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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems


Authors: Donatella Donatelli and Pierangelo Marcati
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2093-2121
MSC (2000): Primary 35L40, 35K40; Secondary 58J45, 58J37
Published electronically: January 6, 2004
MathSciNet review: 2031055
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Abstract: In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form:

\begin{displaymath}W_{t}(x,t) + \frac{1}{\varepsilon}A(x,D)W(x,t)= \frac{1}{\varepsilon ^2} B(x,W(x,t))+\frac{1}{\varepsilon} D(W(x,t))+E(W(x,t)).\end{displaymath}

We analyze the singular convergence, as $\varepsilon \downarrow 0$, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps:
(i)
We single out algebraic ``structure conditions'' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories.
(ii)
We deduce ``energy estimates '', uniformly in $\varepsilon$, by assuming the existence of a symmetrizer having the so-called block structure and by assuming ``dissipativity conditions'' on $B$.
(iii)
We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gérard.
Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.


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Additional Information

Donatella Donatelli
Affiliation: Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, 67100 L’Aquila, Italy
Email: donatell@univaq.it

Pierangelo Marcati
Affiliation: Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, 67100 L’Aquila, Italy
Email: marcati@univaq.it

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03526-3
PII: S 0002-9947(04)03526-3
Keywords: Hyperbolic systems, parabolic systems, pseudodifferential operators, relaxation theory
Received by editor(s): July 15, 2002
Received by editor(s) in revised form: March 26, 2003, and June 18, 2003
Published electronically: January 6, 2004
Additional Notes: This research was partially supported by EU financed network no. HPRN-CT-2002-00282 and by COFIN MIUR 2002 “Equazioni paraboliche e iperboliche nonlineari”
Article copyright: © Copyright 2004 American Mathematical Society