zbMATH — the first resource for mathematics

Hyperbolic systems with discontinuous coefficients: Generalized solutions and a transmission problem in acoustics. (English) Zbl 0705.35146
The paper is concerned with hyperbolic systems of the form \[ (S)\quad (\partial_ t+\Lambda (x,t)\partial_ x)V+F(x,t)V+G(x,t),\quad (x,t)\in R^ 2, \] under initial condition \(V(x,0)=A(x)\), \(x\in R\). Basic problems like existence, uniqueness and regularity of generalized solutions are dealt with for the system (S), in which F and G are n- vectors and the matrices involved in (S) are n by n. The attention is concentrated on the case of coefficients with discontinuities. Sobolev spaces and Colombeau’s algebra \(G(R^ 2)\) are used.
A sample result is the following: Assume \(\Lambda,F,G\in G(R^ 2)\), with \(\Lambda\) globally bounded; moreover, \(\partial_ x\Lambda\) and F are supposed to be of locally of logarithmic growth. Then for every initial function \(A\in G(R)\), the system (S) has a unique solution \(V\in G(R^ 2)\). The second part of the paper is dedicated to a special case, encountered in the acoustics.
Reviewer: C.Corduneanu

35R05 PDEs with low regular coefficients and/or low regular data
35D05 Existence of generalized solutions of PDE (MSC2000)
35L45 Initial value problems for first-order hyperbolic systems
Full Text: DOI
[1] Colombeau, J.F, New generalized functions and multiplication of distributions, (1984), North-Holland Amsterdam/New York/Oxford · Zbl 0761.46021
[2] Colombeau, J.F, Elementary introduction to new generalized functions, (1985), North-Holland Amsterdam/New York/Oxford · Zbl 0627.46049
[3] Hurd, A.E; Sattinger, D.H, Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. amer. math. soc., 132, 159-174, (1968) · Zbl 0155.16401
[4] {\scF. Lafon}, unpublished manuscript, 1987.
[5] Oberguggenberger, M, Generalized solutions to semilinear hyperbolic systems, Monatsh. math., 103, 133-144, (1987) · Zbl 0615.35054
[6] Oberguggenberger, M, Hyperbolic systems with discontinuous coefficients: examples, (), 257-266
[7] Poirée, B, LES équations de l’acoustique linéaire dans un fluide parfait au repos à caracteristiques indéfiniment différentiables par morceaux, Rev. CETHEDEC, 52, 69-79, (1977) · Zbl 0383.76055
[8] Poirée, B, LES équations de l’acoustique linéaire et non linéaire dans LES fluides en mouvement, () · Zbl 0634.76080
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.