| | |
| | |
Stat |
Members: 3667 Articles: 2'599'751 Articles rated: 2609
18 February 2025 |
|
| | | |
|
Article overview
| |
|
2D and 3D convective Brinkman-Forchheimer equations perturbed by a subdifferential and applications to control problems | Sagar Gautam
; Kush Kinra
; Manil T. Mohan
; | Date: |
4 Jan 2023 | Abstract: | The following convective Brinkman-Forchheimer (CBF) equations (or damped
Navier-Stokes (NS) equations) with potential
egin{equation*}
frac{partial oldsymbol{y}}{partial t}-mu
Deltaoldsymbol{y}+(oldsymbol{y}cdot
abla)oldsymbol{y}+alphaoldsymbol{y}+eta|oldsymbol{y}|^{r-1}oldsymbol{y}+
abla
p+Psi(oldsymbol{y})
ioldsymbol{g},
ablacdotoldsymbol{y}=0,
end{equation*}
in a $d$-dimensional torus is considered in this work, where $din{2,3}$,
$mu,alpha,eta>0$ and $rin[1,infty)$. For $d=2$ with $rin[1,infty)$ and
$d=3$ with $rin[3,infty)$ ($2etamugeq 1$ for $d=r=3$), we establish the
existence of extsf{emph{a unique global strong solution}} for the above
multivalued problem with the help of the extsf{emph{abstract theory of
$m$-accretive operators}}. %for nonlinear differential equations of accretive
type in Banach spaces.
Moreover, we demonstrate that the same results hold extsf{emph{local in
time}} for the case $d=3$ with $rin[1,3]$. We explored the $m$-accretivity of
the nonlinear as well as multivalued operators, Yosida approximations and their
properties, and several higher order energy estimates in the proofs. For
$rin[1,3]$, we quantize the NS nonlinearity
$(oldsymbol{y}cdot
abla)oldsymbol{y}$ to establish the existence and
uniqueness results, while for $rin[3,infty)$ ($2etamugeq1$ for $r=3$), we
handle the NS nonlinearity by the nonlinear damping term
$|oldsymbol{y}|^{r-1}oldsymbol{y}$. Finally, we discuss the applications of
the above developed theory in feedback control problems like flow invariance,
time optimal control and stabilization. | Source: | arXiv, 2301.01527 | Services: | Forum | Review | PDF | Favorites |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
|
| |
|
|
|