A fractional degenerate parabolic-hyperbolic Cauchy problem with noise

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy sol...

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Bibliographic Details
Published in:Journal of Differential Equations 2021-05, Vol.284, p.433-521
Main Authors: Bhauryal, Neeraj, Koley, Ujjwal, Vallet, Guy
Format: Article
Language:English
Subjects:
Continuous dependence estimates
Existence & Uniqueness
Mathematics
Nonlinear fractional conservation laws
Rate of convergence
Stochastic conservation laws
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Summary:We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and derive error estimate for the stochastic vanishing viscosity method. In addition, we develop uniqueness method “à la Kružkov” for more general equations where the noise coefficient may depend explicitly on the spatial variable.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2021.02.061