
Milan PokornýCharles University in Prague | CUNI · Mathematical Institute of Charles University
Milan Pokorný
doc., Ph.D.
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Introduction
Skills and Expertise
Publications
Publications (104)
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-rela...
We study the evolutionary compressible Navier–Stokes–Fourier system in a bounded two-dimensional domain with the pressure law p(ϱ,θ)∼ϱθ+ϱlogα(1+ϱ)+θ4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemarg...
We consider a model describing the steady flow of compressible heat‐conducting chemically reacting multicomponent mixture. We show the existence of strong solutions under the additional assumption that the mixture is sufficiently dense. We work in the Lp‐setting combining the methods for the weak solutions with the method of decomposition. The resu...
We study the homogenization of stationary compressible Navier–Stokes–Fourier system in a bounded three dimensional domain perforated with a large number of very tiny holes. Under suitable assumptions imposed on the smallness and distribution of the holes, we show that the homogenized limit system remains the same in the domain without holes.
We study the homogenization of stationary compressible Navier--Stokes--Fourier system in a bounded three dimensional domain perforated with a large number of very tiny holes. Under suitable assumptions imposed on the smallness and distribution of the holes, we show that the homogenized limit system remains the same in the domain without holes.
The existence of large-data weak solutions to a steady compressible Navier--Stokes--Fourier system for chemically reacting fluid mixtures is proved. General free energies are considered satisfying some structural assumptions which include ideal gas mixtures. The model is thermodynamically consistent and contains the Maxwell--Stefan cross-diffusion...
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 closur...
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...
We start with the compressible Oldroyd--B model derived in \cite{Barrett-Lu-Suli} ({\em J. W. Barrett, Y. Lu, E. S\"uli. Existence of large-data finite-energy global weak solutions to a compressible Oldroyd--B model. Comm. Math. Sci. 15 (2017), 1265--1323}), where the existence of global-in-time finite-energy weak solutions was shown in two dimensi...
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-rel...
We establish existence of strong solutions to the stationary Navier-Stokes-Fourier system for compressible flows with density dependent viscosities in regime of heat conducting fluids with very high densities. In comparison to the known results considering the low Mach number case, we work in the Lp-setting combining the methods for the weak soluti...
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 t...
This chapter contains a survey of results in the existence theory of strong solutions to the steady compressible Navier-Stokes system. In the first part, the compressible Navier-Stokes equations are studied in bounded domains, both for homogeneous (no inflow) and inhomogeneous (inflow) boundary conditions. The solutions are constructed in Sobolev s...
The steady compressible Navier-Stokes-Fourier system is considered, with either Dirichlet or Navier boundary conditions for the velocity and the heat flux on the boundary proportional to the difference of the temperature inside and outside. In dependence on several parameters, i.e., the adiabatic constant γ appearing in the pressure law p(ρ υ) ∼ ργ...
We consider a model of chemically reacting heat conducting compressible mixture. We investigate the corresponding system of partial differential equations in the steady regime with slip boundary conditions for the velocity and, in dependence on the model parameters, we establish existence of either weak or variational entropy solutions. The results...
We consider a model of chemically reacting heat conducting compressible mixture. We investigate the corresponding system of partial differential equations in the steady regime with slip boundary conditions for the velocity and, in dependence on the model parameters, we establish existence of either weak or variational entropy solutions. The results...
We deal with a three dimensional model based on the use of barycentric velocity that describes unsteady flows of a heat conducting electrically charged multicomponent chemically reacting non-Newtonian fluid. We show that under certain assumptions on data and the constitutive relations for such a fluid there exists a global in time and large data we...
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, Pokorn\'y, Zatorska (2015) in the sense that we introduce the variational entropy solution for this model and prove existence of a weak solution for $\gamma...
As this book is focused on purely mathematical aspects of the theory of compressible viscous fluids, we omit a detailed derivation of the model in terms of classical continuum mechanics. The interested reader may consult the monographs of Batchelor [6], Lamb [65], or the more recent treatment by Gallavotti [47].
A vast class of nonlinear evolutionary problems arising in mathematical fluid mechanics including the Navier–Stokes system (2. 7) and (2. 8) is not known to admit classical (differentiable, smooth) solutions for all choices of data and on an arbitrary time interval. The existence of classical solutions has been established under rather restrictive...
A priori bounds are natural constraints imposed on the set of (hypothetical) smooth solutions by the data as well as by the differential equations satisfied. A priori bounds determine the function spaces framework the (weak) solutions are looked for. By definition, they are formal, derived under the principal hypothesis of smoothness of all quantit...
In the book, we deliberately focused on purely mathematical aspects omitting the physical background of the modeling of motion of compressible viscous fluids arising from classical continuum mechanics. The interested reader may consult the standard reference material by Batchelor [6], Lamb [65], Landau and Lifshitz [66], or the more recent treatmen...
The property of weak sequential stability plays a crucial role in the analysis of any nonlinear problem. It states that the solution set of a given problem is (weakly) precompact with respect to the topologies induced by the available a priori estimates. In our context, this property can be stated as follows:
This chapter introduces notation as well as the basic mathematical tools used in the book such as the function spaces, embedding theorems, and elementary inequalities. We suppose the reader to be familiar with this material and will refer to it throughout the text without further specification.
In Chap. 3, we have introduced the weak formulation of both the equation of continuity (3. 5) and the momentum balance (3. 11). On the other hand, we have seen in Chap. 4 that regular solutions of the Navier–Stokes system satisfy also the renormalized equation of continuity (4. 8), together with the total energy balance (4. 10). Under the general h...
The last part of this book is devoted to the proof of existence of weak solutions to the compressible Navier–Stokes system for a general pressure law p = p(ϱ).
Stability in numerical analysis means that the approximate solutions admit the same bounds as indicated by the a priori estimates for the original problem. The fact that the numerical solutions satisfy the energy inequality (7. 28) plays a crucial role.
We show that the weak solutions to the Navier–Stokes system exist, globally in time, for any finite energy initial data. The proof will be constructive in the sense that the desired weak solution is obtained as a suitable limit of a numerical scheme. By a numerical scheme we mean a finite number of algebraic equations yielding an approximate soluti...
We derive a consistency formulation of the numerical method (7. 14) and (7. 15). This amounts to rewriting the upwind and other spatial discretization operators in terms of conventional derivatives, extending validity of (7. 14) and (7. 15) to the class of smooth test functions, and identifying the resulting error terms. Consistency formulation the...
We are finally ready to establish convergence of the family [ϱ
h
, u
h
] of approximate (numerical) solutions, the existence of which is guaranteed by Proposition 1. We follow closely the arguments already used in the proof of weak sequential stability developed in Chap. 6 To begin observe that, in view of the uniform bounds (8. 17) and (8. 19)
In this chapter, we establish weak convergence of the densities for the perturbed Navier–Stokes system endowed with the artificial pressure (11. 2). Following the arguments used in Sect. 6. 3 we need the renormalized formulation of the continuity equation (5. 4). Unfortunately, the regularization technique of DiPerna and Lions [26] is no longer app...
We prove the existence of strong solutions to the Cucker-Smale flocking model coupled with an incompressible viscous non-Newtonian fluid with the stress tensor of a powerlaw structure for p ≥ 11/5 . The fluid part of the system admits strong solutions while the solutions to the CS part are weak. The coupling is performed through a drag force on a p...
In this note, we show the existence of regular solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we are able to construct a solution, provided the total mass is sufficiently large. The main mathematical part is...
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 clas...
We consider the compressible Navier - Stokes - Fourier - Poisson system describing the motion of a viscous heat conducting rotating fluid confined to a straight layer $ \Omega_{\epsilon} = \omega \times (0,\epsilon) $, where $\omega$ 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 so...
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 clas...
The steady compressible Navier–Stokes–Fourier system is considered, with either Dirichlet or Navier boundary conditions for the velocity and the heat flux on the boundary proportional to the difference of the temperature inside and outside. In dependence on several parameters, i.e., the adiabatic constant γ appearing in the pressure law p(ϱ, ϑ) ∼ ϱ...
This chapter contains a survey of results in the existence theory of strong solutions to the steady compressible Navier–Stokes system. In the first part, the compressible Navier–Stokes equations are studied in bounded domains, both for homogeneous (no inflow) and inhomogeneous (inflow) boundary conditions. The solutions are constructed in Sobolev s...
The steady compressible Navier--Stokes--Fourier system is considered, with
either Dirichlet or Navier boundary conditions for the velocity and the heat
flux on the boundary proportional to the difference of the temperature inside
and outside. In dependence on several parameters, i.e. the adiabatic constant
$\gamma$ appearing in the pressure law $p(...
The Navier-Stokes-Fourier system is a well established model for describing
the motion of viscous compressible heat-conducting fluids. We study the
existence of time-periodic weak solutions and improve the known result in the
following sense: we extend the class of pressure functions (i.e. consider lower
exponent $\gamma$) and include also the effe...
Consider the ow of a compressible Newtonian uid around or past a rotating rigid obstacle in R3: After a coordinate transform to get a problem in a time-independent domain we assume the new system to be stationary, then linearize and - in this paper dealing with the whole space case only - use Fourier transform to prove the existence of solutions u...
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 e...
We present the study of systems of equations governing a steady flow of poly-atomic, heat-conducting reactive gas mixture. It is shown that the corresponding system of PDEs admits a weak solution and renormalized solution to the continuity equation, provided the adiabatic exponent for the mixture γ is greater than 5 3 .
The goal of this paper is to reconsider the classical elliptic system rot v = f, div v = g in simply connected domains with bounded connected boundaries (bounded and exterior sets). The main result shows solvability of the problem in the maximal regularity regime in the L
p
-framework taking into account the optimal/minimal requirements on the smoo...
We consider a model of motion of binary mixture, based on the compressible Navier-Stokes system. The mass balances of chemically reacting species are described by the reaction-diffusion equations with generalized form of multicomponent difiusion ux. Under a special relation between the two density dependent viscosity coefficients and for singular c...
We study the generalized Oldroyd model with viscosity depending on the shear
stress behaving like $\mu(\mathbf{D}) \sim |\mathbf{D}|^{p-2}$ ($p>\frac 65$)
regularized by a nonlinear stress diffusion. Using the Lipschitz truncation
method we are able to prove global existence of weak solution to the
corresponding system of partial differential equat...
We consider a problem modeling the steady flow of a compressible heat conducting Newtonian fluid subject to the slip boundary condition for the velocity. Assuming the pressure law of the form p(ϱ, ϑ) ~ ϱγ + ϱϑ, we show (under additional assumptions on the heat conductivity and the viscosity) that for any γ > 1 there exists a variational entropy sol...
We investigate a coupling between the compressible Navier-Stokes-Fourier
system and the full Maxwell-Stefan equations. This model describes the motion
of chemically reacting heat-conducting gaseous mixture. The viscosity
coefficients are density-dependent functions vanishing on vacuum and the
internal pressure depends on species concentrations. By...
We consider the incompressible Navier-Stokes equations in the entire three-dimensional space. Assuming additional regularity on the components of the vector field ∂ 3 u we show intermediate anisotropic regularity results between the results by I. Kukavica and M. Ziane [J. Math. Phys. 48, No. 6, 065203, 10 p. (2007; Zbl 1144.81373)] and by C. Cao an...
The paper analyzes basic mathematical questions for a model of chemically reacting mixtures. We derive a model of several (finite) component compressible gas taking rigorously into account the thermodynamical regime. Mathematical description of the model leads to a degenerate parabolic equation with hyperbolic deviation. The thermodynamics implies...
We study the equations describing the steady flow of a compressible radiative gas with newtonian rheology. Under suitable assumptions on the data that include the physically relevant situations (i.e., the pressure law for monoatomic gas, the heat conductivity growing with square root of the temperature), we show the existence of a variational entro...
We study a system of partial differential equations describing the steady flow of a heat conducting incompressible fluid in a bounded three dimensional domain, where the right-hand side of the momentum equation includes the buoyancy force. In the present work we prove the existence of a weak solution under both the smallness and a sign condition on...
We consider the full Navier–Stokes–Fourier system describing the motion of a compressible viscous and heat conducting fluid driven by a time-periodic external force. We show the existence of at least one weak time periodic solution to the problem under the basic hypothesis that the system is allowed to dissipate the thermal energy through the bound...
We examine the conditional regularity of the solutions of Navier-Stokes
equations in the entire three-dimensional space under the assumption
that the data are axially symmetric. We show that if positive part of
the radial component of velocity satisfies a weighted Serrin condition
and in addition angular component satisfies some condition, then the...
We study the steady translational fall of a homogeneous body of revolution around an axis a, with fore-and-aft symmetry, in a second-order liquid at nonzero Reynolds (Re) and Weissenberg (We) numbers. We show that, at first order in these parameters, only two orientations are allowed, namely, those with a either parallel or perpendicular to the dir...
We consider steady compressible Navier–Stokes–Fourier system for a gas with pressure p and internal energy e related by the constitutive law p = (γ − 1)e, γ > 1. We show that for any γ > 3 2 there exists a variational entropy solution (i.e. solution satisfying the weak formulation of balance of mass and momentum, entropy inequality and global balan...
We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain. We show the existence of
a weak solution for arbitrarily large data for the pressure law p(ϱ, ϑ) ∼ ϱ
γ
+ ϱϑ if γ > 1 and p(ϱ, ϑ) ∼ ϱ ln
α
(1 + ϱ) + ϱϑ if γ = 1, α > 0, depending on the model for the heat flux.
Keywordssteady compressible Navier-Stoke...
We study a steady compressible Navier-Stokes-Fourier system in a bounded three-dimensional domain. We consider a general pressure law of the form p=(γ-1)ϱe which includes in particular the case p=a 1 ϱϑ+a 2 ϱ γ . We show the existence of a variational entropy solution (i.e., a solution satisfying balance of mass, momentum, entropy inequality, and g...
In this short note we consider the 3D Navier–Stokes equations in the whole space, for an incompressible fluid. We provide
sufficient conditions for the regularity of strong solutions in terms of certain components of the velocity gradient. Based
on the recent results from Kukavica (J Math Phys 48(6):065203, 2007) we show these conditions as anisotr...
We consider a model for the polymeric fluid which has recently been studied in (12). We show the local-in-time existence of a strong solution to the corresponding system of partial differential equations under less regu- larity assumptions on the initial data than in the mentioned paper. The main difference in our approach is the use of the L p the...
We consider the regularity criteria for the incompressible Navier-Stokes equations connected with one velocity component. Based on the method from Cao and Titi (2008 Indiana Univ. Math. J. 57 2643-61) we prove that the weak solution is regular, provided u3 ∈ Lt(O,T;L s(ℝ3)), 2/t + 3/s ≤ 3/4 + 1/2s, s > 10/3 or provided ∇u3 ∈ Lt(O, T;Ls(ℝ3)), 2/t +...
We improve the regularity criterion for the incompressible Navier–Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi [see
“Regularity criteria for the three dimensional Navier–Stokes equations,” Indiana Univ. Math. J. 57, 2643 (2008)
] and Kuk...
We consider a model for the viscoelastic fluid which has recently been studied in [F.-H. Lin, C. Liu and P. Zhang, Commun. Pure Appl. Math. 58, No. 11, 1437–1471 (2005; Zbl 1076.76006); Y. Chen and P. Zhang, Commun. Partial Differ. Equations 31, No. 12, 1793–1810 (2006; Zbl 1105.76008)]. We show the local-in-time existence of a strong solution to t...
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...
We study convolutions with Oseen kernels (weakly singular and singular) in three-dimensional space. We give a detailed weighted
LP theory for p∈(1; ∞] for anisotropic weights.
We study the steady compressible Navier-Stokes equations in a bounded smooth three-dimensional domain, together with the slip boundary conditions. We show that for a certain class of pressure laws, there exists a weak solution with bounded density (in L ∞ up to boundary).
We consider the compressible Navier–Stokes equations in an exterior three-dimensional domain with non-zero constant density prescribed at infinity. We assume that p(ϱ)=ϱγ, γ>32, and that the force is potential. We show that for time tending to infinity, the density approaches the unique solution to the stationary problem, provided the potential sat...
We study the non-stationary Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axisymmetric. We extend the regularity criterion for axisymmetric weak solutions given in [10].
We consider the steady compressible Navier–Stokes equations of isentropic flow in three-dimensional domains with several exits to infinity with prescribed pressure drops. On the one hand, when each exit is supposed to contain a cone inside, we shall construct bounded energy weak solution for adiabatic constant γ>3. On the other hand, when the exits...
We review several regularity criteria for the Navier-Stokes equations and prove some new ones containing different components of the velocity gradient.
We study the nonstationary Navier-Stokes equations in the entire three-dimensional space and give some criteria on certain components of gradient of the velocity which ensure its global-in-time smoothness.
We improve the regularity criterion for the Navier-Stokes equations proved by He [4]. We show that for the Cauchy problem the Leray-Hopf weak solution is smooth provided $ abla u_3 in L^t(0,T;L^s)$, $frac 2t + frac 3s leq frac 32$.
We consider the steady plane flow of certain classes of viscoelastic fluids in exterior domains with a non-zero velocity prescribed at infinity. We study existence as well as asymptotic behaviour of solutions near infinity and show that for sufficiently small data the solution decays near infinity as fast as the fundamental solution to the Oseen pr...
We consider perturbations to the rest state of heat conductive compressible fluid in a three-dimensional exterior domain. In the regularity class \( {\cal V}_s \), we show stability of L
2
-norms of these perturbations, while in the more restrictive class \( {\cal V}_d \) we show convergence to zero of these perturbations along a time sequence conv...
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 smoo...
We study convolutions with Oseen kernels (weakly singular and singular) in both two- and three-dimensional space. We give a detailed weighted $L^{p}$ theory for $ p\in(1;\infty$ ] for anisotropic weights.
In the paper [3] the authors claim that in order to obtain a particular form of the Green formula for p = 2 it is necessary to use the uniqueness result for the steady transport equation. Our aim is to generalize these results for p ∈ ( 1, ∞) and show the Green formula without the use of the steady transport equation. As a consequence we get the de...
We study axisymmetric solutions to the Navier-Stokes equations in the whole three-dimensional space. We find conditions on the radial and angular components of the velocity field which are sufficient for proving the regularity of weak solutions.
We show that if v is an axially symmetric suitable weak solution to the Navier—Stokes equations (in the sense of L. Caffarelli, R. Kohn & L. Nirenberg — see [2]) such that either \( v_{\rho} \) (the radial component of v) or \( v_{\theta} \) (the tangential component of v) has a higher regularity than is the regularity following from the definition...
We consider the three-dimensional steady flow of certain classes of viscoelastic fluids in exterior domains with non-zero velocity prescribed at infinity. We show that the solution behaves near infinity similarly as the fundamental solution to the Oseen problem.
We study asymptotic properties of the fundamental solution to an Oseen-type system coming from fluid mechanics. We show that the solution has a similar anisotropic structure near infinity as the fundamental solution to the (classical) Oseen problem. We also study integral operators with kernels representing the second gradient of the fundamental so...
In this paper, we exclude the possibility of existence of a singular solution of the selfsimilar type proposed by Jean Leray More precisely, using a slightly stronger hypothesis we give a simpler proof to the analogous result established by J. Nečas, M. Rúžička and V. Šverák. We also discuss the possible existence of a singular solution of pseudo-s...
We prove in a simpler as ususal way global-in-time existence of regular solutions to three-dimensional Navier-Stokes equations under the assumption that the flow is axially symmetric.