4,922 research outputs found
The A-Stokes approximation for non-stationary problems
Let be an elliptic tensor. A function
is a solution to the non-stationary -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 ,
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 -caloric
approximation for the non-stationary -Stokes problem. Precisely,
we show that every almost solution ,
, can be approximated by a solution in the
-sense for all . So, we extend the stationary -Stokes approximation by Breit-Diening-Fuchs to parabolic problems
Regularity theory for nonlinear systems of SPDEs
We consider systems of stochastic evolutionary equations of the type
where is a non-linear
operator, for instance the -Laplacian with and grows linearly.
We extend known results about the deterministic problem to the stochastic
situation. First we verify the natural regularity: where
. If we have Uhlenbeck-structure then
is finite for all
Existence theory for stochastic power law fluids
We consider the equations of motion for an incompressible Non-Newtonian fluid
in a bounded Lipschitz domain during the time intervall
together with a stochastic perturbation driven by a Brownian motion
. The balance of momentum reads as where is the velocity,
the pressure and an external volume force. We assume the common power law
model and
show the existence of weak (martingale) solutions provided
. Our approach is based on the -truncation and a
harmonic pressure decomposition which are adapted to the stochastic setting
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 (as a result of a
random electric field). We show the existence of a weak martingale solution
provided the variable exponent satisfies ( 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 -truncation method
we prove existence of weak solutions for a power-law exponent
,
Compressible fluids interacting with a linear-elastic shell
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
( 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
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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