Existence and uniqueness results for mixed problems for hyperbolic systems of one- dimensional conservation laws. (Resultats d’existence et d’unicite du probleme mixte pour des systemes hyperboliques de lois de conservation monodimensionnels.) (French) Zbl 0735.35092

The authors study nonlinear strictly hyperbolic systems of conservation laws in one space dimension: \[ (1)\;\partial_ t+\partial_ x(f\circ u)=0,\;(t,x)\in\mathbb{R}^ +\times\mathbb{R}^ +;\qquad (2) u(0,\cdot)=u_ 0;\qquad (3) u(\cdot,0)=a_ 0. \] Even in the linear case condition (3) must be relaxed to the ”condition of Kreiss” (*). In the nonlinear case the authors discuss two approaches: i) Connecting (1)-(3) with Riemann problems in \(\mathbb{R}^ +\times\mathbb{R}\); ii) the method of entropy inequalities. Both lead to relaxations of (3) which reduce to (*) in the linear case. For the relaxed problems the authors establish a solution theory (uniqueness, approximation schemes and existence for small data).
Reviewer: N.Weck (Essen)


35L65 Hyperbolic conservation laws
35A35 Theoretical approximation in context of PDEs
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[1] Bardos J. C., Comm. P.D.E. 4 (9) pp 1017– (1979) · Zbl 0418.35024
[2] Benabdallah Le., I 303 (1988)
[3] Benabdallah A., I 305 pp 677– (1987)
[4] Diperna R., Indiana Univ. Math. J. 28 (1) pp 137– (1979) · Zbl 0409.35057
[5] Dubois F., I 304, in: à paraître dans J. Diff. Equations. (1987)
[6] Dubroca B., approximation numérique et application à l’aérodynamique, Thèse, (1988)
[7] Dubroca B., I 306 pp 317– (1988)
[8] Glimm. J., Comm. P.A.M., 18 pp 667– (1965)
[9] Goodman, J. 1982. ”Initial boundary value problems for hyperbolic systems of conservation laws”. PhD Thesis.
[10] Lax P. D., II, Comm. P.A.M., 10 pp 537– (1967)
[11] Lax P. D., Contribution to nonlinear functional analysis pp 603– (1971)
[12] Liu T. P., Arch. Rat. Mech. Anal., 64 pp 137– (1977) · Zbl 0357.35016
[13] Nishida T., J. Diff. Equations 23 pp 244– (1977) · Zbl 0303.35052
[14] A Dubroca and Gallice C141 A. Audounet, P.A. Mazet, Rapport n’l/S255/CERT:DERI-GAN,1986.
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