Local exact controllability to the trajectories of the convective Brinkman-Forchheimer equations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910324166230016 |
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| author | Kumar, Pardeep Mohan, Manil T. |
| author_facet | Kumar, Pardeep Mohan, Manil T. |
| contents | In this article, we discuss the local exact controllability to trajectories of the following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) defined in a bounded domain $Ω
\subset\mathbb{R}^d$ ($d=2,3$) with smooth boundary:
\begin{align*}
\frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{2}\boldsymbol{u}+\nabla p=\boldsymbol{f}+\boldsymbol{\vartheta}, \ \ \ \nabla\cdot\boldsymbol{u}=0,
\end{align*}
where the control $\boldsymbol{\vartheta}$ is distributed in a subdomain $ω\subset Ω$, and the parameters $α,β,μ>0$ are constants. We first present global Carleman estimates and observability inequality for the adjoint problem of a linearized version of CBF equations by using a global Carleman estimate for the Stokes system. This allows us to obtain its null controllability at any time $T>0$. We then use the inverse mapping theorem to deduce local results concerning the exact controllability to the trajectories of CBF equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06335 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local exact controllability to the trajectories of the convective Brinkman-Forchheimer equations Kumar, Pardeep Mohan, Manil T. Analysis of PDEs Optimization and Control In this article, we discuss the local exact controllability to trajectories of the following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) defined in a bounded domain $Ω \subset\mathbb{R}^d$ ($d=2,3$) with smooth boundary: \begin{align*} \frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{2}\boldsymbol{u}+\nabla p=\boldsymbol{f}+\boldsymbol{\vartheta}, \ \ \ \nabla\cdot\boldsymbol{u}=0, \end{align*} where the control $\boldsymbol{\vartheta}$ is distributed in a subdomain $ω\subset Ω$, and the parameters $α,β,μ>0$ are constants. We first present global Carleman estimates and observability inequality for the adjoint problem of a linearized version of CBF equations by using a global Carleman estimate for the Stokes system. This allows us to obtain its null controllability at any time $T>0$. We then use the inverse mapping theorem to deduce local results concerning the exact controllability to the trajectories of CBF equations. |
| title | Local exact controllability to the trajectories of the convective Brinkman-Forchheimer equations |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2402.06335 |