11 research outputs found
The Erpenbeck high frequency instability theorem for ZND detonations
The rigorous study of spectral stability for strong detonations was begun by
J.J. Erpenbeck in [Er1]. Working with the Zeldovitch-von Neumann-D\"oring (ZND)
model, which assumes a finite reaction rate but ignores effects like viscosity
corresponding to second order derivatives, he used a normal mode analysis to
define a stability function V(\tau,\eps) whose zeros in
correspond to multidimensional perturbations of a steady detonation profile
that grow exponentially in time. Later in a remarkable paper [Er3] he provided
strong evidence, by a combination of formal and rigorous arguments, that for
certain classes of steady ZND profiles, unstable zeros of exist for
perturbations of sufficiently large transverse wavenumber \eps, even when the
von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in
the sense defined (nearly twenty years later) by Majda. In spite of a great
deal of later numerical work devoted to computing the zeros of V(\tau,\eps),
the paper \cite{Er3} remains the only work we know of that presents a detailed
and convincing theoretical argument for detecting them.
The analysis in [Er3] points the way toward, but does not constitute, a
mathematical proof that such unstable zeros exist. In this paper we identify
the mathematical issues left unresolved in [Er3] and provide proofs, together
with certain simplifications and extensions, of the main conclusions about
stability and instability of detonations contained in that paper.
The main mathematical problem, and our principal focus here, is to determine
the precise asymptotic behavior as \eps\to \infty of solutions to a linear
system of ODEs in , depending on \eps and a complex frequency as
parameters, with turning points on the half-line
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
On the front-tracking algorithm
AbstractIn this paper we present an improved version of the front-tracking algorithm for systems of conservation laws. The formulation and the theoretical analysis are here somewhat simpler than in previous algorithms. At the same time, our version leads to a more efficient numerical scheme
Blowup asymptotics for scalar conservation laws with a source
Consiglio Nazionale delle Ricerche - Biblioteca Centrale -. P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal
Well-posedness for a class of 2x2 conservation laws with L'infinity' data
Biblioteca Centrale CNR / CNR - Consiglio Nazionale delle RichercheSIGLEITItal