Some Hyperbolic Conservation Laws on $$ {\mathbb {R}} ^{n}$$ R n

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Some Hyperbolic Conservation Laws on Rn Nabila Bedida and Nadji Hermas Abstract. In this paper, we prove the existence and the uniqueness of maximum classical solutions in the temporal variable for some quasilinear hyperbolic systems. Mathematics Subject Classification. 35L40, 35L45, 35L50, 35L60, 35L65, 58J45. Keywords. Maximum solutions, Quasi-linear hyperbolic systems, Hyperbolic laws of conservation.

1. Introduction In this paper, we prove the existence of the maximum solutions in the temporal variable for quasi-linear hyperbolic systems ∂t u (t, x) + ∂x u (t, x) (f (t, u (t, x))) = 0,

(1)

with the initial condition u (t0 , x) = ϕ (x) , ∞

m

n

(2) k

n

m

∗

where t0 ∈ R, f ∈ C (R × R ; R ), ϕ ∈ C (R ; R ) (k ∈ N ∪ {∞}), and ∂x u is the partial derivative of u with respect to the second variable x. According to the physical terminologies, this type of equations of order 1 is called ’hyperbolic laws of conservation’, see [3] and [13]. The mathematical theory of these laws of conservation, which of course has a relation with our equation (1), deals with the existence, the uniqueness and the stability of (classical or variational) solutions of the hyperbolic systems of laws of conservation, and among the tools used in this theory are, for example, geometric methods of characteristic curves invented by W. R. Hamilton, methods based on the formulas of Hopf–Lax and of Lax–Oleinik, methods based on the construction of sequences of approximate solutions, methods that use the approach of viscosity solutions, and ﬁnally, methods based on the notion of variational solutions. Three references [7,10] and [20] present a large part of the theory of hyperbolic laws. Indeed, in [7], we ﬁnd a good study on hyperbolic laws of conservation and their relation with the continuum physics. The author 0123456789().: V,-vol

197

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N. Bedida and N. Hermas

MJOM

discusses, for example, laws of equilibrium in the continuum physics, the deﬁnition of hyperbolic systems of equilibrium laws, the entropy and the stability of classical solutions of hyperbolic systems, the L1 -theory of hyperbolic laws of conservation, and the theory of shock and admissible waves. In reference [10], the authors speak of the phenomenon of decay of solutions of hyperbolic nonlinear systems of conservation laws. Indeed, they study approximate conservation laws and their approximate characteristics, the construction of exact solutions as limits of approximate solutions, and ﬁnally the existence and disintegration of solutions with arbitrary Cauchy data. Reference [20] deals with the hyperbolic systems of conservation laws and the mathematical theory of shock waves. The subject presented by the author of this reference contains the quasi-linear hyperbolic equations, the systems of laws of conservation, the scalar laws of conservation, and the disintegration of solutions when the temporal variable tends towards the inﬁnity. In paper [15], the author gives a founding presentation of the concept of generalized solutions for the ﬁrst order quasi-linea

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Some Hyperbolic Conservation Laws on $$ {\mathbb {R}} ^{n}$$ R n

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