# A Uniqueness Result for Measure Valued

• Thierry Gallou Et, Rapha Ele Herbin
• Published 2007

#### Abstract

We study here the solutions of the nonlinear hyperbolic equation ut + div(vf (u)) = 0 in IR N 0; T ], with given initial condition u(:; 0) = u0(:) in IR N , where v is a given function from IR N 0; T ] to IR N and f is a given function from IR to IR. We prove that if is a measure valued solution satisfying some entropy condition, which we deene, then x;t = u(x;t) for a.e. (x; t) 2 IR N 0; T ], where u is the unique entropy weak solution to the equation. We consider here the following nonlinear hyperbolic equation, with initial condition: (u t (x; t) + div(vf(u(x; t))) = 0; x 2 IR N ; t 2 0; T] u(x; 0) = u 0 (x); x 2 IR N (1) where T > 0, u t denotes the derivative of u with respect to t, divv= N X i=1 N). It is assumed that there exists V > 0 such that supfjv(x; t)j; x 2 IR N ; t 2 0; T]g V , where j:j denotes the Euclidean norm in IR N , that f is a given function of class C 3 from IR to IR, and that u 0 2 L 1 (IR N) is a given function. It is also assumed that supf?div(v(x; t))f 0 (u), x 2 IR N , t 2 0; T], u 2 IRg < 1 (this is true, for instance, if divv = 0). Under these assumptions, Kruzkov 4] gives results of existence (and uniqueness) of the entropy weak solution u, i.e. a function u 2 L 1 (IR N ]0; T) which satisses : Z IR N Z T 0 (u(x; t))' t (x; t) + (u(x; t))v(x; t):grad'(x; t))dtdx + Z IR N Z T 0 divv(x; t)'(x; t) (u(x; t)) ? 0 (u(x; t)f(u(x; t)) dtdx+ Z IR N

### Cite this paper

@inproceedings{Et2007AUR, title={A Uniqueness Result for Measure Valued}, author={Thierry Gallou Et and Rapha Ele Herbin}, year={2007} }