×

zbMATH — the first resource for mathematics

Stability of nonconservative hyperbolic systems and relativistic dissipative fluids. (English) Zbl 1053.35089
Summary: A stability theorem for general quasilinear symmetric hyperbolic systems (not necessarily conservation laws) is proved in this work. The key assumption is the ”stability eigenvalue condition” which requires all the eigenvalues of the constant coefficient system symbol to have negative real part for nonzero Fourier frequency, decaying no faster than \(|\omega|\) when \(|\omega|\to 0\). The decay of the solution to zero, as time grows to infinity, is proved when the space dimension is bigger than or equal to 3. As an application of the general theorem, stability is proved for the equations describing relativistic dissipative fluids.

MSC:
35L60 First-order nonlinear hyperbolic equations
35B35 Stability in context of PDEs
35L45 Initial value problems for first-order hyperbolic systems
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kreiss, SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 30 pp 401– (1999)
[2] Kreiss, J. Diff. Eqns. 142 pp 78– (1998)
[3] Geroch, Ann. Phys. 207 pp 394– (1994) · Zbl 0718.76139 · doi:10.1016/0003-4916(91)90063-E
[4] Kreiss, J. Math. Phys. 38 pp 5272– (1997)
[5] See p. 45, H.-O. Kreiss and J. Lorenz,Initial-Boundary Value Problems and the Navier-Stokes Equations, Vol. 136 of Pure and Applied Mathematics (Academic, New York, 1989).
[6] Shih Liu, Ann. Phys. 169 pp 191– (1986)
[7] Geroch, Phys. Rev. D 41 pp 1885– (1990)
[8] The injectivity condition required by Kreiss et al.4 is the one given here when z is pure imaginary. It is not hard to prove though, using anti-Hermiticity of \^P(i\(\omega\)\^), that the injectivity condition also holds for general complex z.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.