Searcharxiv⌕ Search

arXiv subjects

Šárka Nečasová

Publications and source records attributed to Šárka Nečasová.

At least 37 records · Page 2Linked to original sources
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.

math.AP↗
arXiv2024

Shape Optimization of hemolysis for shear thinning flows in moving domains

We consider the $3$D problem of shape optimization of blood flows in moving domains. Such a geometry is adopted to take into account the modeling of rotating systems and blood pumps for instance. The blood flow is described by generalized Navier-Stokes equations, in the particular case of shear-thinning flows. For a sequence of converging moving domains, we show that a sequence of associated solutions to blood equations converges to a solution of the problem written on the limit moving domain. Thus, we extended the result given in (Sokołowski, Stebel, 2014, in \textit{Evol. Eq. Control Theory}) for $q \geq 11/5$, to the range $6/5< q < 11/5$, where $q$ is the exponent of the rheological law. This shape continuity property allows us to show the existence of minimal shapes for a class of functionals depending on the blood velocity field and its gradient. This allows to consider in particular the problem of hemolysis minimization in blood flows, namely the minimization of red blood cells damage.

math.OC↗
arXiv2024

Global solutions of Euler-Maxwell equations with dissipation

We consider the Cauchy problem for a damped Euler-Maxwell system with no ionic background. For smooth enough data satisfying suitable so-called dispersive conditions, we establish the global in time existence and uniqueness of a strong solution that decays uniformly in time. Our method is inspired by the works of D. Serre and M. Grassin dedicated to the compressible Euler system.

math.AP↗
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.

math.AP↗
arXiv2023

On the interaction between compressible inviscid flow and an elastic plate

We address a free boundary model for the compressible Euler equations where the free boundary, which is elastic, evolves according to a weakly damped fourth order hyperbolic equation forced by the fluid pressure. This system captures the interaction of an inviscid fluid with an elastic plate. We establish a priori estimates on local-in-time solutions in low regularity Sobolev spaces, namely with velocity and density initial data v0,R0 in H3. The main new device is a variable coefficients space tangential-time differential operator of order 1 with non-homogeneous boundary conditions, which captures the hyperbolic nature of the compressible Euler equations as well as the coupling with the structural dynamics.

math.AP↗
arXiv2023

On a nonlocal two-phase flow with convective heat transfer

We study a system describing the dynamics of a two-phase flow of incompressible viscous fluids influenced by the convective heat transfer of Caginalp-type. The separation of the fluids is expressed by the order parameter which is of diffuse interface and is known as the Cahn-Hilliard model. We shall consider a nonlocal version of the Cahn-Hilliard model which replaces the gradient term in the free energy functional into a spatial convolution operator acting on the order parameter and incorporate with it a potential that is assumed to satisfy an arbitrary polynomial growth. The order parameter is influenced by the fluid velocity by means of convection, the temperature affects the interface via a modification of the Landau-Ginzburg free energy. The fluid is governed by the Navier--Stokes equations which is affected by the order parameter and the temperature by virtue of the capillarity between the two fluids. The temperature on the other hand satisfies a parabolic equation that considers latent heat due to phase transition and is influenced by the fluid via convection. The goal of this paper is to prove the global existence of weak solutions and show that, for an appropriate choice of sequence of convolutional kernels, the solutions of the nonlocal system converges to its local version.

math.AP↗
arXiv2023

On the Motion of a Body with a Cavity Filled with Magnetohydrodynamic Fluid

We study the dynamics of a coupled system, formed by a rigid body with a cavity entirely filled with magnetohydrodynamic compressible fluid. Our aim is to derive the global existence of the unique classical solutions and weak solutions to this system. Moreover, we show the weak-strong uniqueness principle which means that a weak solution coincides with a strong solution on the time existence of a strong solution, provided they emanate from the same initial data.

math.AP↗
arXiv2023

On the regularity of weak solutions to the fluid-rigid body interaction problem

We study a 3D fluid-rigid body interaction problem. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations describing conservation of linear and angular momentum. Our aim is to prove that any weak solution satisfying certain regularity conditions is smooth. This is a generalization of the classical result for the $3D$ incompressible Navier-Stokes equations, which says that a weak solution that additionally satisfy Prodi - Serrin $L^r-L^s$ condition is smooth. We show that in the case of fluid - rigid body the Prodi - Serrin conditions imply $W^{2,p}$ and $W^{1,p}$ regularity for the fluid velocity and fluid pressure, respectively. Moreover, we show that solutions are $C^{\infty}$ if additionally we assume that the rigid body acceleration is bounded almost anywhere in time variable.

math.AP↗
arXiv2023

On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams

In this paper, we study a nonlinear fluid-structure interaction problem between a viscoelastic beam and a compressible viscous fluid. The beam is immersed in the fluid which fills a two-dimensional rectangular domain with periodic boundary conditions. Under the effect of periodic forces acting on the beam and the fluid, at least one time-periodic weak solution is constructed which has a bounded energy and a fixed prescribed mass.

math.AP↗
arXiv2023

Analysis of a system modelling the interaction between the motion of a spring and a viscous gas

We are concerned with a one dimensional flow of a compressible fluid which may be seen as a simplification of the flow of fluid in a long thin pipe. We assume that the pipe is on one side ended by a spring. The other side of the pipe is let open -- there we assume either inflow or outflow boundary conditions. Such situation can be understood as a toy model for human lungs. We tackle the question of uniqueness and existence of a strong solution for a system modelling the above process, special emphasis is laid upon the estimate of the maximal time of existence.

math.AP↗
arXiv2023

Existence of weak solution for a compressible multicomponent fluid structure interaction problem

We analyze a system of PDEs governing the interaction between two compressible mutually noninteracting fluids and a shell of Koiter type encompassing a time dependent 3D domain filled by the fluids. The dynamics of the fluids is modelled by a system resembilng compressible Navier-Stokes equations with a physically realistic pressure depending on densities of both the fluids. In fact in the present article the dependence of the fluid pressure on the densities is analogous to the ones considered in \cite{NovoPoko} (where the authors deal with a bi-fluid system in a time independent smooth domain). The shell constitutes the boundary of the fluid domain and it possesses a non-linear, non-convex Koiter energy (of a quite general form). We are interested in the existence of a weak solution to the system until the time-dependent boundary approaches a self-intersection or the Koiter energy degenerates. We first prove a global existence result when the adiabatic exponents solve $\max\{γ, β\}>2$ and $\min\{γ,β\}>0,$ further the densities are comparable and the structure involved is non-dissipative. Next with the assumption that the structure is dissipative we extend our global existence result to the critical case $\max\{γ,β\}\geq 2$ and $\min\{γ,β\}>0.$ The result is achieved in several steps involving, extension of the physical domain, penalization of the interface condition, artificial regularization of the shell energy, added structural dissipation and suitable limit passages depending on uniform estimates. In order to deal with the bi-fluid system we generalize the almost compactness argument developed in \cite{NovoPoko, Vasseur} to the case of time dependent domains with uniform Hölder continuous boundaries. Moreover, the proof of such a compactness result depends on the existence of renormalized continuity equation in time dependent domains.

math.AP↗
arXiv2023

Fluid-rigid body interaction in an incompressible electrically conducting fluid

We analyze a mathematical model that describes the interaction between an insulating rigid body and an incompressible electrically conducting fluid surrounding it. The model as well as the mathematical analysis involve the fields of fluid-structure interaction and of magnetohydrodynamics. Our main result shows the existence of a weak solution to the corresponding system of partial differential equations. The proof relies on a use of a time discretization via the Rothe method to decouple the system. This allows us to deal with test functions, depending on the position of the moving body and therefore on the solution of the system, in the weak formulation of the induction equation. The proof moreover makes use of the Brinkman penalization in order to cope with the mechanical part of the problem.

math.AP↗
arXiv2022

On the trajectory of a light small rigid body in an incompressible viscous fluid

In this paper we study the dynamics of a small rigid body in a viscous incompressible fluid in dimension two and three. More precisely we investigate the trajectory of the rigid body in the limit when the its mass and its size tend to zero. We show that the velocity of the center of mass of the rigid body coincides with the background fluid velocity in the limit. We are able to consider the case where the density of the small rigid body is uniformly bounded respect to its size.

math.AP↗
arXiv2022

On the Motion of a Pendulum with a Cavity Filled with a Compressible Fluid

We study the motion of the coupled system, $\mathscr S$, constituted by a physical pendulum, $\mathscr B$, with an interior cavity entirely filled with a viscous, compressible fluid, $\mathscr F$. The presence of the fluid may strongly affect on the motion of $\mathscr B$. In fact, we prove that, under appropriate assumptions, the fluid acts as a damper, namely, $\mathscr S$ must eventually reach a rest-state. Such a state is characterized by a suitable time-independent density distribution of $\mathscr F$ and a corresponding equilibrium position of the center of mass of $\mathscr S$. These results are proved in the very general class of weak solutions and do not require any restriction on the initial data, other than having a finite energy. We complement our findings with some numerical tests. The latter show, among other things, the interesting property that ``large" compressibility favors the damping effect, since it drastically reduces the time that $\mathscr S$ takes to go to rest.

math.AP↗
arXiv2022

On the velocity of a small rigid body in a viscous incompressible fluid in dimension two and three

In this paper we study the evolution of a small rigid body in a viscous incompressible fluid, in particular we show that a small particle is not accelerated by the fluid in the limit when its size converges to zero under a lower bound on its mass. This result is based on a new a priori estimate on the velocities of the centers of mass of rigid bodies that holds in the case when their masses are also allowed to decrease to zero.

math.AP↗
arXiv2022

Existence of a weak solution to a nonlinear fluid-structure interaction problem with heat exchange

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 to a novel nonlinear moving boundary fluid-structure interaction problem involving heat exchange. The existence of a weak solution is obtained by combining three approximation techniques -- decoupling, penalization and domain extension. This approximation scheme allows us to use the methods already developed for fluids on moving domains, thus drastically simplifying the analysis at hand.

math.AP↗