Regularity theory for nonlinear systems of SPDEs

We consider systems of stochastic evolutionary equations of the type $$du=\mathrm{div}\,S(\nabla u)\,dt+\Phi(u)dW_t$$ where $S$ is a non-linear operator, for instance the $p$-Laplacian $$S(\xi)=(1+|\xi|)^{p-2}\xi,\quad \xi\in\mathbb R^{d\times D},$$ with $p\in(1,\infty)$ and $\Phi$ grows linearly. We extend known results about the deterministic problem to the stochastic situation. First we verify the natural regularity: $$\mathbb E\bigg[\sup_{t\in(0,T)}\int_{G'}|\nabla u(t)|^2\,dx+\int_0^T\int_{G'}|\nabla F(\nabla u)|^2\,dx\,dt\bigg]<\infty,$$ where $F(\xi)=(1+|\xi|)^{\frac{p-2}{2}}\xi$. If we have Uhlenbeck-structure then $\mathbb E\big[\|\nabla u\|_q^q\big]$ is finite for all $q<\infty$

Electro-rheological fluids under random influences: martingale and strong solutions

We study generalised Navier--Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent $p=p(\omega,t,x)$ (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies $p\geq p^->\frac{3n}{n+2}$ ($p^->1$ in two dimensions). Under additional assumptions we obtain also pathwise solutions

Weak Solutions for a Non-Newtonian Diffuse Interface Model with Different Densities

We consider weak solutions for a diffuse interface model of two non-Newtonian viscous, incompressible fluids of power-law type in the case of different densities in a bounded, sufficiently smooth domain. This leads to a coupled system of a nonhomogenouos generalized Navier-Stokes system and a Cahn-Hilliard equation. For the Cahn-Hilliard part a smooth free energy density and a constant, positive mobility is assumed. Using the $L^\infty$-truncation method we prove existence of weak solutions for a power-law exponent $p>\frac{2d+2}{d+2}$, $d=2,3$

Stochastic compressible Euler equations and inviscid limits

We prove the existence of a unique local strong solution to the stochastic compressible Euler system with nonlinear multiplicative noise. This solution exists up to a positive stopping time and is strong in both the PDE and probabilistic sense. Based on this existence result, we study the inviscid limit of the stochastic compressible Navier--Stokes system. As the viscosity tends to zero, any sequence of finite energy weak martingale solutions converges to the compressible Euler system.Comment: 26 page

The A-Stokes approximation for non-stationary problems

Let $\mathcal A$ be an elliptic tensor. A function $v\in L^1(I;LD_{div}(B))$ is a solution to the non-stationary $\mathcal A$-Stokes problem iff \begin{align}\label{abs} \int_Q v\cdot\partial_t\phi\,dx\,dt-\int_Q \mathcal A(\varepsilon(v),\varepsilon(\phi))\,dx\,dt=0\quad\forall\phi\in C^{\infty}_{0,div}(Q), \end{align} where $Q:=I\times B$, $B\subset\mathbb R^d$ bounded. If the l.h.s. is not zero but small we talk about almost solutions. We present an approximation result in the fashion of the $\mathcal A$-caloric approximation for the non-stationary $\mathcal A$-Stokes problem. Precisely, we show that every almost solution $v\in L^p(I;W^{1,p}_{div}(B))$, $1<p<\infty$, can be approximated by a solution in the $L^s(I;W^{1,s}(B))$-sense for all $s<p$. So, we extend the stationary $\mathcal A$-Stokes approximation by Breit-Diening-Fuchs to parabolic problems

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

Convergence Analysis for a Finite Element Approximation of a Steady Model for Electrorheological Fluids

In this paper we study the finite element approximation of systems of $p(\cdot)$-Stokes type, where $p(\cdot)$ is a (non constant) given function of the space variables. We derive --in some cases optimal-- error estimates for finite element approximation of the velocity and of the pressure, in a suitable functional setting