Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

Abstract

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Itô) noise. The Cauchy problem defined on a Riemanian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result ala Perthame is derived, which implies that generalized solutions are kinetic solutions and that kinetic solutions are uniquely determined by their initial data (L 1 contraction principle). Deprived of noise, the equations we consider coincide with those analyzed by , who worked with Kružkov-DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle [15].

Homogenization of Stochastic Conservation Laws with Multiplicative Noise

“…Proof. As in [31,34], we point out that the 1 contraction principle (5.39) is a simple consequence of Proposition 5.1. Indeed, define = 1 2…”

“…Following an approach developed by Perthame [51], later extended to the stochastic case in [16] (see also [16,17,19,36,31,32,43,46]), we establish a rigidity result implying that generalized kinetic solutions are in fact kinetic solutions, at least when the initial function is a kinetic function, 0 = < 0 . The proof herein involves a regularization (via convolution) procedure, the Itô formula, and commutator arguments (going beyond the deterministic one by DiPerna-Lions) [35].…”

“…). Thus, it is not difficult to show that, ̃ , converges a.s. to zero in 1 loc as , → 0 [31]. Choosing (5.33) as test function in (5.36), recalling that ( ) = − 2 , and carrying on as before (5.34), we deliver…”

“…Although we should work with the left/right continuous representatives ± as in [16, Proposition 10] (see also [19]) and make use of the space-weak formulation (5.29), we will not do so in an attempt to save space and keep the presentation as simple as possible. Instead we refer to [16,17,19,36,31,32] for such details, see also [33,34].…”

“…However, working with suitable representatives (versions), we can ensure that , , , are càdlàg / càglàd in time , thereby making the Itô formula available to us, and thus the arguments below can be made rigorous (see e.g. [16,19,31,32,33,34]). In passing, note that , is a measure on [0, ] (depending on the "parameters" , , ).…”