A variational calculus for discontinuous solutions of systems of conservation laws

This paper is concerned with the Cauchy problem for the perturbed system of conservation laws in a single space variable: u{sub t} + [F(u)]{sub x} = h(t,x,u) u(0,x) = u(x), where F:IR{sup n} {r_arrow} IR{sup n} and h : [0, +{infinity}] x IR x IR{sup n} {r_arrow} IR{sup n} are smooth maps. We assume that the system (1) is strictly hyperbolic, and that each characteristic field is either linearly degenerate or genuinely nonlinear in the sense of Lax [8]. Given a piecewise Lipschitz continuous solution u = u(t,x) of (1), we shall study the behavior of first-order variations of u, by defining a suitable class of {open_quotes}generalized tangent vectors{close_quotes} and determining their evolution in time. More precisely, consider a family of initial conditions u(0,x) = {anti u}{sup {var_epsilon}}(x) depending on a small parameter {var_epsilon}(x) depending on a small parameter {var_epsilon}, and assume that {anti u}{sup {var_epsilon}} admits an expansion of the form {anti u}{sup {var_epsilon}}(x) = {anti u}(x) + {var_epsilon}{anti v}(x) + o({var_epsilon}). 13 refs.