6 research outputs found
Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains
In this work, we consider the two and three-dimensional stochastic convective
Brinkman-Forchheimer (2D and 3D SCBF) equations driven by irregular additive
white noise for in
unbounded domains (like Poincar\'e domains)
() where is a Hilbert space valued Wiener process on
some given filtered probability space, and discuss the asymptotic behavior of
its solution. For with and with
(for with ), we first prove the existence and
uniqueness of a weak solution (in the analytic sense) satisfying the energy
equality for SCBF equations driven by an irregular additive white noise in
Poincar\'e domains by using a Faedo-Galerkin approximation technique. Since the
energy equality for SCBF equations is not immediate, we construct a sequence
which converges in Lebesgue and Sobolev spaces simultaneously and it helps us
to demonstrate the energy equality. Then, we establish the existence of random
attractors for the stochastic flow generated by the SCBF equations. One of the
technical difficulties connected with the irregular white noise is overcome
with the help of the corresponding Cameron-Martin space (or Reproducing Kernel
Hilbert space). Finally, we address the existence of a unique invariant measure
for 2D and 3D SCBF equations defined on Poincar\'e domains (bounded or
unbounded). Moreover, we provide a remark on the extension of the above
mentioned results to general unbounded domains also
Bi-spatial random attractor, ergodicity and a random Liouville type theorem for stochastic Navier-Stokes equations on the whole space
This article concerns the random dynamics and asymptotic analysis of the well
known mathematical model,
\begin{align*}
\frac{\partial \boldsymbol{v}}{\partial t}-\nu
\Delta\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+\nabla
p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{v}=0,
\end{align*}
the Navier-Stokes equations. We consider the two-dimensional stochastic
Navier-Stokes equations (SNSE) driven by a linear multiplicative white noise of
It\^o type on the whole space . Firstly, we prove that
non-autonomous 2D SNSE generates a bi-spatial
-continuous random
cocycle. Due to the bi-spatial continuity property of the random cocycle
associated with SNSE, we show that if the initial data is in
, then there exists a unique bi-spatial
-pullback random
attractor for non-autonomous SNSE which is compact and attracting not only in
-norm but also in -norm. Next, we discuss the
existence of an invariant measure for the random cocycle associated with
autonomous SNSE which is a consequence of the existence of random attractors.
We prove the uniqueness of invariant measures for
and for any by using the linear multiplicative structure of the noise
coefficient and exponential stability of solutions. Finally, we prove the
existence of a family of invariant sample measures for 2D autonomous SNSE which
satisfies a random Liouville type theorem
2D and 3D convective Brinkman-Forchheimer equations perturbed by a subdifferential and applications to control problems
The following convective Brinkman-Forchheimer (CBF) equations (or damped
Navier-Stokes equations) with potential
\begin{equation*}
\frac{\partial \boldsymbol{y}}{\partial t}-\mu
\Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta|\boldsymbol{y}|^{r-1}\boldsymbol{y}+\nabla
p+\Psi(\boldsymbol{y})\ni\boldsymbol{g},\ \nabla\cdot\boldsymbol{y}=0,
\end{equation*}
in a -dimensional torus is considered in this work, where ,
and . For with and
with ( for ), we establish the
existence of \textsf{\emph{a unique global strong solution}} for the above
multi-valued problem with the help of the \textsf{\emph{abstract theory of
-accretive operators}}. %for nonlinear differential equations of accretive
type in Banach spaces.
Moreover, we demonstrate that the same results hold \textsf{\emph{local in
time}} for the case with and with . We
explored the -accretivity of the nonlinear as well as multi-valued
operators, Yosida approximations and their properties, and several higher order
energy estimates in the proofs. For , we {quantize (modify)} the
Navier-Stokes nonlinearity to
establish the existence and uniqueness results, while for
( for ), we handle the Navier-Stokes nonlinearity by the
nonlinear damping term . Finally, we
discuss the applications of the above developed theory in feedback control
problems like flow invariance, time optimal control and stabilization