Yan Guo, Ph.D. - US grants

959623253 GUO A plasma is a collection of fast-moving, charged particles. Although most matters in the universe are of forms of plasmas, the main goal of the plasma study is to control fusion. When collisions among particles are ignored, there are many interesting steady states in a collisionless plasma. The proposer studies their dynamical stability which has potential applications in the plasma control problem. %%% A plasma is a collection of fast-moving, charged particles. Although most matters in the universe are of forms of plasmas, the main goal of the plasma study is to control fusion. When collisions among particles are ignored, there are many interesting steady states in a collisionless plasma. The proposer studies their dynamical stability which has potential applications in the plasma control problem. ***

The research that is funded with this award will address problems in
plasma physics, stellar dynamics, and superconductivity. The PI will
construct steady state solutions of the partial differential equations that
model these phenomena and will study their dynamic stability. Specifically,
the Vlasov model for collisionless plasmas, models for polytropic galaxies,
and models for superconductivity and superfluidity that exhibit vortices will
be investigated.

This award will fund research on mathematical models for phenomena in
which huge numbers of individual particles interact. These phenomena
can occur in plasmas, the physical state of the interior of fluorescent lights,
manufacturing reactors for microchips, and of the interior of the sun. In this
case, atomic particles interact by means of electromagnetic forces. Related
phenomena also occur on cosmic scales, for instance in galaxies, where
the interacting particles are entire stars and the interaction is by means of
gravitation. The mathematical models for these very different situations have
similarities. The research will study equilibrium configurations such as the
steady state operation of a plasma-assisted manufacturing device or the steady
rotation of a galaxy and seek to understand its stability or the ways in which
stability is lost. Instability may mean loss of efficiency for a manufacturing
process or the formation of spiral arms in a galaxy.

9815423
Strauss

This three-year award for US-France cooperative research in applied mathematics involves Walter A. Strauss, C. Shu and Yan Guo of Brown University and Pierre Degond of the Universite Paul Sabatier and Frederic Poupaud of the Universite de Nice. The project is aimed at the design, development, analysis and numerical simulation of hybrid quantum kinetic models and their application to the simulation of quantum semiconductor devices. In many applications, the quantum effects are important only in subregions of the device, while the phenomena in a larger domain are governed by classical transport. The objective of this project is to design models that are able to switch from a classical description to a quantum description. The US investigators bring to this collaboration expertise in classical modeling and simulation. This is complemented by French expertise in quantum models.

NSF Award Abstract - DMS-0087338
Mathematical Sciences: Conference in Continuum Mechanics and Conservation Laws

Abstract

0087338 Dafermos

This award supports U.S. participants (invited speakers, postdoctoral researchers, and graduate students) in the Conference in Continuum Mechanics and Conservation Laws on April 27-29, 2001. A major goal of the conference is to bring together leading international researchers in the fields of conservation laws and continuum mechanics for discussion of recent advances at the boundaries of the fields.

Continuum mechanics is a scientific area with well-developed mathematical theory and important applications to engineering. The theory of hyperbolic conservation laws has been developed in conjunction with applications to gas dynamics, magnetohydrodynamics, and nonlinear elasticity. In recent years there has been rapid progress in areas with connections to both subjects, such as the kinetic theory of gases, plasma physics, the mathematical theory of semiconductors, and general relativity. This conference is a timely opportunity for workers in these fields to discuss important recent results.

The study of nonlinear stability and instability of important equilibria in physical and biological systems ultimately relies on rigorous analytical proofs. The Boltzmann equation is the foundation in the kinetic theory for dilute gases. It is well known that many important fluid equations can be formally derived from the Boltzmann equation. We propose to use a nonlinear energy method to prove the validity of diffusive expansion in linear neutron transport theory, of the Navier-Stokes approximation of the Boltzmann theory in the presence of physical boundary conditions, and of the stability of `front' solution for phase segregation in a binary fluid model. We also propose to study pattern formation in various physical and biological applications such as in reaction-diffusion systems and the Benard problem for a heated fluid. It is expected that the pattern of nonlinear instabilities in these system can be characterized by the finitely many fastest growing modes for the corresponding linear system, over the time of instability formation. Finally, we propose to further study nonlinear stability of galaxy configurations.

Kinetic theory is used to describe the dynamics of a large number of dilute `particles'. These `particles' can be as small as gas molecules or charged ions or electrons in a plasma, or enormous objects such as stars in galaxies. Such kind of dilute charged gases (plasma) dominates our outer space, and plays the crucial role in our fusion research. We propose to study the long-time dynamics of these dilute gases form a mathematical standpoint. Furthermore, we propose to study stability of the galaxy models and predict their long-time dynamics. Pattern formation plays an important role in many physical and biological systems. By applying a recent instability method, we propose to develop a mathematical theory to explain these interesting phenomena.

The generation and segregation of magma from the Earth's interior involves interacting physical processes on a wide range of length scales. To better understand the processes of melt migration in a heterogeneous and multiscale mantle and to promote interdisciplinary research and education, a CMG research project that builds around the existing strengths and resources of the Applied Math and Geological Sciences departments at Brown University is proposed. The project will focus on four closely related themes: (1) theoretical and numerical studies of reactive dissolution in multicomponent, viscously deformable, porous media with sharp interfaces; (2) development of numerical methods for solving multi-dimensional and multiscale geologic problems; (3) development of multiscale models for studying partial melting, melt transport and melt-rock reaction at converging and diverging plate boundaries; and (4) comparison with geochemical and geophysical observations. Stability analysis and numerical calculations using adaptive discontinuous Galerkin finite element methods will be used to study the formation of high porosity melt channels in the mantle. The heterogeneous multiscale method (HMM) coupled with discontinuous and finite volume methods will be developed for studying magma transport processes on large length scales. A double porosity model will also be developed for a viscously deformable duo-porosity medium with permeability tensors, and mass, momentum, and energy transfer rates between the high porosity channels and the low porosity matrices being determined by theoretical and numerical calculations. This new generation of models will establish a framework for understanding geochemical and geophysical observations over length scales ranging from millimeters to hundreds of kilometers.

The proposed theoretical and numerical developments, challenging in their own way, are useful to a range of practical applications far beyond the Earth Sciences. Results from this study will provide valuable information for a diverse group of mathematicians, geochemists, and geophysicists, promoting cross-disciplinary integration in applied math and Earth Sciences. The interdisciplinary nature of this proposal will have a great impact on graduate and undergraduate education at Brown University. Graduate and undergraduate students from both Departments will have the opportunity of involvement in a broad multidisciplinary research and educational activities. Projects like the one proposed will create an environment in which to train a new generation of scientists with the broad multidisciplinary skills needed to derive new understanding from the more discipline specific results of the past.

This award provides support for a meeting, held at Brown University in the spring of 2008, on current topics in analysis of nonlinear wave equations, with presentations by leading researchers in the areas of harmonic analysis techniques, dispersive waves, stability analysis, kinetic theory, free boundaries and relativistic equations. The conference encourages and financially supports participation by students and members of groups underrepresented in the mathematical sciences.

Nonlinear waves occur in many fundamental many physical problems and have a central place in modern theory of nonlinear partial differential equations. Historically the foci of research for nonlinear wave equations have been in classical field theories, such as in the study of nonlinear Klein-Gordon equations, the Maxwell- Dirac system, and the Yang-Mills system. More recently there have been explosive and exciting developments in many new directions for nonlinear wave equations, including for dispersive equations such as KdV and nonlinear Schrödinger equations, for kinetic equations including Boltzmann and Vlasov models, in fluid dynamics including the incompressible Euler equations, for free boundary problems, for the Einstein field equations in general relativity, and for geometric wave equations such as wave maps and Schrödinger maps. This conference brings together leading experts in a great variety of different areas of research in nonlinear wave equations, to stimulate interactions between their different points of view that should be of mutual benefit to all the participants. In particular, the conference will be especially beneficial to young researchers, including graduate students and postdocs.

Kinetic theory is at the center of multi-scale modeling, which connects the microscopic particle models to macroscopic fluid models. There are many challenging open problems in kinetic theory which are of great importance from both mathematical and physical standpoints. The main goal of this research is to continue developing new mathematical methods to resolve open problems in partial differential equations arising in the kinetic theory and other fields in mathematical physics. The investigations will include: boundary effects in the Boltzmann theory for dilute gases, derivation of various macroscopic fluid models from the kinetic theory, and nonlinear stability and instability of steady states in a wide range of applied problems.

These research projects will have important impacts in many areas of physical sciences. The study of stable equilibria in the Vlasov theory (collisionless Boltzmann theory) will shed new light on plasma control in nuclear fusion and on galaxy evolution; the study of the Stefan problem will build a mathematical foundation for morphological stability of crystal growth and many other problems arising in materials sciences; and the study of phase-transitions in the Vlasov-Boltzmann model will lead to better understanding of phase segregation in binary fluids. An important objective of the project is to provide training for students and junior scientists involved in carrying out this research.

This award supports travel for participants in the "Conference on Hyperbolic Conservation Laws and Continuum Mechanics," held 12-14 May 2011 at Brown University. The workshop brings together leading theorists in conservation laws and in continuum mechanics, and aims to stimulate work at the interface between these fields and to inform young researchers about recent developments in the area.

The workshop will enhance communication among junior and senior researchers in conservation laws and continuum mechanics. Conference proceedings will be published in the Quarterly of Applied Mathematics of the American Mathematical Society. The conference encourages and supports participation by graduate students, junior researchers, and members of under-represented groups.

Conference web site: http://www.dam.brown.edu/HyperbolicConf/

The project considers boundary effects for several physical models widely used to describe a hot plasma. The questions studied are motivated by, and have implications for, a broad range of applications. As in the case of a nuclear fusion device, where it is important to control and understand the plasma-wall interaction, or in Einstein's theory for general relativity, where a fundamental open question, known as cosmic censorship conjecture, is whether a gravitational collapse of a star cannot (i.e., black holes) or can (i.e., naked singularity) be observed generically. This work will construct examples of exact gravitational collapse in Einstein's theory that can be observed, and will investigate the dynamics of contact lines as well as the effect of a Coriolis force in oceans. This project provides training opportunities for graduate students.

Kinetic theory provides important models for describing a confined plasma in a device. Because of the severe mathematical difficulties caused by the presence of a grazing set at the boundary, questions of well-posedness for kinetic plasma models in the presence of magnetic effect remain open. Motivated by applications such as contact-line dynamics and the effects of a constant rotation or a constant magnetic field on a fluid, the investigator will pursue several lines of research. The problems investigated are the asymptotical stability of BGK waves; the rigorous analysis of numerical evidence that indicates the existence of a relativistic Larson-Penston self-similar gravitational collapse, leading to formation of a stable naked singularity and the violation of the cosmic censorship hypothesis; and the well-posedness of a hydrodynamic model describing contact line dynamics. The project will also develop a new mechanism to construct long time (global) smooth inviscid fluid flows based on the dispersive effect induced by a constant rotation or a magnetic field.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.