×

zbMATH — the first resource for mathematics

Uniqueness for discontinuous ODE and conservation laws. (English) Zbl 0948.34006
The paper is concerned with the Cauchy problem \[ x'= f(t,x),\quad x(0)=\overline x,\tag{1} \] where \(f:[0,T]\times \mathbb{R}\to\mathbb{R}\) is a measurable function and the solutions are understood in Carathéodory sense. The main result ensures the existence of a unique solution to (1) under the following assumptions:
(i) For every point \((\overline t,\overline x)\) there exists a slope \(\lambda(\overline t,\overline x)\) such that the function \(f\) is constant along the segment \(s(\overline t,\overline x)\), with \(s(\overline t,\overline x)= \{(t,x): t\in(0,\overline t)\), \(x=\overline x+(t-\overline t)\lambda(\overline t,\overline x)\}\).
(ii) There exist disjoint intervals \([a,b]\) and \([c,d]\) such that \(f(t,x)\in [a,b]\) and \(\lambda(\overline t,\overline x)\in [c,d]\) for all \((\overline t,\overline x)\in[0, T]\times\mathbb{R}\).

MSC:
34A36 Discontinuous ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baiti, P.; Bressan, A., The semigroup generated by a temple class system with large data, Differential integral equations, 10, 401-418, (1997) · Zbl 0890.35083
[2] Baiti, P.; Jenssen, H-K., Well-posedness for a class of 2×2 conservation laws with L∞ data, J. differential equations, 140, 161-185, (1997) · Zbl 0892.35097
[3] Bressan, A., Unique solutions for a class of discontinuous differential equations, Proc. amer. math. soc., 104, 772-778, (1988) · Zbl 0692.34004
[4] Bressan, A., The semigroup approach to systems of conservation laws, Mathematica contemp., 10, 21-74, (1996) · Zbl 0866.35064
[5] Bressan, A.; Colombo, R.M., The semigroup generated by 2×2 conservation laws, Arch. rat. mech. anal., 133, 1-75, (1995) · Zbl 0849.35068
[6] A. Bressan, G. Crasta, B. Piccoli, Well posedness of the Cauchy problem for n×n systems of conservation laws, Amer. Math. Soc. Memoir, to appear. · Zbl 0958.35001
[7] Crandall, M., The semigroup approach to first-order quasilinear equations in several space variables, Israel J. math., 12, 108-132, (1972) · Zbl 0246.35018
[8] Filippov, A.F., Differential equations with discontinuous right hand sides, (1988), Kluwer Dordrecht · Zbl 0664.34001
[9] P. Hartman, Ordinary Differential Equations, 2n ed., Birkhäuser, Basal, 1982. · Zbl 0476.34002
[10] Kruzkov, S., First order quasi linear equations with several space variables, Math. USSR. sb., 10, 217-243, (1970)
[11] Serre, D., Solutions á variations bornées pour certains systèmes hyperboliques de lois de conservation, J. differential equations, 68, 137-168, (1987) · Zbl 0627.35062
[12] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer, Berlin, 1994. · Zbl 0807.35002
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.