Existence and Uniqueness of the Entropy Solution to a Nonlinear Hyperbolic Equation

This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: u t + div(vf (u)) = 0 in IR N 0; T ], with initial data u(:; 0) = u 0 (:) in IR N. where u 0 2 L 1 (IR N) is a given function, v is a divergence-free bounded function of class C 1 from IR N 0; T ] to IR N , and f is a function of class C 1 from IR to IR. It also gives a result of convergence of a numerical scheme for the discretization of theis equation. We rst show the existence of a "process" solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions of the numerical scheme). The uniqueness of this entropy process solution is then proved; it is also proven that the entropy process solution is in fact a entropy weak solution, hence the existence and uniqueness of the entropy weak solution, and the convergence of the numerical scheme. The present work is concerned with the existence of an entropy weak solution u 2 L 1 (IR N ]0; T) to the following nonlinear hyperbolic equation with initial data: u t (x; t) + div(vf(u(x; t))) = 0; x 2 IR N ; t 2 0; T] (1) u(x; 0) = u 0 (x); x 2 IR N : (2) where T > 0, u t denotes the partial derivative of u with respect to time variable t, div denotes the divergence operator with respect to the space variable x = (x 1