Lois de conservation scalaires avec poids. (Scalar conservation laws with weight). (French) Zbl 0612.35084

Le Floch, Philippe; Nedelec, Jean-Claude

C. R. Acad. Sci., Paris, Sér. I 301, 793-796 (1985).

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

Keywords:

weak entropy solutions; equations with weight; Dirichlet condition; well- posed; existence; uniqueness; explicit formula; representation of Lax; semi-group property