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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 \((\overline t,\overline x)\in[0, T]\times\mathbb{R}\).

34A36 Discontinuous ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
Full Text: DOI
[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
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