arXiv subjects

Debayan Maity

Publications and source records attributed to Debayan Maity.

5 recordsLinked to original sources

arXiv2022

Control of the Stefan problem in a periodic box

In this paper we consider the one-phase Stefan problem with surface tension, set in a two-dimensional strip-like geometry, with periodic boundary conditions respect to the horizontal direction $x_1\in\mathbb{T}$. We prove that the system is locally null-controllable in any positive time, by means of a control supported within an arbitrary open and non-empty subset. We proceed by a linear test and duality, but quickly find that the linearized system is not symmetric and the adjoint has a dynamic coupling between the two states through the (fixed) boundary. Hence, motivated by a Fourier decomposition with respect to $x_1$, we consider a family of one-dimensional systems and prove observability results which are uniform with respect to the Fourier frequency parameter. The latter results are also novel, as we compute the full spectrum of the underlying operator for the non-zero Fourier modes. The zeroth mode system, on the other hand, is seen as a controllability problem for the linear heat equation with a finite-dimensional constraint. The complete observability of the adjoint is derived by using a Lebeau-Robbiano strategy, and the local controllability of the nonlinear system is then shown by combining an adaptation of the source term method introduced in \cite{tucsnak_burgers} and a Banach fixed point argument. Numerical experiments motivate several challenging open problems, foraying even beyond the specific setting we deal with herein.

math.OC↗

arXiv2020

Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier-Stokes-Fourier fluid and a damped plate equation

The article is devoted to the mathematical analysis of a fluid-structure interaction system where the fluid is compressible and heat conducting and where the structure is deformable and located on a part of the boundary of the fluid domain. The fluid motion is modeled by the compressible Navier-Stokes-Fourier system and the structure displacement is described by a structurally damped plate equation. Our main results are the existence of strong solutions in an $L^p-L^q$ setting for small time or for small data. Through a change of variables and a fixed point argument, the proof of the main results is mainly based on the maximal regularity property of the corresponding linear systems. For small time existence, this property is obtained by decoupling the linear system into several standard linear systems whereas for global existence and for small data, the maximal regularity property is proved by showing that the corresponding linear coupled {\em fluid-structure} operator is $\mathcal{R}-$sectorial.

math.AP↗

arXiv2017

$L^p$-$L^q$ Maximal Regularity for some Operators Associated with Linearized Incompressible Fluid-Rigid Body Problems

We study an unbounded operator arising naturally after linearizing the system modelling the motion of a rigid body in a viscous incompressible fluid. We show that this operator is $\mathcal{R}$ sectorial in $L^q$ for every $q\in (1,\infty)$, thus it has the maximal $L^p$-$L^q$ regularity property. Moreover, we show that the generated semigroup is exponentially stable with respect to the $L^q$ norm. Finally, we use the results to prove the global existence for small initial data, in an $L^p$-$L^q$ setting, for the original nonlinear problem.

math.AP↗

arXiv2017

Mathematical Analysis of the Motion of a Rigid Body in a Compressible Navier-Stokes-Fourier Fluid

We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier-Stokes-Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an L p-L q setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the R-sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.

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arXiv2014

Some Controllability Results for Linearized Compressible Navier-Stokes System

In this article, we study the null controllability of linearized compressible Navier-Stokes system in one and two dimension. We first study the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state. We prove that the linearized system around $(\bar \rho,0,\bar \theta)$, with $\bar \rho > 0,$ $ \bar \theta > 0$ is not null controllable by localized interior control or by boundary control. But the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition. We also prove that the the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state $(\bar \rho,\bar v ,\bar \theta)$, with $\bar \rho > 0,$ $\bar v > 0,$ $\bar \theta > 0$ is not null controllable by localized interior control or by boundary control for small time $T.$ Next we consider two-dimensional compressible Navier-Stokes system for barotropic fluid linearized around a constant steady state $(\bar \rho, {\bf 0}).$ We prove that this system is also not null controllable by localized interior control.

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