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Manil T. Mohan

Publications and source records attributed to Manil T. Mohan.

At least 55 records · Page 3Linked to original sources
arXiv2025

Feedback Stabilizability of a generalized Burgers-Huxley equation with kernel around a non-constant steady state

In this article, we investigate a generalized Burgers-Huxley equation with a smooth kernel defined in a bounded domain $Ω\subset\mathbb{R}^d$, $d\in\{1,2,3\}$, focusing on feedback stabilizability around a non-constant steady state. Initially, employing the Banach fixed point theorem, we establish the local existence and uniqueness of a strong solution, which is subsequently extended globally using an energy estimate. To analyze stabilizability, we linearize the model around a non-constant steady state and examine the stabilizability of the principal system. For the principal system, we develop a feedback control operator by solving an appropriate algebraic Riccati equation. This allows for the construction of both finite and infinite-dimensional feedback operators. By applying this feedback operator and establishing necessary regularity results, we utilize the Banach fixed point theorem to demonstrate the stabilizability of the entire system. Furthermore, we also explore the stabilizability of the model problem around the zero steady state and validate our findings through numerical simulations using the finite element method for both zero and non-constant steady states.

math.OC↗
arXiv2025

Stabilizability of 2D and 3D Navier-Stokes equations with memory around a non-constant steady state

In this article, we investigate the stabilizability of the two- and three-dimensional Navier-Stokes equations with memory effects around a non-constant steady state using a localized interior control. The system is first linearized around a non-constant steady state and then reformulated into a coupled system by introducing a new variable to handle the integral term. Due to the presence of variable coefficients in the linear operator, the rigorous computation of eigenvalues and eigenfunctions becomes infeasible. Therefore, we concentrate on the principal operator, and investigate its analyticity and spectral properties. We establish a feedback stabilization result for the principal system, ensuring a specific decay rate. Using the feedback operator derived from this analysis, we extend the approach to the full system, constructing a closed-loop system. By proving a suitable regularity result and applying a fixed-point argument, we ultimately demonstrate the stabilizability of the full system. We also discuss the stabilizability of the corresponding vorticity equation around a non-constant steady state.

math.AP↗
arXiv2024

On the convective Brinkman-Forchheimer equations

The convective Brinkman--Forchheimer equations or the Navier--Stokes equations with damping in bounded or periodic domains $\subset\mathbb{R}^d$, $2\leq d\leq 4$ are considered in this work. The existence and uniqueness of a global weak solution in the Leray-Hopf sense satisfying the energy equality to the system: $$\partial_t\boldsymbol{u}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ (for all values of $β>0$ and $μ>0$, whenever the absorption exponent $r>3$ and $2βμ\geq 1$, for the critical case $r=3$) is proved. We exploit the monotonicity as well as the demicontinuity properties of the linear and nonlinear operators and the Minty-Browder technique in the proofs. Finally, we discuss the existence of global-in-time strong solutions to such systems in periodic domains.

math.AP↗
arXiv2024

Kolmogorov equations for 2D stochastic convective Brinkman-Forchheimer equations: Analysis and Applications

In this work, we consider the following 2D stochastic convective Brinkman-Forchheimer (SCBF) equations in a bounded smooth domain $\mathcal{O}$: \begin{align*} \mathrm{d}\boldsymbol{u}+\left[-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p\right]\mathrm{d}t=\sqrt{\mathrm{Q}}\mathrm{W}, \ \nabla\cdot\boldsymbol{u}=0, \end{align*} where $μ,α,β>0$, $r\in\{1,2,3\}$, $\mathrm{Q}$ is a non-negative operator of trace class, $\mathrm{W}$ is a cylindrical Wiener process in a Hilbert space $\mathbb{H}$. Under the following assumption on the viscosity co-efficient $μ$ and the Darcy co-efficient $α$: for some positive constant $γ_1$, \begin{equation*} μ(μ+α)^2>γ_1\max\{4\mathrm{Tr}(\mathrm{Q}),\mathrm{Tr}(\mathrm{A}^{2δ}\mathrm{Q})\}, \end{equation*} where $\mathrm{A}$ is the Stokes operator and $δ\in(0,\frac{1}{2})$, our primary goal is to solve the corresponding Kolmogorov equation in the space $\mathbb{L}^2(\mathbb{H};η),$ where $η$ is the unique invariant measure associated with 2D SCBF equations. Then, we establish the well-known ``carré du champs'' identity. Some sharp estimates on the derivatives of the solution constitute the key component of the proofs. We take into consideration two control problems from the application point of view. The first is an infinite horizon control problem for which we establish the existence of a solution for the Hamilton-Jacobi-Bellman equation associated with it. Finally, by exploiting $m$-accretive theory, we demonstrate the existence of a unique solution for an obstacle problem associated with the Kolmogorov operator corresponding to the stopping-time problem for 2D SCBF equations.

math.OC↗
arXiv2024

Random dynamics of solutions for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded Poincaré domains

In this article, we consider a novel version of three-dimensional (3D) globally modified Navier-Stokes (GMNS) system introduced by [Caraballo et. al., Adv. Nonlinear Stud. (2006), 6:411-436], which is very significant from the perspective of deterministic as well as stochastic partial differential equations. Our focus is on examining a stochastic version of the suggested 3D GMNS equations that are perturbed by an infinite-dimensional additive noise. We can consider a rough additive noise (Lebesgue space valued) with this model, which is not appropriate to consider with the system presented in [Caraballo et. al., Adv. Nonlinear Stud. (2006), 6:411-436]. One of the technical problems associated with the rough noise is overcome by the use of the corresponding Cameron-Martin (or reproducing kernel Hilbert) space. This article aims to accomplish three objectives. Firstly, we establish the existence and uniqueness of weak solutions (in the analytic sense) of the underlying stochastic system using a Doss-Sussman transformation and a primary ingredient Minty-Browder technique. Secondly, we demonstrate the existence of random attractors for the underlying stochastic system in the natural space of square integrable divergence-free functions. Finally, we show the existence of an invariant measure for the underlying stochastic system for any viscosity coefficient $ ν> 0 $ and uniqueness of invariant measure for sufficiently large $ν$ by using the exponential stability of solutions. A validation of the proposed version of 3D GMNS equations has also been discussed in the appendix by establishing that the sequence of weak solutions of 3D GMNS equations converges to a weak solution of 3D Navier-Stokes equations as the modification parameter goes to infinity.

math.PR↗
arXiv2024

Central limit theorem and moderate deviation principle for the stochastic generalized Burgers-Huxley equation with multiplicative noise

In this work, we investigate the Central Limit Theorem (CLT) and Moderate Deviation Principle (MDP) for the stochastic generalized Burgers-Huxley (SGBH) equation with multiplicative Gaussian noise. The SGBH equation is a diffusion-convection-reaction type equation which consists a nonlinearity of polynomial order, and we take into account of an infinite-dimensional noise having a coefficient that has linear growth. We first prove the CLT which allows us to establish the convergence of the distribution of the solution to a re-scaled SGBH equation to a desired distribution function. Furthermore, we extend our asymptotic analysis by investigating the MDP for the SGBH equation. Using the weak convergence method, we establish the MDP and derive the corresponding rate function.

math.PR↗
arXiv2024

Blow-up estimates for a system of semilinear SPDEs driven by mixed fractional Brownian motions

In this paper, we obtain the existence and finite-time blow-up for the solution to a system of semilinear stochastic partial differential equations driven by a combination of Brownian and fractional Brownian motions. Under suitable assumptions, lower and upper bounds for the finite-time blow-up solution are obtained. We provide sufficient conditions for the existence of a global weak solution to the system. Further, a lower bound for the probability of the finite-time blow-up solution of the considered system is provided by using Malliavin calculus.

math.PR↗
arXiv2024

Existence of global and explosive mild solutions of fractional reaction-diffusion system of semilinear SPDEs with fractional noise

In this paper, we investigate the existence and finite-time blow-up for the solution of a reaction-diffusion system of semilinear stochastic partial differential equations (SPDEs) subjected to a two-dimensional fractional Brownian motion given by \begin{eqnarray*} du_{1}(t,x)&=&\left[ Δ_αu_{1}(t,x)+γ_{1}u_{1}(t,x)+u^{1+β_{1}}_{2}(t,x) \right]dt &\qquad \ \ +k_{11}u_{1}(t,x)dB^{H}_{1}(t)+k_{12}u_{1}(t,x)dB^{H}_{2}(t), du_{2}(t,x)&=&\left[ Δ_αu_{2}(t,x)+γ_{2}u_{2}(t,x)+u^{1+β_{2}}_{1}(t,x) \right]dt &\qquad \ \ +k_{21}u_{2}(t,x)dB^{H}_{1}(t)+k_{22}u_{2}(t,x)dB^{H}_{2}(t), \end{eqnarray*} for $x \in \mathbb{R}^{d},\ t \geq 0$, along with \begin{equation*} \begin{array}{ll} u_{i}(0,x)=f_{i}(x), &x \in \mathbb{R}^{d}, \nonumber \end{array} \end{equation*} where $Δ_α$ is the fractional power $-(-Δ)^{\fracα{2}}$ of the Laplacian, $0<α\leq 2$ and $β_{i}>0,\ γ_{i}>0$ and $k_{ij}\geq 0, i,j=1,2$ are constants. We provide sufficient conditions for the existence of a global weak solution. Under the assumption that $β_{1}\geq β_{2}>0$ with Hurst index $ 1/2 \leq H < 1,$ we obtain the blow-up times for an associated system of random partial differential equations in terms of an integral representation of exponential functions of Brownian motions. Moreover, we provide lower and upper bounds for the finite-time blow-up of the above system of SPDEs and obtain the upper bounds for the probability of non-explosive solutions to our considered system.

math.AP↗
arXiv2024

Wong-Zakai approximation for a class of SPDEs with fully local monotone coefficients and its application

In this article, we establish the \textsl{Wong-Zakai approximation} result for a class of stochastic partial differential equations (SPDEs) with fully local monotone coefficients perturbed by a multiplicative Wiener noise. This class of SPDEs encompasses various fluid dynamic models and also includes quasi-linear SPDEs, the convection-diffusion equation, the Cahn-Hilliard equation, and the two-dimensional liquid crystal model. It has been established that the class of SPDEs in question is well-posed, however, the existence of a unique solution to the associated approximating system cannot be inferred from the solvability of the original system. We employ a Faedo-Galerkin approximation method, compactness arguments, and Prokhorov's and Skorokhod's representation theorems to ensure the existence of a \textsl{probabilistically weak solution} for the approximating system. Furthermore, we also demonstrate that the solution is pathwise unique. Moreover, the classical Yamada-Watanabe theorem allows us to conclude the existence of a \textsl{probabilistically strong solution} (analytically weak solution) for the approximating system. Subsequently, we establish the Wong-Zakai approximation result for a class of SPDEs with fully local monotone coefficients. We utilize the Wong-Zakai approximation to establish the topological support of the distribution of solutions to the SPDEs with fully local monotone coefficients. Finally, we explore the physically relevant stochastic fluid dynamics models that are covered by this work's functional framework.

math.PR↗
arXiv2024

Blow-up of stochastic semilinear parabolic equations driven by Lévy noise

The blow-up phenomena of stochastic semilinear parabolic equations with additive as well as linear multiplicative Lévy noises are investigated in this work. By suitably modifying the concavity method in the stochastic context, we establish the blow-up phenomena of such systems defined on bounded domains.

math.PR↗
arXiv2024

Existence and uniqueness of weak solutions for the generalized stochastic Navier-Stokes-Voigt equations

In this work, we consider the incompressible generalized Navier-Stokes-Voigt equations in a bounded domain $\mathcal{O}\subset\mathbb{R}^d$, $d\geq 2$, driven by a multiplicative Gaussian noise. The considered momentum equation is given by: \begin{align*} \mathrm{d}\left(\boldsymbol{u} - κΔ\boldsymbol{u}\right) = \left[\boldsymbol{f} +\operatorname{div} \left(-π\mathbf{I}+ν|\mathbf{D}(\boldsymbol{u})|^{p-2}\mathbf{D}(\boldsymbol{u})-\boldsymbol{u}\otimes \boldsymbol{u}\right)\right]\mathrm{d} t + Φ(\boldsymbol{u})\mathrm{d} \mathrm{W}(t). \end{align*} In the case of $d=2,3$, $\boldsymbol{u}$ accounts for the velocity field, $π$ is the pressure, $\boldsymbol{f}$ is a body force and the final term stay for the stochastic forces. Here, $κ$ and $ν$ are given positive constants that account for the kinematic viscosity and relaxation time, and the power-law index $p$ is another constant (assumed $p>1$) that characterizes the flow. We use the usual notation $\mathbf{I}$ for the unit tensor and $\mathbf{D}(\boldsymbol{u}):=\frac{1}{2}\left(\nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^{\top}\right)$ for the symmetric part of velocity gradient. For $p\in\big(\frac{2d}{d+2},\infty\big)$, we first prove the existence of a martingale solution. Then we show the pathwise uniqueness of solutions. We employ the classical Yamada-Watanabe theorem to ensure the existence of a unique probabilistic strong solution.Then we show the pathwise uniqueness of solutions. We employ the classical Yamada-Watanabe theorem to ensure the existence of a unique probabilistic strong solution.

math.PR↗
arXiv2024

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

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.

math.PR↗
arXiv2024

Lower and upper bounds for the explosion times of a system of semilinear SPDEs

In this paper, we obtain lower and upper bounds for the blow-up times to a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of explosion times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We also provide an estimate for the probability of the finite-time blow-up. With a suitable choice of parameters, the impact of the noise on the solution is investigated. The above-obtained results are also extended for semilinear SPDEs forced by two dimensional Brownian motions.

math.AP↗
arXiv2024

Local exact controllability to the trajectories of the convective Brinkman-Forchheimer equations

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.

math.AP↗
arXiv2024

Boundary control of generalized Korteweg-de Vries-Burgers-Huxley equation: Well-Posedness, Stabilization and Numerical Studies

A boundary control problem for the following generalized Korteweg-de Vries-Burgers-Huxley equation: $$u_t=νu_{xx}-μu_{xxx}-αu^δu_x+βu(1-u^δ)(u^δ-γ), \ x\in[0,1], \ t>0,$$ where $ν,μ,α,β>0,$ $δ\in[1,\infty)$, $γ\in(0,1)$ subject to Neumann boundary conditions is considered in this work. We first establish the well-posedness of the Neumann boundary value problem by an application of monotonicity arguments, the Hartman-Stampacchia theorem, the Minty-Browder theorem, and the Crandall-Liggett theorem. The additional difficulties caused by the third order linear term is successfully handled by proving a proper version of the Minty-Browder theorem. By using suitable feedback boundary controls, we demonstrate $\mathrm{L}^2$- and $\mathrm{H}^1$-stability properties of the closed-loop system for sufficiently large $ν>0$. The analytical conclusions from this work are supported and validated by numerical investigations.

math.AP↗
arXiv2023

Feedback stabilization of Convective Brinkman-Forchheimer Extended Darcy equations

In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a $d$-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 $d\in\{2,3\}$, $μ,α,β>0$, $γ\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$. We prove the exponential stabilization of CBFeD system by finite- and infinite-dimensional feedback controllers. The solvability of the controlled problem is achieved by using the abstract theory of $m$-accretive operators and density arguments. As an application of the above solvability result, by using infinite-dimensional feedback controllers, we demonstrate exponential stability results such that the solution preserves an invariance condition for a given closed and convex set. By utilizing the unique continuation property of controllability for finite-dimensional systems, we construct a finite-dimensional feedback controller which exponentially stabilizes CBFeD system locally, where the control is localized in a smaller subdomain. Furthermore, we establish the local exponential stability of CBFeD system via proportional controllers.

math.OC↗
arXiv2023

Analysis and Numerical Study of Boundary control of generalized Burgers-Huxley equation

In this work, a boundary control problem for the following generalized Burgers-Huxley (GBH) equation: $$u_t=νu_{xx}-αu^δu_x+βu(1-u^δ)(u^δ-γ), $$ where $ν,α,β>0,$ $1\leqδ<\infty$, $γ\in(0,1)$ subject to Neumann boundary conditions is analyzed. Using the Minty-Browder theorem, standard elliptic partial differential equations theory, the maximum principle and the Crandall-Liggett theorem, we first address the global existence of a unique strong solution to GBH equation. Then, for the boundary control problem, we prove that the controlled GBH equation (that is, the closed loop system) is exponentially stable in the $\mathrm{H}^1$-norm (hence pointwise) when the viscosity $ν$ is known (non-adaptive control). Moreover, we show that a damped version of GBH equation is globally asymptotically stable (in the $\mathrm{L}^2$-norm), when $ν$ is unknown (adaptive control). Using the Chebychev collocation method with the backward Euler method as a temporal scheme, numerical findings are reported for both the non-adaptive and adaptive situations, supporting and confirming the analytical results of both the controlled and uncontrolled systems.

math.AP↗
arXiv2023

Global existence and non-existence of weak solutions for non-local stochastic semilinear reaction-diffusion equations driven by a fractional noise

In the present paper, we study the existence and blow-up behavior to the following stochastic non-local reaction-diffusion equation: \begin{equation*} \left\{ \begin{aligned} du(t,x)&=\left[(Δ+γ) u(t,x)+\int_{D}u^{q}(t,y)dy -ku^{p}(t,x)+δu^{m}(t,x)\int_{D}u^{n}(t,y)dy \right]dt &\quad+ηu(t,x)dB^{H}(t), u(t,x)&=0, \ \ t>0, \ \ x\in \partial D, u(0,x)&=f(x) \geq 0, \ \ x\in D, \end{aligned} \right. \end{equation*} where $D\subset \mathbb{R}^{d}\ (d \geq 1)$ is a bounded domain with smooth boundary $\partial D$. Here, $k>0, γ, δ, η\geq 0$ and $p,q,n>1,\ m\geq 0$ with $m+n \geq q\geq p$. The initial data $f $ is a non-negative bounded measurable function in class $C^{2}$ which is not identically zero. Here, $\left\{ B^{H}(t) \right\}_{t \geq 0} $ is a one-dimensional fractional Brownian motion with Hurst parameter $\frac{1}{2} \leq H<1$ defined on a filtered probability space $\left( Ω, \mathcal{F}, (\mathcal{F}_{t})_{t \geq 0}, \mathbb{P} \right)$. First, we estimate a lower bound for the finite-time blow-up and by choosing a suitable initial data, we obtain the upper bound for the finite-time blow-up of the above equation. Next, we provide a sufficient condition for the global existence of a weak solution of the above equation. Further, we obtain the bounds for the probability of blow-up solution.

math.PR↗