We establish the well-posedness of stochastic scalar conservation laws on moving hypersurfaces driven by Brownian motion. To handle the interaction between stochastic forcing and evolving geometry, we derive an It\^{o} formula on moving surfaces and introduce the notion of generalized entropy solutions incorporating the relevant stochastic interaction terms. A martingale entropy solution is constructed via the vanishing-viscosity method, based on a uniform $L^\infty$-bound in space and time, an $L^1$-estimate for the spatial gradient, an $L^1$-continuity estimate in time, and a suitable tightness argument. Pathwise uniqueness is established by adapting Kruzhkov's doubling-of-variables method to moving hypersurfaces, yielding an $L^1$-contraction property. Finally, together with the Yamada-Watanabe theorem, these results yield the well-posedness of the problem.
We develop an intrinsic well-posedness theory for nonlinear backward stochastic differential equations (BSDEs) driven simultaneously by Brownian motion and a (level-$2$) rough path of finite $p$-variation. Unlike earlier approaches based on smooth approximation or transformation methods, we formulate the equation direc...
P. Friz, Jian Song, Hui-Lin Zhang et al.· 0 citations
Consider the three-dimensional primitive equations subject to Neumann wind stress driven by finitely many Brownian motions. For arbitrary smooth spatial wind profiles, global pathwise existence and uniqueness without assuming a vertical derivative of the initial datum is proved, extending the local theory of Binz, Hieb...
This work investigates the mean-field stochastic PDEs involving a class of pseudo-monotone kernels. We first study the well-posedness -- in both the strong and weak sense -- within the variational framework by introducing a notion of measure-dependent pseudo-monotone operators, which generalizes the classical framework...
We study a first-order stochastic transport equation driven by Brownian transport noise and a nonlinear state-dependent Poisson jump term. The drift vector field is merely integrable and satisfies the subcritical Krylov--R\"ockner condition. The dependence of the jump coefficient on the solution creates a nontrivial co...
This paper studies doubly nonlinear parabolic obstacle problems. We introduce a renormalized entropy formulation in which the reaction measure is encoded by evaluating the entropy multiplier at the obstacle. For diffusion depending only on the gradient and continuous time-independent obstacles, we prove a global $L^1$...
This paper studies the obstacle problem for first-order scalar conservation laws driven by multiplicative noise. By adapting a barrier-substitution strategy to the kinetic formulation, we permit the reflection measure to be a general Radon measure and eliminate the usual obstacle-noise compatibility condition. We estab...
Ruoyang Liu, Rangrang Zhang· 0 citations
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