Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition

We consider the stochastic divergence-free continuity equations with Ladyzhenskaya-Prodi-Serrin condition. Wellposedness is proved meanwhile uniqueness may fail for the deterministic PDE. The main issue of uniqueness realies on stochastic characteristic method and the generalized Ito-Ventzel-Kunita formula. Moreover, we prove a stability property for the solution with respect to the initial datum.Comment: To appears in Nonlinear Differential Equations and Applications NoDE

Some regularity results for the stochastic pressure equation of Wick-type

We prove that the solution of the stochastic pressure equation of Wick-type belongs to a space of generalized random fields having square integrable homogeneous chaos kernels. We find the chaos expansion, and calculate its stochastic regularity in distributional sense. Furthermore, we show that the solution is stable under perturbations of the permeability field and the source term

Existence theory for stochastic power law fluids

We consider the equations of motion for an incompressible Non-Newtonian fluid in a bounded Lipschitz domain $G\subset\mathbb R^d$ during the time intervall $(0,T)$ together with a stochastic perturbation driven by a Brownian motion $W$. The balance of momentum reads as $$dv=\mathrm{div}\, S\,dt-(\nabla v)v\,dt+\nabla\pi \,dt+f\,dt+\Phi(v)\,dW_t,$$ where $v$ is the velocity, $\pi$ the pressure and $f$ an external volume force. We assume the common power law model $S(\varepsilon(v))=\big(1+|\varepsilon(v)|\big)^{p-2} \varepsilon(v)$ and show the existence of weak (martingale) solutions provided $p>\tfrac{2d+2}{d+2}$. Our approach is based on the $L^\infty$-truncation and a harmonic pressure decomposition which are adapted to the stochastic setting