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Oleinik type estimates and uniqueness for \(n \times n\) conservation laws. (English) Zbl 0990.35095
It is investigated the strictly hyperbolic \(n\times n\) system of conservation laws \(u_t+f(u)_x=0\), \(t>0\), \(x\in\mathbb{R}\), with the initial condition \(u(0,x)= \overline u(x)\). The authors prove that the above Cauchy problem has a unique weak solution, which satisfies some assumptions (entropy condition, tame oscillation or decay estimate). So, they generalize a classical result proved by O. A. Oleinik in the scalar case [in Am. Math. Soc., Transl. II. Ser. 26, 95-172 (1963); translation from Usp. Mat. Nauk 12, No. 3, 3-73 (1957; Zbl 0080.07701)].
Reviewer: R.Luca (Iaşi)

MSC:
35L65 Hyperbolic conservation laws
35B45 A priori estimates in context of PDEs
35L45 Initial value problems for first-order hyperbolic systems
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