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arXiv · 1904.03623

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

Luca Galimberti·Kenneth H. Karlsen
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Abstract

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Ito) noise. The Cauchy problem defined on a Riemannian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result 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 Ben-Artzi and LeFloch (2007), who worked with Kruzkov-DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle (2010).

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Luca Galimberti, Kenneth H. Karlsen. 2019-06-27. Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds. https://arxiv.org/abs/1904.03623

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