4,922 research outputs found

    The A-Stokes approximation for non-stationary problems

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    Let A\mathcal A be an elliptic tensor. A function vL1(I;LDdiv(B))v\in L^1(I;LD_{div}(B)) is a solution to the non-stationary A\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×BQ:=I\times B, BRdB\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 A\mathcal A-caloric approximation for the non-stationary A\mathcal A -Stokes problem. Precisely, we show that every almost solution vLp(I;Wdiv1,p(B))v\in L^p(I;W^{1,p}_{div}(B)), 1<p<1<p<\infty, can be approximated by a solution in the Ls(I;W1,s(B))L^s(I;W^{1,s}(B))-sense for all s<ps<p. So, we extend the stationary A\mathcal A-Stokes approximation by Breit-Diening-Fuchs to parabolic problems

    Regularity theory for nonlinear systems of SPDEs

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    We consider systems of stochastic evolutionary equations of the type du=divS(u)dt+Φ(u)dWtdu=\mathrm{div}\,S(\nabla u)\,dt+\Phi(u)dW_t where SS is a non-linear operator, for instance the pp-Laplacian S(ξ)=(1+ξ)p2ξ,ξRd×D,S(\xi)=(1+|\xi|)^{p-2}\xi,\quad \xi\in\mathbb R^{d\times D}, with p(1,)p\in(1,\infty) and Φ\Phi grows linearly. We extend known results about the deterministic problem to the stochastic situation. First we verify the natural regularity: E[supt(0,T)Gu(t)2dx+0TGF(u)2dxdt]<,\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(ξ)=(1+ξ)p22ξF(\xi)=(1+|\xi|)^{\frac{p-2}{2}}\xi. If we have Uhlenbeck-structure then E[uqq]\mathbb E\big[\|\nabla u\|_q^q\big] is finite for all q<q<\infty

    Existence theory for stochastic power law fluids

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    We consider the equations of motion for an incompressible Non-Newtonian fluid in a bounded Lipschitz domain GRdG\subset\mathbb R^d during the time intervall (0,T)(0,T) together with a stochastic perturbation driven by a Brownian motion WW. The balance of momentum reads as dv=divSdt(v)vdt+πdt+fdt+Φ(v)dWt,dv=\mathrm{div}\, S\,dt-(\nabla v)v\,dt+\nabla\pi \,dt+f\,dt+\Phi(v)\,dW_t, where vv is the velocity, π\pi the pressure and ff an external volume force. We assume the common power law model S(ε(v))=(1+ε(v))p2ε(v)S(\varepsilon(v))=\big(1+|\varepsilon(v)|\big)^{p-2} \varepsilon(v) and show the existence of weak (martingale) solutions provided p>2d+2d+2p>\tfrac{2d+2}{d+2}. Our approach is based on the LL^\infty-truncation and a harmonic pressure decomposition which are adapted to the stochastic setting

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

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    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(ω,t,x)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 pp>3nn+2p\geq p^->\frac{3n}{n+2} (p>1p^->1 in two dimensions). Under additional assumptions we obtain also pathwise solutions

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

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    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 LL^\infty-truncation method we prove existence of weak solutions for a power-law exponent p>2d+2d+2p>\frac{2d+2}{d+2}, d=2,3d=2,3

    Compressible fluids interacting with a linear-elastic shell

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    We study the Navier--Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter's elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies γ>127\gamma>\frac{12}{7} (γ>1\gamma>1 in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in [D. Lengeler, M. \Ruzicka, Weak Solutions for an Incompressible Newtonian Fluid Interacting with a Koiter Type Shell. Arch. Ration. Mech. Anal. 211 (2014), no. 1, 205--255] on incompressible Navier--Stokes equations

    Stochastic compressible Euler equations and inviscid limits

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    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
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