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Approximate controllability and Irreducibility of the transition semigroup associated with Convective Brinkman-Forchheimer extended Darcy Equations

Sagar Gautam, Manil T. Mohan

Abstract

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.

Approximate controllability and Irreducibility of the transition semigroup associated with Convective Brinkman-Forchheimer extended Darcy Equations

Abstract

In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a -dimensional torus : \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 , , , with and is the control. For the super critical () and critical ( with ) cases, we first show the approximate controllability of the above system in the usual energy space (divergence-free 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 a density argument and monotonicity properties of the linear and nonlinear operators.
Paper Structure (17 sections, 9 theorems, 176 equations)

This paper contains 17 sections, 9 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. The Model
  3. Literature review
  4. The deterministic problem.
  5. The stochastic problem.
  6. The control problem.
  7. Irreducibility
  8. Difficulties, novelties and approaches
  9. Organization of the paper
  10. Functional Settings and Preliminaries
  11. Function spaces
  12. Linear operator
  13. Bilinear operator
  14. Nonlinear operator
  15. Approximate controllability
  16. ...and 2 more sections

Key Result

Proposition 2.1

For $d=2,3$ with $r>3$, define the operator $\Gamma(\cdot):\mathrm{D}(\mathcal{M})\to\mathbb{H}$ by where $\mathrm{D}(\Gamma)=\{\boldsymbol{y}\in\mathbb{V}\cap\widetilde{\mathbb{L}}^{r+1}:\mathrm{A}\boldsymbol{y}\in\mathbb{H}\}.$ Then, $\mathrm{D}(\Gamma)=\mathrm{D}(\mathrm{A})$ and $\Gamma$ is quasi-$m$-accretive in $\mathbb{H}$ for sufficiently large $\kappa\geq\eta_1+\eta_2+\eta_3$ such that

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 11 more