1987 — 1990 
Tzavaras, Athanasios 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Mathematical Sciences: Systems of Evolution Equations in Thermomechanical Processes @ University of WisconsinMadison
This research program investigates the competition between dissipative mechanisms such as frictions and the destabilizing action of nonlinearity of the material response. The corresponding nonlinear evolution (partial differential) equations exhibit both hyperbolic and parabolic characters. Their analysis for the qualitative behavior of the physical system poses new and difficult mathematical problems. Specific results from the research program are expected to provide insight to physical problems such as shear band formation for strainrate dependent materials and the motion of viscoelastic materials with fading memory.

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1992 — 1995 
Tzavaras, Athanasios 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Mathematical Sciences: Nonlinear Dynamics in Continuum Mechanics @ University of WisconsinMadison
The objective of this research is to understand the dynamic response of continuous media via the study of singularities for the governing partial differential equations of motion. Three topics are proposed: (i) Formation of shear bands at high strain rates; (ii) Study of selfsimilar viscous limits and of radial solutions for hyperbolic systems of conservation laws; (iii) Hydrodynamic limits for discrete velocity models in the case of Riemann data. Shear bands are regions of intensely localized strain that appear during high speed deformations of metals and often precede rupture. For that reason their study is critical for the design of improved materials in situations of high speed plastic flow. The theoretical understanding of structures such as shock waves or shear bands is critical for designing improved algorithms in technological applications where such phenomena are dominant, for instance in ballistic penetration of metals and in various manufacturing processes involving metals.

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1995 — 1999 
Tzavaras, Athanasios 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Mathematical Sciences: "Nonlinear Dynamics in Continuum Mechanics." @ University of WisconsinMadison
9505342 Tzavaras We propose to study various topics related to the formation, propagation and resolution of singularities during the dynamic deformation of continuous media. The topics are: (i) Effect of viscosity on shock propagation for hyperbolic conservation laws. (ii) Fluidmechanic limits for discrete velocity models for Riemann data. (iii) Formation of shear bands at high strain rates. The proposed techniques are from the domain of qualitative theory of partial differential equations, with the use of computational tools when necessary to guide the theory. The proposed problems vary from concrete models for specific phenomena, to general theory hyperbolic systems of conservation laws, to developing techniques for studying the transition from discrete to continuum theories. The theoretical understanding of structures such as shock waves and shear bands is critical for designing improved algorithms in technological applications where such phenomena are dominant. Shear bands are regions of intensely localized strain that appear during high speed deformations of metals and often precede rupture. Shear bands play a major role in ballistic penetration of metals and in various manufacturing processes involving metals.

0.969 
1998 — 2002 
Chen, GuiQiang [⬀] Greenberg, James Tzavaras, Athanasios Slemrod, Marshall (coPI) [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
U.S.France Cooperative Research: Mathematical Problems in Continuum Mechanics and Related Equations @ Northwestern University
This threeyear award provides support for USFrance cooperative research on mathematical problems in continuum modeling and analysis for physical systems. The collaboration involves the US research team led by GuiQiang 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.

0.955 
1999 — 2002 
Tzavaras, Athanasios 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Viscosity and Relaxation Approximations of Hyperbolic Systems @ University of WisconsinMadison
This research is concerned with several aspects of the theory of weak solutions for hyperbolic systems, with the objective to develop techniques for understanding the emergence of shock waves from smooth approximate solutions in the small viscosity and small relaxationtime limits. Such questions are intimately tied to the mechanical issue of the passage from one thermomechanical theory to another, and appear in models of continuum physics, kinetic theory, and in the interface of the two. The goal is on the one hand to exploit ideas from the kinetic theory of gases in developing theory for hyperbolic systems, on the other hand in applying recent advances from the theory of hyperbolic equations in studying the passage from microscopic theories (of interacting particles) to mesoscopic theories (at the kinetic level) to macroscopic (continuum) theories.
Shock waves are coherent structures that appear in situations involving high speed supersonic flows. Their mathematical modeling involves nonlinear hyperbolic partial differential equations. Understanding the theory of these equations is very important in the design and implementation of numerical algorithms for computing. The emphasis of this proposal is in situations involving shock waves in transitions from rarefied to dense gases. A typical application is air flow in high altitude supersonic flight, for example during reentry of a space shuttle into the atmosphere.

0.969 
2002 — 2006 
Tzavaras, Athanasios 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Hyperbolic and Kinetic Partial Differential Equations @ University of WisconsinMadison
NSF Award Abstract  DMS0205032 Mathematical Sciences: Hyperbolic and kinetic partial differential equations
Abstract
0205032 Tzavaras
This project deals with several aspects of the theory of weak solutions for hyperbolic systems, and the mathematical theory of transport equations that arise in kinetic theory of gases. Specific themes are: (i) To exploit the interface between the theory of weak solutions for hyperbolic systems and the theory of transport equations in the kinetic theory of gases, particularly with regard to issues of propagation and cancellation of oscillations. (ii) To study wellposedness and hydrodynamic limits for certain collisional kinetic models, problems that are intimately connected to the thermomechanical issues arising in the passage from microscopic to continuum theories. (iii) To exploit variational techniques in the study of the structural properties for the equations of multidimensional elastodynamics and viscoelasticity. (iv) To analyze various instances of diffusionsensitive dynamics, like the effect of smallviscosity on the longtime evolution of hyperbolic systems and the notion of graph solutions for diffusion sensitive systems.
The mathematical research on hyperbolic systems of conservation laws is to a large extent motivated by the fundamental conservation laws in physics and continuum mechanics. From its early stages, analytical and numerical methods in this field have developed together, and analytical understanding contributes in the design of high performance computational algorithms. In recent years there has seen a very fruitful exchange between ideas in the theory of kinetic equations and the theory of weak solutions for hyperbolic systems. At the core of this exchange is the issue of deriving continuum theories from microscopic models of kinetic theory of gases or statistical physics. This project will make use of this exchange of ideas to develop mathematical techniques to better understand the wide variety of important physical systems that are modeled by conservation laws.

0.969 
2005 — 2009 
Tzavaras, Athanasios 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Kinetic Techniques For Hyperbolic and Multiscale Problems @ University of WisconsinMadison
In recent years there has seen a very fruitful exchange between ideas in the theory of kinetic equations and the theory of weak solutions for hyperbolic systems. At the core of this exchange lies the issue of deriving continuum theories from microscopic models of kinetic theory of gases or statistical physics. In this context transport properties play a crucial role whether in a framework of kinetic equations, or in a context of nonlinear transport as it appears with differential constraints in the context of polyconvex elastodynamics or nonlinear models for Maxwell?s equations. This proposal has the objectives to perform analytical and modeling work on the topics: (i) transport and oscillations in systems of two conservation laws, (ii) collisional kinetic models and their hydrodynamic limits, (iii) effect of differential constraints on the equations of polyconvex elastodynamics, (iv) mathematical aspects of kinetic theory of dilute polymers, and (v) development of kinetic techniques for homogenization problems.
Hyperbolic systems of conservation laws express the basic laws of continuum physics and as such are central in modeling in the sciences. Kinetic modeling is becoming all the more pronounced, as microscopic modeling and the associated derivation of mesoscopic equations is commonplace in today?s engineering applications. Understanding of homogenization issues and the couplings in problems where multiple scales interact are necessary ingredients for the design of e cient computational algorithms that proceed without resolving all microscopic information of the problem. The mathematical analysis of issues related to the passage from small to coarser scales has significant implications on the design of highperformance computing algorithms, particularly when treating problems where scale interactions occur. Such problems are at the center of modern material science with several applications in chemical and materials engineering.

0.969 
2008 — 2012 
Tzavaras, Athanasios Tadmor, Eitan [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Frg: Collaborative Research: Kinetic Description of Multiscale Phenomena: Modeling, Theory and Computation @ University of Maryland College Park
Kinetic equations play a central role in many areas of mathematical physics, from micro and nanophysics to continuum mechanics. They are an indispensable tool in the mathematical description of applications in physical and social sciences, from semiconductors, polymers and plasma to traffic networking and swarming. The ultimate goal of this proposal is to develop novel analytical and numerical methods based on kinetic descriptions of complex phenomena with multiple scales and with a wide range of applications. The objective is to achieve a better understanding of problems which are in the forefront of current research and to contribute to the solution of longstanding problems by synergetic collaboration of theory, modeling and numerics. To this end, this Focus Research Group (FRG) will provide a platform, led by leading researchers from Universities of Maryland, Brown, Iowa State, WisconsinMadison, Arizona State, AustinTexas and Toulouse, France, who will merge their expertise in the construction, analysis and implementation of kinetic descriptions for a selected suite of problems with crossing scales from quantum and micro scales to the macro scales. Topics to be discussed include kinetic descriptions of microscopic and quantum phenomena, and kinetic descriptions of macroscopic phenomena. As a recent novel example for the kinetic methodology we will use kinetic descriptions to study hyperbolic flows for complex supply chains. The theoretical and modeling aspects of this research program, on both microscopic and macroscopic scales, will be integrated with kineticbased numerical methods for capturing ``smaller scales phenomena".
The rationale behind this proposal is a timely effort to address several important issues in modern applied mathematics. Kinetic theories are not new. Yet, there have been many major developments in kinetic modeling, kinetic theories and related numerical methods, with the potential for a considerable impact on emerging new fields in physical and social sciences. The proposed effort will significantly strengthen the leading role that the US researchers can play in pursuing cuttingedge research and training a new generation of applied mathematicians in this important field. We expect this project to contribute to the development of scientific workforce by advanced training for doctoral and postdoctoral researchers and by providing a platform for interdisciplinary interactions with researchers from related disciplines. Internal and external interactions will be maintained through synergetic collaborations which will come to fruition during the three annual workshops to be held in Maryland (Year 1), France and Brown (Year 2), and Wisconsin (Year 3). International meetings will be held as part of a series of interdisciplinary workshops organized by the Center for Scientific Computation and Mathematical Modeling (CSCAMM) at the University of Maryland. Project researchers will collaborate with the DOE Center for Multiscale Plasma Dynamics in CSCAMM, the DOE Ames Laboratory at Iowa State University, and the Institute for Computational Engineering and Sciences (ICES) at UT Austin.

0.969 
2008 — 2009 
Tzavaras, Athanasios Tadmor, Eitan [⬀] Liu, JianGuo (coPI) [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
International Conference On Hyperbolic Problems: Theory, Numerics & Applications @ University of Maryland College Park
This award supports participation in the International Conference on Hyperbolic Problems: Theory, Numerics & Applications, to be held at the University of Maryland, College Park, in June 2008.
The conference, the twelfth meeting in a biannual series, brings together researchers, students, and practitioners with interest in the theoretical, computational, and applied aspects of hyperbolic timedependent problems. Topics of the conference include the study of nonlinear wave patterns in multiple dimensions, passage from microscopic to macroscopic in models for particle dynamics, theory and simulation of interfaces, and application of transport theory in complex environments including scattering in random media, biological applications, and traffic flow.
The organizers encourage participation by junior researchers and members of groups underrepresented in the mathematical sciences. The meeting provides an excellent opportunity for junior and senior participants to exchange ideas and to learn about recent research in emerging areas of theory, application, and numerical simulation connected with nonlinear hyperbolic evolution equations. Conference proceedings will be published.
Conference web site: http://hyp2008.umd.edu

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