Measure and continuous vector field at a boundary II: geodesics and support propagation

Abstract

Nonnegative measures that are solutions to a transport equation with continuous coefficients have been widely studied. Because of the low regularity of the associated vector field, there is no natural flow since nonuniqueness of integral curves is the general rule. It has been known since the works by L. Ambrosio [2] and L. Ambrosio and G. Crippa [3, 4] that such measures can be described as a superposition of $\delta$-measures supported on integral curves. In this article, motivated by some observability questions for the wave equation, we are interested in such transport equations in the case of domains with boundary. Associated with a wave equation with C^1-coefficients are bicharacteristics that are integral curves of a continuous Hamiltonian vector field. We first study in details their behaviour in the presence of a boundary and define their natural generalisation that follows the laws of geometric optics. Then, we introduce a natural class of transport equations with a source term on the boundary, and we prove that any nonnegative measure satisfying such an equation has a union of maximal generalized bicharacteristics for support. This result is a weak form of the superposition principle in the presence of a boundary. With its companion article [7], this study completes the proof of wave observability generalizing the celebrated result of Bardos, Lebeau, and Rauch [5] in a low regularity framework where coefficients of the wave equation (and associated metric) are C^1 and the boundary and the manifold are C^2.