arXiv subjects

Manika Bag

Publications and source records attributed to Manika Bag.

8 recordsLinked to original sources

arXiv2026

Local controllability of the Cahn-Hilliard-Burgers' equation around certain steady states

In this article we study the local controllability of the one-dimensional Cahn-Hilliard-Navier-Stokes equation, that is Cahn-Hilliard-Burgers' equation, around a certain steady state using a localized interior control acting only in the concentration equation. To do it, we first linearize the nonlinear equation around the steady state. The linearized system turns out to be a system coupled between second order and fourth order parabolic equations and the control acts in the fourth order parabolic equation. The null controllability of the linearized system is obtained by a duality argument proving an observability inequality. To prove the observability inequality, a new Carleman inequality for the coupled system is derived. Next, using the source term method, it is shown that the null controllability of the linearized system with non-homogeneous terms persists provided the non-homogeneous terms satisfy certain estimates in a suitable weighted space. Finally, using a Banach fixed point theorem in a suitable weighted space, the local controllability of the nonlinear system is obtained.

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arXiv2025

Optimal Control of a Higher-Order Cahn-Hilliard Equation Coupled with Brinkman Equation

In this work, we investigate optimal control of a Brinkman equation couple with sixth-order Cahn-Hilliard equation. The Cahn-Hilliard equation is endowed with a source term accounting for mass exchange and the velocity equation contains a non divergence-free forcing term, which act as distributed control variable. We consider the aforementioned system with constant mobility, viscosity and nonlinearity of double-well shape is regular. The cost functional of the optimal control problem contains a nondifferentiable term like the $L^1$-norm with sparsity constant $\kappa$, which leads to sparsity of optimal controls. We study the first order necessary optimality condition for both the case $\kappa=0$ and $\kappa>0.$ When the cost functional is differentiable, first order necessary optimality conditions are characterized by Lagrange multiplier method and for nondifferentiable case we have used the idea of Casas and Tr\"oltzsch from the paper (Math. control Relat. Fields, 10(3):527-546, 2020).

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arXiv2025

Boundary Stabilizability of Generalized Burgers-Huxley Equation with Memory

In this paper, we study a generalized Burgers-Huxley equation with memory, subject to nonhomogeneous Dirichlet boundary conditions. We construct a linear, finite-dimensional Dirichlet boundary feedback controller aimed at stabilizing the stationary solution corresponding to the homogeneous boundary condition. This controller is designed using eigenfunctions of the Laplace operator. We begin by analyzing the stabilization of a linear system under the proposed feedback law. Subsequently, we demonstrate that the same controller also stabilizes the full nonlinear system by applying the Banach fixed point theorem. Finally, we provide a remark on the stabilization of the generalized Burgers-Huxley equation with memory around the zero solution under nonhomogeneous Dirichlet boundary conditions.

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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 $\Omega\subset\mathbb{R}^d$, $d\in\{1,2,3\}$, 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.

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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.

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arXiv2024

Well-posedness of three-dimensional Damped Cahn-Hilliard-Navier-Stokes Equations

This paper presents a mathematical analysis of the evolution of a mixture of two incompressible, isothermal fluids flowing through a porous medium in a three dimensional bounded domain. The model is governed by a coupled system of convective Brinkman Forchheimer equations and the Cahn Hilliard equation, considering a regular potential and non degenerate mobility. We first establish the existence of a Leray Hopf weak solution for the coupled system when the absorption exponent r greater than or equal to 1. Additionally, we prove that every weak solution satisfies the energy equality for greater than 3. This further leads to the uniqueness of weak solutions in three-dimensional bounded domains, subject to certain restrictions on the viscosity and the Forchheimer coefficient in the critical case r=3. Moreover, we provide an alternative simplified proof for the uniqueness of weak solutions for r greater than or equal to 3 that holds without imposing any restrictions on viscosity or Forchheimer coefficient. Similar results are also obtained for the case of degenerate mobility and singular potential.

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arXiv2023

Optimal boundary control for the Cahn-Hilliard-Navier-Stokes Equations

In this work, we study an optimal boundary control problem for a Cahn - Hilliard -Navier-Stokes (CHNS) system in a two dimensional bounded domain. The CHNS system consists of a Navier-Stokes equation governing the fluid velocity field coupled with a convective Cahn - Hilliard equation for the relative concentration of the fluids. An optimal control problem is formulated as the minimization of a cost functional subject to the controlled CHNS system where the control acts on the boundary of the Navier-Stokes equations. We first prove that there exists an optimal boundary control. Then we establish that the control-to-state operator is Frechet differentiable and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint system.

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arXiv2022

On Cahn-Hilliard-Navier-Stokes equations with Nonhomogeneous Boundary

The evolution of two isothermal, incompressible, immiscible fluids in a bounded domain is governed by Cahn-Hilliard-Navier-Stokes equations (CHNS System). In this work, we study the well-posedness results for the CHNS system with nonhomogeneous boundary condition for the velocity equation. We obtain the existence of global weak solutions in the two-dimensional bounded domain. We further prove the continuous dependence of the solution on initial conditions and boundary data that will provide the uniqueness of the weak solution. The existence of strong solutions is also established in this work. Furthermore, we show that in the two-dimensional case, each global weak solution converges to a stationary solution.

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