Energy estimates and uniqueness of the weak solutions of initial-boundary value problems for semilinear hyperbolic systems

Abstract.

Bounds in energy for the smooth solutions of initial and boundary value problems for semilinear hyperbolic systems are given. We prove that the integral energy identities for smooth solutions are still valid for weak solutions (i.e., piecewise continuous and piecewise continuously differentiable solutions separated by shock and/or acceleration waves). Energy estimates of the weak solutions of the same problems are given. The way weak and strong discontinuities in the input data affect the regularity of the total energy of the solutions is discussed. The uniqueness in L ² sense of the weak solutions is derived.