Approximate controllability and Irreducibility of the transition semigroup associated with Convective Brinkman-Forchheimer extended Darcy Equations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Gautam, Sagar Mohan, Manil T. |
| author_facet | Gautam, Sagar Mohan, Manil T. |
| contents | In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a $d$-dimensional torus $\mathbb{T}^d$:
\begin{align*}
\frac{\partial\boldsymbol{y}}{\partial t}-μΔ\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+α\boldsymbol{y}+β\vert \boldsymbol{y}\vert^{r-1}\boldsymbol{y}+γ\vert \boldsymbol{y}\vert ^{q-1}\boldsymbol{y}+\nabla p=\boldsymbol{g}+\boldsymbol{u},\ \nabla\cdot\boldsymbol{y}=0,
\end{align*}
where $d\in\{2,3\}$, $μ,α,β>0$, $γ\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$ and $\boldsymbol{u}$ is the control. For the super critical ($r>3$) and critical ($r=3$ with $2βμ>1$) cases, we first show the approximate controllability of the above system in the usual energy space (divergence-free $\mathbb{L}^2(\mathbb{T}^d)$ space). As an application of the approximate controllability result, we establish the irreducibility of the transition semigroup associated with stochastic CBFeD system perturbed by non-degenerate Gaussian noise in the usual energy space by exploiting the regularity of solutions, smooth approximation of the multi-valued map $\mathrm{sgn}(\cdot)$ a density argument and monotonicity properties of the linear and nonlinear operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_19363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximate controllability and Irreducibility of the transition semigroup associated with Convective Brinkman-Forchheimer extended Darcy Equations Gautam, Sagar Mohan, Manil T. Probability Analysis of PDEs Optimization and Control In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a $d$-dimensional torus $\mathbb{T}^d$: \begin{align*} \frac{\partial\boldsymbol{y}}{\partial t}-μΔ\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+α\boldsymbol{y}+β\vert \boldsymbol{y}\vert^{r-1}\boldsymbol{y}+γ\vert \boldsymbol{y}\vert ^{q-1}\boldsymbol{y}+\nabla p=\boldsymbol{g}+\boldsymbol{u},\ \nabla\cdot\boldsymbol{y}=0, \end{align*} where $d\in\{2,3\}$, $μ,α,β>0$, $γ\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$ and $\boldsymbol{u}$ is the control. For the super critical ($r>3$) and critical ($r=3$ with $2βμ>1$) cases, we first show the approximate controllability of the above system in the usual energy space (divergence-free $\mathbb{L}^2(\mathbb{T}^d)$ space). As an application of the approximate controllability result, we establish the irreducibility of the transition semigroup associated with stochastic CBFeD system perturbed by non-degenerate Gaussian noise in the usual energy space by exploiting the regularity of solutions, smooth approximation of the multi-valued map $\mathrm{sgn}(\cdot)$ a density argument and monotonicity properties of the linear and nonlinear operators. |
| title | Approximate controllability and Irreducibility of the transition semigroup associated with Convective Brinkman-Forchheimer extended Darcy Equations |
| topic | Probability Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2402.19363 |