arXiv subjects

Kuntal Bhandari

Publications and source records attributed to Kuntal Bhandari.

13 recordsLinked to original sources

arXiv2026

Homogenization of the compressible Navier-Stokes equations via two-scale convergence in perforated domains

We study the homogenization of the compressible isentropic Navier-Stokes equations in periodically perforated domains where the size of the obstacles is of the same order as the distance between neighboring obstacles. Using the two-scale convergence method, which can be characterized via the unfolding operator, we derive the corresponding macroscopic model determined by Darcy's law. In particular, the macroscopic density satisfies the porous medium equation. The main challenge lies in identifying the pressure term in the limit. We overcome this by establishing the strong two-scale convergence of the densities, which is achieved by controlling the oscillation defect measure of the unfolded densities. A crucial contribution of our work is the development of a methodological framework applicable to more complex compressible fluid models. Furthermore, regarding conservative forces, we extend existing results from the literature to adiabatic constants $γ> \frac95$.

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arXiv2026

Maximal regularity for a compressible fluid-structure interaction system with Navier-slip boundary conditions

We investigate a fluid-structure interaction system in which the dynamics of the fluid is described by the compressible Navier-Stokes equations, while the elastic structure is modeled by a damped plate equation. The fluid evolves in a three-dimensional bounded domain, with the structure occupies a part of its boundary. Instead of standard no-slip boundary conditions, we consider the Navier-slip boundary conditions at the fluid-structure interface as well as at the fixed boundary. We establish the local-in-time existence and uniqueness of strong solutions within $L^{p}-L^{q}$ framework. The existence result is obtained for small time by decoupling the linearized system and employing a cascade strategy combined with the Tikhonov fixed point theorem, whereas the uniqueness is shown by deriving weak regularity properties for the associated linear coupled operator in a Hilbert space setting. It is the first result addressing strong solutions for a compressible fluid interacting with a damped plate under Navier-slip boundary conditions.

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arXiv2026

Dissipative solutions to a Beris-Edwards type model for compressible active nematic liquid crystals

We study the hydrodynamics of compressible active nematic liquid crystals in a three-dimensional and bounded domain, with a nonlinear viscosity tensor and nonhomogeneous boundary data, in a Landau-de Gennes framework. We prove the existence of dissipative solutions within a Beris-Edwards type model for active nematodynamics, which are weak solutions satisfying the underlying equations modulo a defect measure. The proof follows from a three level approximation scheme -- the Galerkin approximation, the classical parabolic regularization of the continuity equation, and the convex regularization of the potential generating the viscous stress. New techniques are required to deal with non-Newtonian stress tensor, larger classes of admissible pressure potentials and nonhomogeneous boundary conditions.

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arXiv2025

Weak solutions to a full compressible magnetohydrodynamic flow interacting with thermoelastic structure

This paper is concerned with an interaction problem between a full compressible, electrically conducting fluid and a thermoelastic shell in a two-dimensional setting. The shell is modelled by linear thermoelasticity equations, and encompasses a time-dependent domain which is filled with a fluid described by full compressible (non-resistive) magnetohydrodynamic equations. The magnetohydrodynamic flow and the shell are fully coupled, resulting in a fluid-structure interaction problem that involves heat exchange. We establish the existence of weak solutions through domain extension, operator splitting, decoupling, penalization of the interface condition, and appropriate limit passages.

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arXiv2025

Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations

This paper deals with the null-controllability of a system of {\em mixed parabolic-elliptic pdes} at any given time $T>0$. More precisely, we consider the \textit{Kuramoto-Sivashinsky--Korteweg-de Vries equation} coupled with a second order elliptic equation posed in the interval $(0,1)$. We first show that the linearized system is globally null-controllable by means of a localized interior control acting on either the KS-KdV or the elliptic equation. Using the \textit{Carleman approach}, we provide the existence of a control with the explicit cost $Ce^{C/T}$ with some constant $C>0$ independent in $T$. Then, applying the source term method followed by the \textit{Banach fixed point theorem}, we conclude the small-time local null-controllability result of the nonlinear systems.

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arXiv2025

On an Euler-Schrödinger system appearing in laser-plasma interaction

We consider the Cauchy problem for the barotropic Euler system coupled to a vector Schrödinger equation in the whole space. Assuming that the initial density and vector potential are small enough, and that the initial velocity is close to some reference vector field $u_0$ such that the spectrum of $Du_0$ is bounded away from zero, we prove the existence of a global-in-time unique solution with (fractional) Sobolev regularity. Moreover, we obtain some algebraic time decay estimates of the solution. Our work extends the papers by D. Serre and M. Grassin [11, 13, 19] and previous works by B. Ducomet and co-authors [4, 8] dedicated to the compressible Euler-Poisson system.

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arXiv2024

On the multicomponent reactive flows in moving domains

This paper is concerned with the existence of global-in-time weak solutions to the multicomponent reactive flows inside a moving domain whose shape in time is prescribed. The flow is governed by the 3D compressible Navier-Stokes-Fourier system coupled with the equations of species mass fractions. The fluid velocity is supposed to fulfill the complete slip boundary condition, whereas the heat flux and species diffusion fluxes satisfy the conservative boundary conditions. The existence of weak solutions is obtained by means of suitable approximation techniques. To this end, we need to rigorously analyze the penalization of the boundary behavior, viscosity and the pressure in the weak formulation.

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arXiv2024

Asymptotic limit of the compressible Navier-Stokes system on domains with rough boundaries

In this paper, we study the asymptotic behavior of solutions to the compressible Navier-Stokes system considered on a sequence of spatial domains, whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter. Imposing the full-slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary, provided the oscillations are non-degenerate, meaning not oriented in a single direction.

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arXiv2024

Weak solutions to the heat conducting compressible self-gravitating flows in time-dependent domains

In this paper, we consider the heat-conducting compressible self-gravitating fluids in time-dependent domains, which typically describe the motion of viscous gaseous stars. The flow is governed by the 3-D Navier-Stokes-Fourier-Poisson equations where the velocity is supposed to fulfil the full-slip boundary condition and the temperature on the boundary is given by a non-homogeneous Dirichlet condition. We establish the global-in-time weak solution to the system. Our approach is based on the penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Moreover, to accommodate the non-homogeneous boundary heat flux, the concept of {\em ballistic energy} is utilized in this work.

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arXiv2023

Controllability issues for parabolic-elliptic systems involving nonlocal couplings

This work addresses controllability properties for some systems of partial differential equations in which the main feature is the coupling through nonlocal integral terms. In the first part, we study a nonlinear parabolic-elliptic system arising in mathematical biology and, using recently developed techniques, we show how Carleman estimates can be directly used to handle the nonlocal terms, allowing us to implement well-known strategies for controlling coupled systems and nonlinear problems. In the second part, we investigate fine controllability properties of a 1-d linear nonlocal parabolic-parabolic system. In this case, we will see that the controllability of the model can fail and it will depend on particular choices and combinations of local and nonlocal couplings.

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arXiv2023

Insensitizing control problem for the Hirota-Satsuma system of KdV-KdV type

This paper is concerned with the existence of insensitizing controls for a nonlinear coupled system of two Korteweg-de Vries (KdV) equations, typically known as the Hirota-Satsuma system. The idea is to look for controls such that some functional of the states (the so-called sentinel) is insensitive to the small perturbations of initial data. Since the system is coupled, we consider a sentinel in which we observe both components of the system in a localized observation set. By some classical argument, the insensitizing problem is then reduced to a null-control problem for an extended system where the number of equations is doubled. We study the null-controllability for the linearized model associated to that extended system by means of a suitable Carleman estimate which is proved in this paper. Finally, the local null-controllability of the extended (nonlinear) system is obtained by applying the inverse mapping theorem, and this implies the required insensitizing property for the concerned model.

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arXiv2023

Insensitizing control problems for the stabilized Kuramoto-Sivashinsky system

In this work, we address the existence of insensitizing controls for a nonlinear coupled system of fourth- and second-order parabolic equations known as the stabilized Kuramoto-Sivashinsky model. The main idea is to look for controls such that some functional of the state (the so-called sentinel) is locally insensitive to the perturbations of the initial data. Since the underlying model is coupled, we shall consider a sentinel in which we may observe one or two components of the system in a localized observation set. By some classical arguments, the insensitizing problem can be reduced to a null-controllability one for a cascade system where the number of equations is doubled. Upon linearization, the null-controllability for this new system is studied by means of Carleman estimates but unlike other insensitizing problems for scalar models, the election of the Carleman tools and the overall control strategy depends on the initial choice of the sentinel due to the (lack of) couplings arising in the extended system. Finally, the local null-controllability of the extended (nonlinear) system (and thus the insensitizing property) is obtained by applying a local inversion theorem.

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arXiv2022

Boundary null-controllability of 1d linearized compressible Navier-Stokes system by one control force

In this article, we study the boundary null-controllability properties of the one-dimensional linearized (around $(Q_0,V_0)$ with constants $Q_0>0, V_0>0$) compressible Navier-Stokes equations in the interval $(0,1)$ when a control function is acting either on the density or velocity component at one end of the interval. We first prove that the linearized system, with a Dirichlet boundary control on the density component and homogeneous Dirichlet boundary conditions on the velocity component, is null-controllable in $H^s_{per}(0,1) \times L^2(0,1)$ for any $s > 1/2$ provided the time $T > 1$, where $H^s_{per}(0,1)$ denotes the Sobolev space of periodic functions. The proof is based on solving a mixed parabolic-hyperbolic moments problem and to do so, we perform a spectral analysis for the associated adjoint operator which is the main involved part of this work. As a corollary, we also prove that the system is approximately controllable in $L^2(0,1) \times L^2(0,1)$ when $T>1$. On the other hand, assuming that the density is equal on the two boundary points w.r.t. time, when a control is applied on the velocity part through a Dirichlet condition, we can only able to prove that the system is null-controllable in a strict subspace of finite codimension $\mathcal{H}\subset H^s_{per}(0,1) \times L^2(0,1)$ for $s>1/2$ when $T>1$. More precisely, in this case we are able to show that all the eigenfunctions of the associated adjoint operator are observable for higher frequencies whereas for the lower frequencies it is hard to conclude anything. A parabolic-hyperbolic joint Ingham-type inequality which we prove in this article, leads to an observability inequality in the space $\mathcal{H}^*$ and the controllability result follows. The significant point is that the moments method does not yield a better space for the null-controllability when a control acts on the velocity part.

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