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Martin Kalousek

Publications and source records attributed to Martin Kalousek.

15 recordsLinked to original sources
arXiv2024

On existence of weak solutions to a Baer-Nunziato type system

In this paper, a dissipative version of a compressible one velocity Baer--Nunziato type system for a mixture of two compressible heat conducting gases is considered. The complete existence proof for weak solutions to this system was addressed as an open problem in [6, Section 5]. The purpose of this paper is to prove the global in time existence of weak solutions to the one velocity Baer--Nunziato type system for arbitrary large initial data. The goal is achieved in three steps. Firstly, the given system is transformed into a new one which possesses the "Navier-Stokes-Fourier" structure. Secondly, the new system is solved by an adaptation of the Feireisl--Lions approach for solving the compressible Navier--Stokes equations. Eventually, the existence of a weak solution to the original one velocity Baer--Nunziato system using the almost uniqueness property of renormalized solutions to pure transport equations.

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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 $\max\{\gamma, \beta\}>2$ and $\min\{\gamma,\beta\}>0,$ 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 $\max\{\gamma,\beta\}\geq 2$ and $\min\{\gamma,\beta\}>0.$ 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 H\"{o}lder continuous boundaries. Moreover, the proof of such a compactness result depends on the existence of renormalized continuity equation in time dependent domains.

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arXiv2022

Singular limit for the compressible Navier--Stokes equations with the hard sphere pressure law on expanding domains

The article is devoted to the asymptotic limit of the compressible Navier-Stokes system with a pressure obeying a hard--sphere equation of state on a domain expanding to the whole physical space $R^3$. Under the assumptions that acoustic waves generated in the case of ill-prepared data do not reach the boundary of the expanding domain in the given time interval and a certain relation between the Reynolds and Mach numbers and the radius of the expanding domain, we prove that the target system is the incompressible Euler system on $R^3$. We also provide an estimate of the rate of convergence expressed in terms of characteristic numbers and the radius of domains.

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arXiv2021

Existence of weak solutions to a diffuse interface model involving magnetic fluids with unmatched densities

In this article we prove the global existence of weak solutions for a diffuse interface model in a bounded domain (both in 2D and 3D) involving incompressible magnetic fluids with unmatched densities. The model couples the incompressible Navier-Stokes equations, gradient flow of the magnetization vector and the Cahn-Hilliard dynamics describing the partial mixing of two fluids. The density of the mixture depends on an order parameter and the modelling, specifically the density dependence, is inspired from Abels, Garcke and Grün 2011.

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arXiv2020

Local-in-time existence of strong solutions to a class of compressible non-Newtonian Navier-Stokes equations

The aim of this article is to show a local-in-time existence of a strong solution to the generalized compressible Navier-Stokes equation for arbitrarily large initial data. The goal is reached by $L^p$-theory for linearized equations which are obtained with help of the Weis multiplier theorem and can be seen as generalization of the work of Shibata and Enomoto (devoted to compressible fluids) to compressible non-Newtonian fluids.

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arXiv2020

Global existence of weak solutions to a diffuse interface model for magnetic fluids

This article is devoted to the derivation and analysis of a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier-Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn-Hilliard equations. We show global in time existence of weak solutions to the system using the time discretization method.

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arXiv2019

Dissipative solutions to a system for the flow of magnetoviscoelastic materials

We address the question of global in time existence of solutions to a magnetoviscoelastic system with general initial data. We show that the notion of dissipative solutions allows to prove such an existence in two and three dimensions. This extends an earlier result for the viscoelastic subsystem to the setting which includes the magnetization vector and its evolution in terms of a Landau-Lifshitz-Gilbert equation.

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arXiv2019

On dissipative solutions to a system arising in viscoelasticity

We consider a model for an incompressible visoelastic fluid. It consists of the Navier-Stokes equations involving an elastic term in the stress tensor and a transport equation for the evolution of the deformation gradient. The novel feature of the paper is the introduction of the notion of a dissipative solution and its analysis. We show that dissipative solutions exist globally in time for arbitrary finite energy initial data and that a dissipative solution and a strong solution emanating from the same initial data coincide as long as the latter exists.

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arXiv2019

Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity

The paper is concerned with the analysis of an evolutionary model for magnetoviscoelastic materials in two dimensions. The model consists of a Navier-Stokes system featuring a dependence of the stress tensor on elastic and magnetic terms, a regularized system for the evolution of the deformation gradient and the Landau-Lifshitz-Gilbert system for the dynamics of the magnetization. First, we show that our model possesses global in time weak solutions, thus extending work by Benešová et al. 2018. Compared to that work, we include the stray field energy and relax the assumptions on the elastic energy density. Second, we prove the local in time existence of strong solutions. Both existence results are based on the Galerkin method. Finally, we show a weak-strong uniqueness property.

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arXiv2018

Existence and homogenization of nonlinear elliptic systems in nonreflexive spaces

We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic $\mathcal{N}$-function. First, an existence result is shown under the assumption that the $\mathcal{N}$-function or its convex conjugate satisfies $Δ_2$-condition. The second result concerns the homogenization process for families of strongly nonlinear elliptic problems with the homogeneous Dirichlet boundary condition under above stated conditions on the elliptic operator, which is additionally assumed to be periodic in the spatial variable.

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arXiv2017

Gradient $L^q$ theory for a class of non-diagonal nonlinear elliptic systems

We consider a class of nonlinear non-diagonal elliptic systems with $p$-growth and establish the $L^q$-integrability for all $q\in [p,p+2]$ of any weak solution provided the corresponding right hand side belongs to the corresponding Lebesgue space and the involved elliptic operator asymptotically satisfies the $p$-uniform ellipticity, the so-called splitting condition and it is continuous with respect to the spatial variable. For operators satisfying the uniform $p$-ellipticity condition the higher integrability is known for $q\in[p,dp/(d-2)]$ and for operators having the so-called Uhlenbeck structure, the theory is valid for all $q\in [p,\infty)$. The key novelty of the paper is twofold. First, the statement uses only the information coming from the asymptotic operator and second, and more importantly, by using the splitting condition, we are able to extend the range of possible $q$'s significantly whenever $p<d-2$.

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arXiv2017

Homogenization of nonlinear elliptic systems in nonreflexive Musielak-Orlicz spaces

We study the homogenization process for families of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operator is assumed to be indicated by a general inhomogeneous anisotropic $N-$function, which may be possibly also dependent on the spatial variable, i.e., the homogenization process will change the characteristic function spaces at each step. Such a problem is well known and there exists many positive results for the function satisfying $Δ_2$ and $\nabla_2$ conditions an being in addition Hölder continuous with respect to the spatial variable. We shall show that cases these conditions can be neglected and will deal with a rather general problem in general function space setting.

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arXiv2016

Homogenization of a stationary flow of a generalized Newtonian fluid

We perform the homogenization process avoiding the necessity of testing the weak formulation of the initial and homogenized systems by corresponding weak solutions. We show that the stress tensor for homogenized problem depends on the gradient involving the limit of a sequence selected from a family of solutions of initial problems.

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