Weakly Coupled Systems of Conservation Laws on Moving Surfaces

Abstract

We consider weakly coupled systems of nonlinear hyperbolic conservation laws on moving surfaces. As in the Euclidean space, see, for example, (Levy, Commun Partial Differ Equ 17(3–4):657–698, 1992, [9], Rohde, Weakly coupled systems of hyperbolic conservation laws. PhD thesis, Mathematische Fakultat der Albert-Ludwigs-Universitat Freiburg 1996, [16]), the coupling is realized by a source term, which only depends on (x, t) and the unknown function u(x, t) but not on its derivatives. Scalar conservation laws on moving surfaces were considered in Dziuk et al., Interfaces Free Boundaries 15:202–236, 2013, [4]), Lengeler and Muller (J Differ Equ 254(4):1705–1727, 2013, [10]). The velocity of the surface is given by a smooth function, and we assume the surface to be compact. We prove the existence for an entropy solution. First, we consider the regularized parabolic problem with viscosity parameter \(\varepsilon \) and show that there exists a weak solution by decoupling and linearizing the problem. Then, we prove the boundedness of this solution in \(L^\infty (G_T)\), use standard regularity results to prove that this solution is a solution in the classical sense, and show uniform boundedness in \(L^\infty (G_T)\) and \(W^{1,1}(G_T)\) with respect to \(\varepsilon \).