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- W3196471330 abstract "We consider the initial-value problem for stochastic continuity equations of the form ∂ t ρ + div h ρ u ( t , x ) + ∑ i = 1 N a i ( x ) ∘ d W i d t = 0 , defined on a smooth closed Riemannian manifold M with metric h , where the Sobolev regular velocity field u is perturbed by Gaussian noise terms W ̇ i ( t ) driven by smooth spatially dependent vector fields a i ( x ) on M . Our main result is that weak ( L 2 ) solutions are renormalized solutions, that is, if ρ is a weak solution, then the nonlinear composition S ( ρ ) is a weak solution as well, for any “reasonable” function S : R → R . The proof consists of a systematic procedure for regularizing tensor fields on a manifold, a convenient choice of atlas to simplify technical computations linked to the Christoffel symbols, and several DiPerna–Lions type commutators C ɛ ( ρ , D ) between (first/second order) geometric differential operators D and the regularization device ( ɛ is the scaling parameter). This work, which is related to the “Euclidean” result in Punshon-Smith (0000), reveals some structural effects that noise and nonlinear domains have on the dynamics of weak solutions." @default.
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- W3196471330 date "2021-12-01" @default.
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- W3196471330 title "Renormalization of stochastic continuity equations on Riemannian manifolds" @default.
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- W3196471330 doi "https://doi.org/10.1016/j.spa.2021.08.009" @default.
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