arXiv subjects

Manil T. Mohan

Publications and source records attributed to Manil T. Mohan.

At least 91 records · Page 5Linked to original sources

arXiv2022

Wong-Zakai approximation and support theorem for 2D and 3D stochastic convective Brinkman-Forchheimer equations

In this work, we demonstrate the Wong-Zakai approximation results for two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations forced by Hilbert space valued Wiener noise on bounded domains. Even though the existence and uniqueness of a pathwise strong solution to SCBF equations is known, the existence of a unique solution to the approximating system is not immediate from the solvability results of SCBF equations, and we prove it by using Faedo-Galerkin approximation technique and monotonicity arguments. Moreover, as an application of the Wong-Zakai approximation, we obtain the support of the distribution of solutions to SCBF equations.

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arXiv2021

Approximate controllability of non-autonomous second order impulsive functional evolution equations in Banach spaces

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder's fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we verify our results to examine the approximate controllability of the non-autonomous wave equation with non-instantaneous impulses and finite delay in the application section.

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arXiv2021

$\mathbb{H}^1$-Random attractors for 2D stochastic convective Brinkman-Forchheimer equations in unbounded domains

The asymptotic behavior of solutions of two dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations in unbounded domains is discussed in this work (for example, Poincaré domains). We first prove the existence of $\mathbb{H}^1$-random attractors for the stochastic flow generated by 2D SCBF equations (for the absorption exponent $r\in[1,3]$) perturbed by an additive noise on Poincaré domains. Furthermore, we deduce the existence of a unique invariant measure in $\mathbb{H}^1$ for the 2D SCBF equations defined on Poincaré domains. In addition, a remark on the extension of these results to general unbounded domains is also discussed. Finally, for 2D SCBF equations forced by additive one-dimensional Wiener noise, we prove the upper semicontinuity of the random attractors, when the domain changes from bounded to unbounded (Poincaré).

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arXiv2021

Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise

The long time behavior of Wong-Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman-Forchheimer (CBF) equations with non-linear diffusion terms on bounded and unbounded ($\mathbb{R}^d$ for $d=2,3$) domains is discussed in this work. To establish the existence of random pullback attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail estimates are exploited to prove AC. In the literature, CBF equations are also known as \emph{Navier-Stokes equations (NSE) with damping}, and it is interesting to see that the modification in NSE by linear and nonlinear damping provides better results than that available for NSE. The presence of linear damping term helps to establish the results in the whole domain $\mathbb{R}^d$. The nonlinear damping term supports to obtain better results in 3D and also for a large class of nonlinear diffusion terms. Moreover, we prove the existence of a unique random pullback attractor for stochastic CBF equations with additive white noise. Finally, for additive as well as multiplicative noise case, we establish the convergence of solutions and upper semicontinuity of random pullback attractors for Wong-Zakai approximations of stochastic CBF equations towards the random pullback attractors for stochastic CBF equations when correlation time of colored noise converges to zero.

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arXiv2021

Well-posedness of an inverse problem for two and three dimensional convective Brinkman-Forchheimer equations with the final overdetermination

In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations: \begin{align*} \boldsymbol{u}_t-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{F}:=\boldsymbol{f} g, \ \ \ \nabla\cdot\boldsymbol{u}=0, \end{align*} in bounded domains $Ω\subset\mathbb{R}^d$ ($d=2,3$) with smooth boundary, where $α,β,μ>0$ and $r\in[1,\infty)$. The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function $\boldsymbol{u}$, the pressure field $p$ and the vector-valued function $\boldsymbol{f}$. We have proved the well-posedness result (existence, uniqueness and stablility) for the inverse problem for the 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for $r\geq 1$ in two dimensions and for $r \geq 3$ in three dimensions. The global solvability results available in the literature helped us to obtain the uniqueness and stability results for the model with fast growing nonlinearities.

math.AP↗

arXiv2021

Large deviation principle for occupation measures of stochastic generalized Burgers-Huxley equation

The present work deals with the global solvability as well as asymptotic analysis of stochastic generalized Burgers-Huxley (SGBH) equation perturbed by space-time white noise in a bounded interval of $\mathbb{R}$. We first prove the existence of unique mild as well as strong solution to SGBH equation and then obtain the existence of an invariant measure. Later, we establish two major properties of the Markovian semigroup associated with the solutions of SGBH equation, that is, irreducibility and strong Feller property. These two properties guarantees the uniqueness of invariant measures and ergodicity also. Then, under further assumptions on the noise coefficient, we discuss the ergodic behavior of the solution of SGBH equation by providing a Large Deviation Principle (LDP) for the occupation measure for large time (Donsker-Varadhan), which describes the exact rate of exponential convergence.

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arXiv2021

Approximate controllability of non-instantaneous impulsive fractional evolution equations of order $1<α<2$ with state-dependent delay in Banach spaces

The current article examines the approximate controllability problem for non-instantaneous impulsive fractional evolution equations of order $1<α<2$ with state-dependent delay in separable reflexive Banach spaces. In order to establish sufficient conditions for the approximate controllability of our problem, we first formulate the linear-regulator problem and obtain the optimal control in feedback form. By using this optimal control, we deduce the approximate controllability of the linear fractional control system of order $1<α<2$. Further, we derive sufficient conditions for the approximate controllability of the nonlinear problem. Finally, we provide a concrete example to validate the efficiency of the derived results.

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arXiv2021

Existence and upper semicontinuity of random pullback attractors for 2D and 3D non-autonomous stochastic convective Brinkman-Forchheimer equations on whole domain

In this work, we analyze the long time behavior of 2D as well as 3D convective Brinkman-Forchheimer (CBF) equations and its stochastic counter part with non-autonomous deterministic forcing term in $\mathbb{R}^d$ $ (d=2, 3)$: $$\frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\quad \nabla\cdot\boldsymbol{u}=0,$$ where $r\geq1$. We prove the existence of a unique global pullback attractor for non-autonomous CBF equations, for $d=2$ with $r\geq1$, $d=3$ with $r>3$ and $d=r=3$ with $2βμ\geq1$. For the same cases, we show the existence of a unique random pullback attractor for non-autonomous stochastic CBF equations with multiplicative white noise. Finally, we establish the upper semicontinuity of the random pullback attractor, that is, the random pullback attractor converges towards the global pullback attractor when the noise intensity approaches to zero. Since we do not have compact Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of the solution is proved by the method of energy equations given by Ball. For the case of Navier-Stokes equations defined on $\mathbb{R}^d$, such results are not available and the presence of Darcy term $α\boldsymbol{u}$ helps us to establish the above mentioned results for CBF equations.

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arXiv2021

A local in time existence and uniqueness result of an inverse problem for the Kelvin-Voigt fluids

In this paper, we consider an inverse problem for three dimensional viscoelastic fluid flow equations, which arises from the motion of Kelvin-Voigt fluids in bounded domains (a hyperbolic type problem). This inverse problem aims to reconstruct the velocity and kernel of the memory term simultaneously, from the measurement described as the integral over determination condition. By using the contraction mapping principle in an appropriate space, a local in time existence and uniqueness result for the inverse problem of Kelvin-Voigt fluids are obtained. Furthermore, using similar arguments, a global in time existence and uniqueness results for an inverse problem of Oseen type equations are also achieved.

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arXiv2021

Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in periodic domains: $$\frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ where $r\geq1$. We prove that the global attractor of the above system is a singleton under small forcing intensity ($r\geq 1$ for $n=2$ and $r\geq 3$ for $n=3$ with $2βμ\geq 1$ for $r=n=3$). After perturbing the above system with additive or multiplicative white noise, the random attractor does not have a singleton structure. But we obtain that the random attractor for 2D stochastic CBF equations with additive and multiplicative white noise converges towards the deterministic singleton attractor for $1\leq r\leq 2$ and $1\leq r<\infty$, respectively, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). Interestingly in the case of 3D stochastic CBF equations perturbed by multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for $3\leq r\leq 5$ ($2βμ\geq 1$ for $r=3$), when the coefficient of random perturbation converges to zero.

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arXiv2021

$\mathbb{L}^p$-solutions of deterministic and stochastic convective Brinkman-Forchheimer equations

In the first part of this work, we establish the existence and uniqueness of a local mild solution to the deterministic convective Brinkman-Forchheimer (CBF) equations defined on the whole space, by using properties of the heat semigroup and fixed point arguments based on an iterative technique. The second part is devoted for establishing the existence and uniqueness of a pathwise mild solution upto a random time to the stochastic CBF equations perturbed by Lévy noise by exploiting the contraction mapping principle. We also discuss the local solvability of the stochastic CBF equations subjected to fractional Brownian noise.

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arXiv2021

Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations

The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. In this work, we consider some distributed optimal control problems like total energy minimization, minimization of enstrophy, etc governed by the two dimensional CBF equations with the absorption exponent $r=1,2$ and $3$. We show the existence of an optimal solution and the first order necessary conditions of optimality for such optimal control problems in terms of the Euler-Lagrange system. Furthermore, for the case $r=3$, we show the second order necessary and sufficient conditions of optimality. We also investigate an another control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is 2D CBF equations, using optimal control techniques.

math.OC↗

arXiv2021

Large deviation principle for occupation measures of two dimensional stochastic convective Brinkman-Forchheimer equations

The present work is concerned about two-dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations perturbed by a white noise (non degenerate) in smooth bounded domains in $\R^{2}$. We establish two important properties of the Markov semigroup associated with the solutions of 2D SCBF equations (for the absorption exponent $r=1,2,3$), that is, irreducibility and strong Feller property. These two properties implies the uniqueness of invariant measures and ergodicity also. Then, we discuss about the ergodic behavior of 2D SCBF equations by providing a Large Deviation Principle (LDP) for the occupation measure for large time (Donsker-Varadhan), which describes the exact rate of exponential convergence.

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arXiv2020

Large time behavior of the deterministic and stochastic 3D convective Brinkman-Forchheimer equations in periodic domains

The large time behavior of the deterministic and stochastic three dimensional convective Brinkman-Forchheimer (CBF) equations for $r\geq3$ ($r>3$, for any $μ$ and $β$, and $r=3$ for $2βμ\geq1$), in periodic domains is carried out in this work. Our first goal is to prove the existence of global attractors for the 3D deterministic CBF equations. Then, we show the existence of random attractors for the 3D stochastic CBF equations perturbed by small additive smooth noise. Finally, we establish the upper semicontinuity of random attractor for the 3D stochastic CBF equations (stability of attractors), when the coefficient of random perturbation approaches to zero.

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arXiv2020

Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equations

This work is concerned about the asymptotic behavior of the solutions of the two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations driven by white noise with nonlinear diffusion terms. We prove the existence and uniqueness of weak pullback mean random attractors for the 2D SCBF equations (for $r\geq1$) as well as 3D SCBF equations (for $r>3$, any $μ,β>0$ and for $r=3$, $2μβ\geq1$) in Bochner spaces, when the diffusion terms are Lipschitz nonlinear functions. Furthermore, we establish the existence of weak pullback mean random attractors for a class of locally monotone stochastic partial differential equations.

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arXiv2020

Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity

In this work, we carry out the asymptotic analysis of the two dimensional convective Brinkman-Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and Poincaré domains (using asymptotic compactness property). In Poincaré domains, for $r=1,2$ and $3$, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors for the 2D CBF equations. We consider an expanding sequence of simply connected, bounded and smooth subdomains $Ω_m$ of the Poincaré domain $Ω$ such that $Ω_m\toΩ$ as $m\to\infty$. If $\mathscr{A}_m$ and $\mathscr{A}$ are the global attractors of the 2D CBF equations corresponding to $Ω$ and $Ω_m$, respectively, then we show that for large enough $m$, the global attractor $\mathscr{A}_m$ enters into any neighborhood $\mathcal{U}(\mathscr{A})$ of $\mathscr{A}$. The presence of Darcy term in the CBF equations helps us to obtain the above mentioned results in general unbounded domains also. Finally, we discuss about the quasi-stability property of the semigroup associated with the 2D CBF equations in bounded domains and establish the existence of finite fractal dimensional global as well as exponential attractors for $r\in[1,\infty)$.

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arXiv2020

Wentzell-Freidlin Large Deviation Principle for the stochastic convective Brinkman-Forchheimer equations

This work addresses some asymptotic behavior of solutions to the stochastic convective Brinkman-Forchheimer (SCBF) equations perturbed by multiplicative Gaussian noise in bounded domains. Using a weak convergence approach of Budhiraja and Dupuis, we establish the Laplace principle for the strong solution to the SCBF equations in a suitable Polish space. Then, the Wentzell-Freidlin large deviation principle is derived using the well known results of Varadhan and Bryc. The large deviations for short time are also considered in this work. Furthermore, we study the exponential estimates on certain exit times associated with the solution trajectory of the SCBF equations. Using contraction principle, we study these exponential estimates of exit times from the frame of reference of Freidlin-Wentzell type large deviations principle. This work also improves several LDP results available in the literature for the tamed Navier-Stokes equations as well as Navier-Stokes equations with damping in bounded domains.

math.PR↗

arXiv2020

Existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in bounded domains

In this work, we discuss the large time behavior of the solutions of the two dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations in bounded domains. Under the functional setting $\V\hookrightarrow\H\hookrightarrow\V'$, where $\H$ and $\V$ are appropriate separable Hilbert spaces and the embedding $\V\hookrightarrow\H$ is compact, we establish the existence of random attractors in $\H$ for the stochastic flow generated by the 2D SCBF equations perturbed by small additive noise. We prove the upper semicontinuity of the random attractors for the 2D SCBF equations in $\H$, when the coefficient of random term approaches zero. Moreover, we obtain the existence of random attractors in a more regular space $\V$, using the pullback flattening property. The existence of random attractors ensures the existence of invariant compact random set and hence we show the existence of an invariant measure for the 2D SCBF equations.

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