Arnab Roy

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• Doctor of Philosophy
• Ikerbasque Researcher at Basque Center for Applied Mathematics

We study the feedback stabilization of the Boussinesq system in a two dimensional domain, with mixed boundary conditions. After ascertaining the precise loss of regularity of the solution in such models, we prove first Green’s formulas for functions belonging to weighted Sobolev spaces and then correctly define the underlying control system. This p...

In this work, we study a system coupling the incompressible Navier–Stokes equations in a cylindrical type domain with an elastic structure, governed by a damped shell equation, located at the lateral boundary of the domain occupied by the fluid. We prove the existence of a unique maximal strong solution.

In this article, we consider a fluid-structure interaction system where the fluid is viscous and com-pressible and where the structure is a part of the boundary of the fluid domain and is deformable. The fluid is governed by the barotropic compressible Navier-Stokes system, whereas the structure displacement is described by a wave equation. We show...

We consider the motion of a small rigid object immersed in a viscous compressible fluid in the 3-dimensional Eucleidean space. Assuming the object is a ball of a small radius ε we show that the behavior of the fluid is not influenced by the object in the asymptotic limit ε→0. The result holds for the isentropic pressure law p(ϱ)=aϱγ for any γ>32 un...

We consider several rigid bodies immersed in a viscous Newtonian fluid contained in a bounded domain in $R^3$. We introduce a new concept of dissipative weak solution of the problem based on a combination of the approach proposed by Judakov with a suitable form of energy inequality. We show that global--in--time dissipative solutions always exist a...

We consider the viscous, incompressible, nonlinear Boussinesq system. We study the existence and regularity of solutions to the Boussinesq system with nonhomogeneous boundary conditions for which the normal component for the velocity is not necessary equal to zero. We establish the existence of a global weak solutions to the three-dimensional insta...

This paper investigates for the first time a rigorous mathematical framework for a stochastic fluid-rigid body interaction problem. It is shown that the coupled problem of a rigid ball immersed in a Newtonian fluid filling the entire $\mathbb{R}^3$ and subject to additive noise in the fluid and rigid body equation as well as transport noise in the...

This article develops the hydrostatic Lagrangian approach to the compressible primitive equations. A fundamental aspect in the analysis is the investigation of the compressible hydrostatic Lamé and Stokes operators. Local strong well-posedness for large data and global strong well-posedness for small data are established under various assumptions o...

This article develops the hydrostatic Lagrangian approach to the compressible primitive equations. A fundamental aspect in the analysis is the investigation of the compressible hydrostatic Lamé and Stokes operators. Local strong well-posedness for large data and global strong well-posedness for small data are established under various assumptions o...

We consider a bounded domain Ω ⊂ R 3 and a rigid body S(t) ⊂ Ω moving inside a viscous compressible Newtonian fluid. We exploit the body's roughness to establish that the solid collides with its container within a finite time. We investigate the case when the boundary of the body is of C 1,α-regularity and show that collision can happen for some su...

This article considers fluid structure interaction describing the motion of a fluid contained in a porous medium. The fluid is modelled by Navier-Stokes equations and the coupling between fluid and the porous medium is described by the classical Beaver-Joseph or the Beaver-Joseph-Saffman interface condition. In contrast to previous work these condi...

This article considers fluid structure interaction describing the motion of a fluid contained in a porous medium. The fluid is modelled by Navier-Stokes equations and the coupling between fluid and the porous medium is described by the classical Beaver-Joseph or the Beaver-Joseph-Saffman interface condition. In contrast to previous work these condi...

We study the interaction between incompressible viscous fluids and multilayered elastic structures in a 3D/2D/3D framework, where a 3D fluid interacts with a 2D thin elastic layer, coupled to a 3D thick elastic solid. The system is driven by time-periodic boundary conditions involving Bernoulli pressure. We prove the existence of at least one time-...

In this article, the fluid-rigid body interaction problem of nematic liquid crystals described by the general Beris-Edwards $Q$-tensor model is studied. It is proved first that the total energy of this problem decreases in time. The associated mathematical problem is a quasilinear mixed-order system with moving boundary. After the transformation to...

We consider the interaction of a compressible fluid with a flexible plate in two space dimensions. The fluid is described by the Navier--Stokes equations in a domain that is changing in accordance with the motion of the structure. The displacement of the latter evolves according to a beam equation. Both are coupled through kinematic boundary condit...

We consider the motion of a large number of heavy particles in a Newtonian fluid occupying a bounded spatial domain. When we say "heavy", we mean a particle with a mass density that approaches infinity at an appropriate rate as its radius vanishes. We show that the collective effect of heavy particles on the fluid motion is similar to the Brinkman...

This article investigates the interaction of nematic liquid crystals modeled by a simplified Ericksen-Leslie model with a rigid body. It is shown that this problem is locally strongly well-posed, and that it also admits a unique, global strong solution for initial data close to constant equilibria. The proof of the global strong solution relies on...

In this paper we study a mathematical model describing the movement of a colloidal particle in a fixed, bounded three dimensional container filled with a nematic liquid crystal fluid. The motion of the fluid is governed by the Beris-Edwards model for nematohydrodynamics equations, which couples the incompressible Navier-Stokes equations with a para...

This article investigates the interaction of nematic liquid crystals modeled by a simplified Ericksen-Leslie model with a rigid body. It is shown that this problem is locally strongly well-posed, and that it also admits a unique, global strong solution for initial data close to constant equilibria. The proof of the global strong solution relies on...

We show that the collective effect of N rigid bodies (S n,N) N n=1 of diameters (r n,N) N n=1 immersed in an incompressible non-Newtonian fluid is negligible in the asymptotic limit N → ∞ as long as their total packing volume N n=1 r d n,N , d = 2, 3 tends to zero exponentially-N n=1 r d n,N ≈ A −N-for a certain constant A > 1. The result is rather...

This paper investigates the interaction of nematic liquid crystals modeled by a simplified Ericksen-Leslie model with a colloidal particle. It is shown that this problem is locally strongly well-posed, and that it also admits a unique global strong solution for initial data close to constant equilibria. The proof relies on the property of maximal r...

In this paper we study a mathematical model describing the movement of a colloidal particle in a fixed, bounded three dimensional container filled with a nematic liquid crystal fluid. The motion of the fluid is governed by the Beris–Edwards model for nematohydrodynamics equations, which couples the incompressible Navier-Stokes equations with a para...

We consider the motion of a compressible viscous fluid containing a moving rigid body confined to a planar domain Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\O...

In this paper, we consider the viscous, incompressible, nonlinear Boussinesq system in two and three spatial dimension. We study the existence and regularity of solutions to the Boussinesq system with nonhomogeneous boundary conditions for which the normal component for the velocity is not necessary equal to zero. We establish the existence of glob...

We consider the stabilizability of a fluid-structure interaction system where a rigid ball is moving in a viscous compressible fluid occupying a bounded domain. We have developed a control law (a switching feedback law) with one end of the spring and damper at the centre of the ball while the other end is not fixed but instead it jumps between a fi...

We consider a bounded domain Ω ⊂ R 3 and a rigid body S(t) ⊂ Ω moving inside a viscous compressible Newtonian fluid. We exploit the roughness of the body to show that the solid collides its container in finite time. We investigate the case when the boundary of the body is of C 1,α-regularity and show that collision can happen for some suitable rang...

We consider the motion of $N$ rigid bodies -- compact sets $(\mathcal{S}^1_\ep, \cdots, \mathcal{S}^N_\ep )_{\ep > 0}$ -- immersed in a viscous incompressible fluid contained in a {domain in} the Euclidean space $\mathbb{R}^d$, $d=2,3$. We show the fluid flow is not influenced by the presence of the bodies in the asymptotic limit as $\ep \to 0$...

In this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions...

We consider the motion of a small rigid object immersed in a viscous compressible fluid in the 3-dimensional Eucleidean space. Assuming the object is a ball of a small radius $\varepsilon$ we show that the behavior of the fluid is not influenced by the object in the asymptotic limit $\varepsilon \to 0$. The result holds for the isentropic pressure...

We consider the motion of compressible Navier-Stokes fluids with the hard sphere pressure law around a rigid obstacle when the velocity and the density at infinity are non zero. This kind of pressure model is largely employed in various physical and industrial applications. We prove the existence of weak solution to the system in the exterior domai...

We consider the motion of a compressible viscous fluid containing a moving rigid body confined to a planar domain Ω ⊂ R 2. The main result states that the influence of the body on the fluid is negligible if (i) the diameter of the body is small and (ii) the fluid is nearly incompressible (the low Mach number regime). The specific shape of the body...

In this article we show local-in-time existence of a weak solution to a system of partial differential equations describing the evolution of a compressible isentropic fluid which contains several rigid bodies. The fluid-structure interaction is incorporated by the Navier-slip boundary condition at the interface of the fluid and the rigid bodies. At...

In this paper, we study a nonlinear interaction problem between a thermoelastic shell and a heat-conducting fluid. The shell is governed by linear thermoelasticity equations and encompasses a time-dependent domain which is filled with a fluid governed by the full Navier-Stokes-Fourier system. The fluid and the shell are fully coupled, giving rise t...

We study the well-posedness of a system of one-dimensional partial differential equations modeling blood flows in a network of vessels with viscoelastic walls. We prove the existence and uniqueness of maximal strong solution for this type of hyperbolic/parabolic model. We also prove a stability estimate under suitable nonlinear Robin boundary condi...

We study control properties of a linearized fluid–structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to...

In this paper, we study a nonlinear interaction problem between a thermoelastic shell and a heat-conducting fluid. The shell is governed by linear thermoelasticity equations and encompasses a time-dependent domain which is filled with a fluid governed by the full Navier-Stokes-Fourier system. The fluid and the shell are fully coupled, giving rise t...

We consider the motion of compressible Navier-Stokes fluids with the hard sphere pressure law around a rigid obstacle when the velocity and the density at infinity are non zero. This kind of pressure model is largely employed in various physical and industrial applications. We prove the existence of weak solution to the system in the exterior domai...

We consider a coupled system of partial and ordinary differential equations describing the interaction between an incompressible inviscid fluid and a rigid body moving freely inside the fluid. We prove the existence of measure-valued solutions which is generated by the vanishing viscosity limit of incompressible fluid–rigid body interaction system...

In this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions...

We consider the stabilizability of a fluid-structure interaction system where the fluid is viscous and compressible and the structure is a rigid ball. The feedback control of the system acts on the ball and corresponds to a force that would be produced by a spring and a damper connecting the center of the ball to a fixed point \(h_1\). We prove the...

This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole space. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and angular mom...

We study control properties of a linearized fluid-structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to...

In this article, we consider a fluid-structure interaction system where the fluid is viscous and com-pressible and where the structure is a part of the boundary of the fluid domain and is deformable. The fluid is governed by the barotropic compressible Navier-Stokes system whereas the structure displacement is described by a wave equation. We show...

We study the well-posedness of a system of one-dimensional partial differential equations modeling blood flows in a network of vessels with viscoelastic walls. We prove the existence and uniqueness of maximal strong solution for this type of hyperbolic/parabolic model. We also prove a stability estimate under suitable nonlinear Robin boundary condi...

We consider a coupled system of partial and ordinary differential equations describing the interaction between an isentropic inviscid fluid and a rigid body moving freely inside the fluid. We prove the existence of measure-valued solutions which is generated by the vanishing viscosity limit of incompressible fluid-rigid body interaction system unde...

This paper is devoted to study the controllability of a one-dimensional fluid-particle interaction model where the fluid follows the viscous Burgers equation and the point mass obeys Newton’s second law. We prove the null controllability for the velocity of the fluid and the particle and an approximate controllability for the position of the partic...

This paper is devoted to study the controllability of a one-dimensional fluid-particle interaction model where the fluid follows the viscous Burgers equation and the point mass obeys Newton's second law. We prove the null controllability for the velocity of the fluid and the particle and an approximate controllability for the position of the partic...

We consider the stabilizability of a fluid-structure interaction system where the fluid is viscous and compressible and the structure is a rigid ball. The feedback control of the system acts on the ball and corresponds to a force that would be produced by a spring and a damper connecting the center of the ball to a fixed point $h_1$. We prove the g...

This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole $\mathbb{R}^3$. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and an...

In this paper, we study the controllability of a fluid-structure interaction system. We consider a viscous and incompressible fluid modeled by the Boussinesq system and the structure is a rigid body with arbitrary shape which satisfies Newton’s laws of motion. We assume that the motion of this system is bidimensional in space. We prove the local nu...

In this work, we study a system coupling the incompressible Navier-Stokes equations in a cylindrical type domain with an elastic structure, governed by a damped shell equation, located at the lateral boundary of the domain occupied by the fluid. We prove the existence of a unique maximal strong solution.