330 research outputs found
Pointwise Green's function bounds and stability of relaxation shocks
We establish sharp pointwise Green's function bounds and consequent
linearized and nonlinear stability for smooth traveling front solutions, or
relaxation shocks, of general hyperbolic relaxation systems of dissipative
type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability,
i.e., stable point spectrum of the linearized operator about the wave, and
hyperbolic stability of the corresponding ideal shock of the associated
equilibrium system. This yields, in particular, nonlinear stability of weak
relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The
techniques of this paper should have further application in the closely related
case of traveling waves of systems with partial viscosity, for example in
compressible gas dynamics or MHD.Comment: 120 pages. Changes since original submission. Corrected typos, esp.
energy estimates of Section 7, corrected bad forward references, expanded
Remark 1.17, end of introductio
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding
open problems are discussed, and a perspective is presented for the
mathematical theory of multidimensional conservation laws. Some basic features
and phenomena of multidimensional hyperbolic conservation laws are revealed,
and some samples of multidimensional systems/models and related important
problems are presented and analyzed with emphasis on the prototypes that have
been solved or may be expected to be solved rigorously at least for some cases.
In particular, multidimensional steady supersonic problems and transonic
problems, shock reflection-diffraction problems, and related effective
nonlinear approaches are analyzed. A theory of divergence-measure vector fields
and related analytical frameworks for the analysis of entropy solutions are
discussed.Comment: 43 pages, 3 figure
Diffusion of energy in chains of oscillators with bulk noise
These notes are based on a mini-course given during the conference Particle
systems and PDE's - II which held at the Center of Mathematics of the
University of Minho in December 2013. We discuss the problem of normal and
anomalous diffusion of energy in systems of coupled oscillators perturbed by a
stochastic noise conserving energy
Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)
This workshop brought together
leading experts, as well as the most
promising young researchers, working on nonlinear
hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in
modeling, analysis, and numerics. Particular topics included ill-/well-posedness,
randomness and multiscale modeling, flows in a moving domain, free boundary problems,
games and control
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