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Lois de conservation scalaires avec poids. (Scalar conservation laws with weight). (French) Zbl 0612.35084
Let us consider weak entropy solutions (in the sense of Lax) of the nonlinear hyperbolic equations with weight $(*)\quad \partial_ t(r(x)u)+\partial_ x(r(x)f(u))=0,\quad u(x,t)\in {\mathbb{R}},\quad x>0,\quad t>0$ with $$r\in {\mathcal C}^ 1(]0,\infty [)$$, $$f\in {\mathcal C}^ 2({\mathbb{R}})$$, $$f''>0$$, $$\lim_{\infty}(| f| /| u|)=\infty$$ and $$f(0)=f'(0)=0$$. With the usual Dirichlet condition, the initial and boundary value problem is not well-posed. In this paper, a new boundary condition (at $$x=0)$$ is defined for the equations (*). Then, an existence and uniqueness result is obtained: an explicit formula for the solution, which reduces to the explicit representation of Lax (1957) when $$r\equiv 1$$, is given; and for the sake of uniqueness, a semi-group property of Keyfitz (1972) is generalized. The proofs are detailed by the author [Trans. Am. Math. Soc. (1988)]. Concerning the boundary condition for nonlinear hyperbolic equations, some extensions are developed by F. Dubois and the first author [”Boundary conditions for the nonlinear hyperbolic systems of conservation laws”, J. Differ. Equations (1988)].

##### MSC:
 35L65 Hyperbolic conservation laws 76N15 Gas dynamics, general 35D05 Existence of generalized solutions of PDE (MSC2000) 35L67 Shocks and singularities for hyperbolic equations