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

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 àla...

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Bibliographic Details
Published in:Journal of hyperbolic differential equations 2019-09, Vol.16 (3), p.519-593
Main Authors: Galimberti, L., Karlsen, K. H.
Format: Article
Language:English
Subjects:
Cauchy problems
Conservation laws
Mathematical analysis
Riemann manifold
Well posed problems
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Summary: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 àla 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 Ben-Artzi and LeFloch [Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 24(6) (2007) 989–1008], who worked with Kružkov–DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle [Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259(4) (2010) 1014–1042].
ISSN:0219-8916
1793-6993
DOI:10.1142/S0219891619500188