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Milan Pokorny

Publications and source records attributed to Milan Pokorny.

13 recordsLinked to original sources
arXiv2025

Weak solutions and weak-strong uniqueness for a compressible power-law-Oldroyd--B fluid model

We consider a model of a viscoelastic compressible flow in $R^{3}$ which is additionally shear thickening (the stress tensor corresponds to the power law model, however, the divergence of the velocity is due to the model bounded). We prove existence of a weak solution to this model provided the growth in the power law model is larger or equal than $\frac 52$. We also prove that any sufficiently smooth solution of this model is unique in the class of weak solution, provided extra integrability of the initial value for the extra stress tensor is assumed.

math.AP↗
arXiv2019

Steady solutions to a model of compressible chemically reacting fluid with high density

We consider a model describing the steady flow of compressible heat-conducting chemically-reacting multi-component mixture. We show the existence of strong solutions under the additional assumption that the mixture is sufficiently dense. We work in the $L^p$-setting combining the methods for the weak solutions with the method of decomposition. The result is a generalization of our previous papers, where the case of single-constituted fluid was studied.

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arXiv2019

Continuity equation and vacuum regions in compressible flows

We investigate the creation and properties of eventual vacuum regions in the weak solutions of the continuity equation, in general, and in the weak solutions of compressible Navier--Stokes equations, in particular. The main results are based on the analysis of renormalized solutions to the continuity and pure transport equations and their inter-relations which are of independent interest.

math.AP↗
arXiv2018

Weak solutions for some compressible multicomponent fluid models

The principle purpose of this work is to investigate a "viscous" version of a "simple" but still realistic bi-fluid model described in [Bresch, Desjardin, Ghidaglia, Grenier, Hillairet] whose "non-viscous" version is derived from physical considerations in \cite[Ishii, Hibiki]{ISHI} as a particular sample of a multifluid model with algebraic closure. The goal is to show existence of weak solutions for large initial data on an arbitrarily large time interval. We achieve this goal by transforming the model to an academic system which resembles to the compressible Navier-Stokes equations, with however two continuity equations and a momentum equation endowed with pressure of complicated structure dependent on two variable densities. The new "academic system" is then solved by an adaptation of the Lions--Feireisl approach for solving compressible Navier--Stokes equation, completed with several observations related to the DiPerna--Lions transport theory inspired by [Maltese, Michalek, Mucha, Novotny, Pokorny, Zatorska] and [Vasseur, Wen, Yu]. We also explain how these techniques can be generalized to a model of mixtures with more then two species. This is the first result on the existence of weak solutions for any realistic multifluid system.

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arXiv2017

Derivation of the Navier - Stokes - Poisson system with radiation for an accretion disk

We study the 3-D compressible barotropic radiation fluid dynamics system describing the motion of the compressible rotating viscous fluid with gravitation and radiation confined to a straight layer. We show that weak solutions in the 3-D domain converge to the strong solution of the rotating 2-D Navier-Stokes-Poisson system with radiation for all times less than the maximal life time of the strong solution of the 2-D system when the Froude number is small or to the strong solution of the rotating pure 2-D Navier- Stokes system with radiation.

math.AP↗
arXiv2017

Flocking particles in a non-Newtonian shear thickening fluid

We prove existence and uniqueness of strong solutions to the Cucker--Smale flocking model coupled with an incompressible viscous non-Newtonian fluid, with the stress tensor of a power--law structure for $p\geq\frac{11}{5}$. The coupling is performed through a drag force on a periodic spatial domain $T^3$.

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arXiv2016

Weak and variational entropy solutions to the system describing steady flow of a compressible reactive mixture

We consider a system of partial differential equations which describes steady flow of a compressible heat conducting chemically reacting gaseous mixture. We extend the result from Giovangigli, Pokorný, Zatorska (2015) in the sense that we introduce the variational entropy solution for this model and prove existence of a weak solution for $γ>\frac 43$ and existence of a variational entropy solution for any $γ>1$. The proof is based on improved density estimates.

math.AP↗
arXiv2016

The rotating Navier- Stokes- Fourier- Poisson system on thin domains

We consider the compressible Navier - Stokes - Fourier - Poisson system describing the motion of a viscous heat conducting rotating fluid confined to a straight layer $ Ω_ε = ω\times (0,ε) $, where $ω$ is a 2-D domain. The aim of this paper is to show that the weak solutions in the 3D domain converge to the strong solution of the 2-D Navier - Stokes - Fourier - Poisson system $ω$ as $ε\to 0$ on the time interval, where the strong solution exists. We consider two different regimes in dependence on the asymptotic behaviour of the Froude number.

math.AP↗
arXiv2016

Existence of weak solutions for compressible Navier-Stokes equations with entropy transport

We consider the compressible Navier-Stokes system with variable entropy. The pressure is a nonlinear function of the density and the entropy/potential temperature which, unlike in the Navier-Stokes-Fourier system, satisfies only the transport equation. We provide existence results within three alternative weak formulations of the corresponding classical problem. Our constructions hold for the optimal range of the adiabatic coefficients from the point of view of the nowadays existence theory.

math.AP↗
arXiv2012

Strong solutions to the Navier-Stokes-Fourier system with slip-inflow boundary conditions

We consider a system of partial differential equations describing the steady flow of a compressible heat conducting Newtonian fluid in a three-dimensional channel with inflow and outflow part. We show the existence of a strong solution provided the data are close to a constant, but nontrivial flow with sufficiently large dissipation in the energy equation.

math.AP↗
arXiv2007

On the steady compressible Navier-Stokes-Fourier system

We study the motion of the steady compressible heat conducting viscous fluid in a bounded three dimensional domain governed by the compressible Navier-Stokes-Fourier system. Our main result is the existence of a weak solution to these equations for arbitrarily large data. A key element of the proof is a special approximation of the original system guaranteeing pointwise uniform boundedness of the density. Therefore the passage to the limit omits tedious technical tricks required by the standard theory. Basic estimates on the solutions are possible to obtain by a suitable choice of physically reasonable boundary conditions.

math.AP↗
arXiv2002

A counterexample to the smoothness of the solution to an equation arising in fluid mechanics

We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the ``back to coordinates map'' may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for the Navier-Stokes equation.

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