An existence and uniqueness result for two nonstrictly hyperbolic systems. (English) Zbl 0727.35083

Le Floch, Philippe

Nonlinear evolution equations that change type, Proc. Workshop IMA Nonlinear Waves, Minneapolis/MN (USA) 1988-89, IMA Vol. Math. Appl. 27, 126-138 (1990).

Summary: [For the entire collection see Zbl 0712.00019.]
We prove a result of existence and uniqueness of entropy weak solutions for two nonstrictly hyperbolic systems, both a nonconservative system of two equations \[ \partial_ tu+\partial_ xf(u)=0,\quad \partial_ tw+a(u)\partial_ xw=0, \] and a conservative system of two equations \[ \partial_ tu+\partial_ xf(u)=0,\quad \partial_ tv+\partial_ x(a(u)v)=0, \] where f: \({\mathbb{R}}\to {\mathbb{R}}\) is a given strictly convex function and \(a=(d/du)f\). We use the Volpert’s product and find entropy weak solutions u and w which have bounded variation while the solutions v are Borel measures. The equations for w and v can be viewed as linear hyperbolic equations with discontinuous coefficients.

MSC:

35L60	First-order nonlinear hyperbolic equations
35D05	Existence of generalized solutions of PDE (MSC2000)

Keywords:

existence; uniqueness; entropy weak solutions; Borel measures

Citations:

Zbl 0712.00019