research and education in pure and applied
mathematics, at the undergraduate, graduate, and post-graduate level
in order to train mathematical scientists capable of handling the
above noted tasks.
Highlights of our new initiatives include:
1. Laboratory Components of Undergraduate Courses A laboratory component in one or more courses will be developed that will clearly illustrate the successes and limitations of Applied Mathematics Modeling of physical problems. The experiments would be demonstrations carried out to clearly illustrate points and not to train students for experimentation. Potential courses for such a component are: Applied Dynamical Systems and
Chaos, PDEs, Introduction to Applied Mathematics, and undergraduate and graduate Fluid Mechanics.

2. Integrated Undergraduate, Graduate, and Faculty Research Lab in Spatial Systems\noindent Individually and in groups, students will study various spatial systems (mostly on lattices) from combinatorial and probabilistic perspectives, with some use of computer simulation. Research questions that arise in faculty and postdoctoral fellowsresearch will be described to students who investigate these and other questions that surface in their course of investigation. This Lab will meet in late afternoon, and we shall seek out one or two bright high school students as possible participants, e.g. students from local high schools who are among the winners of the Wisconsin Math., Engin., & Science Talent Search.

3. Mathematics PhD Program and GPAMS. With the effective use of VIGRE
funds, we will:
(a) Recruit quality American graduate students with offers of
traineeships (VIGRE fellowships and Teaching Assistantships);
(b) Provide graduate students with a quality graduate program,
including interdisciplinary training that giving experience
in solving practical mathematical problems;
(c) Reduce the time-to-degree by targeting substantial fellowship
money on a small group of students who have the potential of becoming
research mathematicians at top universities and labs.
Cognizant of the fact that many graduates do not work in academic
settings, we plan to widen the scope of the current Mathematics PhD
program in the Department of Mathematics, supplying our graduate
students with the necessary experience in solving practical
mathematical problems. To these ends, we are planning to start an
Applied Mathematics Consulting . This would take the
form of nonstandard graduate courses for mathematics graduate
students whose main goal is to set up interdisciplinary cooperation,
at a research level, between mathematics graduate students and
researchers in other fields at the University. Students would take
this two-semester course in their second year, supervised by a
faculty member in the Department of Mathematics, with the help of
post--doctoral fellows funded through VIGRE. The goals of this
project are to foster new connections between researchers in
mathematics and other disciplines and to encourage interaction
between mathematics graduate students and faculty in other
disciplines.

4. Post-Doctoral Training ProgramOur post-doctoral program - the Van Vleck (V^2) program has attracted some of the best recent PhDs: each V^2-postdoc has a faculty mentor for both research and teaching, and teaches two courses a semester. They collaborate with faculty, work with PhD students, participate in seminars, and teach elementary and advanced
undergraduate courses and, at least once, a graduate course. The
V^2--program is an extension of our graduate program providing
continuing (collaborative) research and teacher training for new
PhDs.

We will extend and strengthen our postdoctoral program, to
include a Vigre Van Vleck component, the V^3--program. We propose
to award several V^3--postdocs. The post--doctoral fellows will
provide essential mentoring to the graduate students with
interdisciplinary postdocs mentoring students in GPAMS. The
interdisciplinary post--doctoral fellows will also participate in the Applied Mathematics Consulting Program previously described.
V^3--postdocs will be supported in part by regular teaching duties
(one course each semester). They will be: (1) mentored by local
faculty, develop an independent research program, and improve
teaching skills; (2) develop communication skills through
presentation of research seminars, including one colloquium talk for
a general audience; (3) prepare papers for publication in
professional journals; (4) prepare a research proposal for
submission to a funding agency at the conclusion of her or his
appointment.

As part of our VIGRE program, we will, in addition, develop several new undergraduate and graduate courses, including a Mathematics College Teaching Course (How to teach Math at the College level).

Funding for this award is provided by the Division of Mathematical Sciences
and the MPS Directorate's Office of Multidisciplinary Activities.

NSF Award Abstract - DMS-0071463
Mathematical Sciences: Research on Plasma Sheaths: An Interdisciplinary Mathematical-Experimental Program

Abstract

0071463 Slemrod




The Euler-Poisson equations describe a plasma consisting of ions and electrons. A universal property of these equations in a bounded domain is the existence of a bulk quasi-neutral plasma domain and a thin space charge sheath near the boundary. This project studies various aspects of sheath formation: rigorous justification of the quasi-neutral limit for two- and also multi-species plasmas, rigorous derivation of limit equations, rigorous derivation of rules for defining the sheath boundary, connections with kinetic theory, and development of relaxation schemes for solving the Euler-Poisson equations on the various scales to be found in sheath formation problems.

Confined plasmas form space charge sheaths, relatively thin zones where the ion density is greater than the electron density. The phenomenon is important in various technological applications (plasma etching, microelectronics, gaseous lasers). This project is an interdisciplinary mathematical-experimental program for studying space charge sheaths. The objectives of the research are to:
(1) develop new tools, including mathematical models, numerical methods, and analytical techniques, for prediction of plasma sheath formation and dynamics;
(2) perform new laboratory experiments to examine the formation, behavior, and
various properties of plasma sheaths;
(3) integrate the results of (1) and (2) to provide input for both the mathematical and experimental aspects of the research.
The research will bring to the attention of the U.S. mathematical community the myriad of open problems surrounding the analysis of plasma sheaths.

NSF Award Abstract-DMS-0203858 Mathematical Sciences: $L^1$ stability of hyperbolic conservation laws with geometrical sources and kinetic equations

Abstract

0203858 Ha

This research addresses the stability of weak solutions of hyperbolic conservation laws and related problems in kinetic theory. Stability will be studied by constructing explicit Lyapunov functionals. Specific goals are: (i) establish the stability of weak solutions to hyperbolic conservation laws with geometric source terms and certain kinetic models with collision terms; (ii) study the nonlinear stability of shock waves of the Boltzmann equation with boundary effects, and hydrodynamic limits of some collisional kinetic equations.

Hyperbolic conservation laws with geometric source terms appear in many physical situations, such as shallow water flow through a channel, nozzle flow through a duct, and self-similar gas flow in multi-dimensional Euler equations. The issue of stability of solutions is important in the design of systems modeled by these equations, which include aircraft and space shuttle engines. The Boltzmann equation and the Smoluchowski equation are fundamental equations in kinetic theory. Stability analysis for these equations can be used in development of accurate methods for numerical simulation of the corresponding physical systems.

ABSTRACT

FRG: Multi-Dimensional Problems for the Euler Equations
of Compressible Fluid Flow and Related Problems
in Hyperbolic Conservation Laws

Historically, fluid and solid mechanics study the motion of
incompressible and compressible materials, with or without internal
dissipation. For gases and solids with internal dissipation as a
secondary effect, the gross wave dynamics is governed by inviscid,
thermal diffusionless, dynamics. Within these categories, compressible
motion for solids corresponds to the study of elastic waves and their
propagation; compressible motion for fluids is usually associated with
inviscid gas dynamics. Furthermore both compressible solids and
fluids exhibit shock waves and hence we must search for discontinous
solutions to the underlying equations of motion.
Incompressible motion on the other hand concerns
itself with the motion of denser fluids where the idealization of
incompressibility is useful, e.g. water or oil, as well as the motion of
certain solids like rubber. While there are still many important
mathematical issues to be resolved for incompressible fluids, for example,
the well-posedness of the Navier-Stokes equations in three space
dimensions, the mathematical study of compressible
solids (as represented by the equations of nonlinear elastodynamics) and
fluids (as represented by the Euler equations of inviscid flows)
in two and three space dimensions is even less developed.
This provides the motivation to the proposers to collaborate in a
three year effort to advance the mathematical understanding of the
multi-dimensional equations of inviscid compressible fluid dynamics
and related problems in elastodynamics.
The core of our plan is to arrange a sustained interaction between and
around the members of the group, who will
(1) collaborate scientifically, focusing on the advancement of the
analysis of multi-dimensional compressible flows by developing new
theoretical techniques and by using and designing effective, robust and
reliable numerical methods;
(2) work together over the next several years to create the environment
and manpower necessary for the research on multi-dimensional compressible
Euler equations and related problems to flourish; and in the meantime,
(3) share the responsibility of training graduate students and
postdoctoral fellows.

The project is devoted to a mathematical study of the Euler equations
governing the motion of an inviscid compressible fluid and related
problems. Compressible fluids occur all around us in nature, e.g. gases
and plasmas, whose study is crucial to understanding aerodyanmics,
atmospheric sciences, thermodynamics, etc.
While the one-dimensional fluid flows are rather well understood, the
general theory for multi-dimensional flows is comparatively mathematically
underdeveloped. The proposers will collaborate in a three
year effort to advance the mathematical understanding of the
multi-dimensional equations of inviscid compressible fluid dynamics.
Success in this project will advance knowledge of this fundamental area of
mathematics and mechanics and will introduce a new generation of
researchers to the outstanding problems in the field.

Slemrod will continue his research into conservation laws of mixed hyperbolic- elliptic type arising in continuum mechanics and especially compressible gas dynamics. The classical example of such flows occurs in transonic gas dynamics for flight near the speed of sound. Slemrod is developing methods for proving existence of solutions to the relevant system of partial differential equations in various flow geometries. A surprising feature of his approach is that it also leads to a new approach for proving existence of solutions to the Gauss-Codazzi system describing the problem of isometric embedding of a two dimensional Riemannian manifold in three dimensional Euclidean space with Gauss curvature having both positive and negative sign.

The implications of this research are quite striking. First of all the methods given provide a new way for engineers to solve problems of transonic flight on computers. Secondly the geometry problem while of interest in its own right in mathematics also occurs in problems arising in the structure of thin shells and and fiber re-enforced materials. Again the research will provide engineers new tools for solving such problems on computers.