arXiv subjects

Milan Pokorný

Publications and source records attributed to Milan Pokorný.

18 recordsLinked to original sources

arXiv2026

Dissipative measure-valued solutions and weak--strong uniqueness for a viscous Baer--Nunziato system with pressure relaxation

We study a viscous one-velocity Baer--Nunziato system for two barotropic compressible fluids in a bounded three-dimensional domain. In contrast with models in which the volume fraction is merely transported or determined by an algebraic equilibrium constraint, it satisfies a differential closure driven by the pressure gap between the phases. For arbitrary finite-energy initial data and adiabatic exponents $γ^\pm>1$, we construct global-in-time dissipative measure-valued solutions and prove dissipative measure-valued--strong uniqueness relative to any sufficiently regular solution with the same initial data. The principal difficulties are the singular and non-continuous behavior of the pressure-relaxation source at vanishing volume fractions, the associated concentration defects, and the lack of direct coercivity of the standard thermodynamic relative energy with respect to the volume fraction. These are resolved by an endpoint cutoff argument and an augmented relative energy coupled to the renormalized volume-fraction equation.

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arXiv2026

Steady compressible Navier-Stokes-Fourier system with general temperature dependent viscosities and hard sphere pressure law

We study the existence theory for steady compressible Navier-Stokes-Fourier system in a three dimensional bounded domain for the case of viscosities depending on the temperature in the form $(1+\vartheta)^α$ for $0\leq α\leq 1$ and the hard sphere pressure $p(\varrho,\vartheta) = \vartheta\varrho h(\varrho)$, $h(\varrho)$ is increasing and singular at $\varrho = a > 0$. This paper considers both the heat-flux and Dirichlet boundary conditions for the temperature and Dirichlet boundary conditions for the velocity. The key point to controlling the pressure is based on some estimates of the Bogovskii operator.

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arXiv2026

Steady weak solutions to an inflow/outflow driven compressible fluid-structure interaction problem

We study a stationary 3D/2D fluid-structure interaction problem between an elastic structure described by the linear plate equation and a fluid described by the compressible Navier-Stokes equations with hard-sphere pressure and inflow/outflow boundary data. This problem is motivated by wind-tunnel configuration and by the need for physically relevant steady states about which compressible flow-plate dynamics can be linearized. The main difficulty in the analysis is the lack of uniform estimates, both for approximate and weak solutions. In particular, the fixed-point construction for approximate solution yields a density estimate depending on approximate parameter, while the pressure estimate for the weak solution is only finite and non-quantifiable. As a result, large pressure loads can drive outward volume growth, while low pressure regions may lead to contact and therefore domain degeneration. This necessitates a novel approach based on a Lipschitz \emph{domain-correction} (barrier) mechanism that provides a framework in which solutions can be constructed without volume blow-up or degeneration of the domain. Constrained by the possibly very large fluid pressure load, our main result is the existence of a weak solution for a sufficiently large plate stiffness. Keywords: fluid-structure interaction, compressible Navier-Stokes, stationary weak solutions, hard-sphere pressure, inflow/outflow, linear plate, mathematical aeroelasticity

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arXiv2026

Weak-strong uniqueness for bi-fluid compressible system with algebraic closure

We consider a real two-fluid system of compressible viscous fluids with a common velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The existence of global-in-time finite energy weak solutions to this system is known since the work of Novotný and Pokorný [Arch. Rational Mech. Anal., 2020]. On the other hand, existence of local-in-time strong solutions is due to Piasecki and Zatorska [J. Math Fluid Mech., 2022]. In this paper, we establish the weak-strong uniqueness principle using the relative entropy method. In sharp contrast to the two-phase model of Baer-Nunziato type, the volume fraction of phase $+$ obeys a transport equation with an additional nonlinear term. This gives rise to troublesome terms in the relative entropy inequality. We are able to close the estimate by making an elaborate use of the structure of the system.

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arXiv2025

On a few counterexamples to solvability of the $div$ equation in domains with external cusps

This paper examines the solvability of the equation $\mathrm{div} \ \mathbf{u} = f$ with a zero Dirichlet boundary condition for $\mathbf{u}$. A classical result establishes that for a bounded domain $Ω\subset \mathbb{R}^N$ with a Lipschitz boundary and for $f \in L^p(Ω)$ with zero mean value there exists a solution $\mathbf{u} \in (W_0^{1, p}(Ω))^N$ for $1 < p < \infty$ with the $W^{1,p}$ norm controlled by the $L^p$ norm of the right-hand side $f$. The results were extended to John domains and excluded the existence of the solution operator in domains with external cusps. Our aim is to specify at least some classes of the right-hand sides for which the problem cannot have a solution in the space $W^{1,p}_0(Ω)$. We first extend the counterexample by Luc Tartar originally formulated for right-hand side functions in $\overline{L^2}$ in two space dimensions to a more general class of functions in $\overline{L^p}$ spaces and a more general type of singular domains. We then generalize this result to an arbitrary dimension $N$. Returning to two space dimensions, we investigate domains with boundary properties superior to those of previously studied Hölder continuous domains and construct counterexamples also in this situation.

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arXiv2025

Steady compressible Navier-Stokes-Fourier system with general temperature dependent viscosities I: density estimates based on Bogovskii operator

The aim of this paper is to reconsider the existence theory for steady compressible Navier--Stokes--Fourier system assuming more general condition of the dependence of the viscosities on the temperature in the form $μ(\vartheta)$, $ξ(\vartheta) \sim (1+\vartheta)^α$ for $0\leq α\leq 1$. This extends the known theory for $α=1$ from and improves significantly the results for $α=0$. This paper is the first of a series of two papers dealing with this problem and is connected with the Bogovskii-type estimates of the sequence of densities. This leads, among others, to the limitation $γ>\frac 32$ for the pressure law $p(\varrho,\vartheta) \sim \varrho^γ+ \varrho\vartheta$. The paper considers both the heat-flux (Robin) and Dirichlet boundary conditions for the temperature as well as both the homogeneous Dirichlet and zero inflow/outflow Navier boundary conditions for the velocity. Further extension for $γ>1$ only is based on different type of pressure estimates and will be the content of the subsequent paper.

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arXiv2024

Construction of weak solutions to the equations of a compressible viscous model

The paper aims on the construction of weak solutions to equations of a model of compressible viscous fluids, being a simplification of the classical compressible Navier-Stokes system. We present a novel scheme for approximating systems that preserves structural integrity by avoiding classical regularization with $ - \varepsilon Δ\varrho $, thus maintaining the transport character of the continuity equation. Our approach, which necessitates specific conditions on the constitutive equation, accommodates physically relevant models such as isentropic and van der Waals gases, and globally handles non-monotone pressures. From an analytical perspective, our method synthesizes techniques from Feireisl-Lions and Bresch-Jabin to demonstrate the convergence of approximate densities using compensated compactness techniques. We also apply renormalization of the continuity equations and utilize weight techniques to manage unfavorable terms.

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arXiv2023

Homogenization of the unsteady compressible Navier-Stokes equations for adiabatic exponent $γ>3$

We consider the unsteady compressible Navier-Stokes equations in a perforated three-dimensional domain, and show that the limit system for the diameter of the holes going to zero is the same as in the perforated domain provided the perforations are small enough. The novelty of this result is the lower adiabatic exponent $γ>3$ instead of the known value $γ>6$. The proof is based on the use of two different restriction operators leading to two different types of pressure estimates. We also discuss the extension of this result for the unsteady Navier-Stokes-Fourier system as well as the optimality of the known results in arbitrary space dimension for both steady and unsteady problems.

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arXiv2022

Weak solutions for the Stokes system for compressible non-Newtonian fluids with unbounded divergence

We investigate the existence of weak solutions to a certain system of partial differential equations, modelling the behaviour of a compressible non-Newtonian fluid for small Reynolds number. We construct the weak solutions despite the lack of the $L^\infty$ estimate on the divergence of the velocity field. The result was obtained by combining the regularity theory for singular operators with a certain logarithmic integral inequality for $BMO$ functions, which allowed us to adjust the method from (Feireisl et al., 2015) to more relaxed conditions on the velocity.

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arXiv2021

Existence analysis of a stationary compressible fluid model for heat-conducting and chemically reacting mixtures

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, with a pressure containing a $γ$-power law. The model is thermodynamically consistent and contains the Maxwell-Stefan cross-diffusion equations in the Fick-Onsager form as a special case. Compared to previous works, a very general model class is analyzed, including cross-diffusion effects, temperature gradients, compressible fluids, and different molar masses. A priori estimates are derived from the entropy balance and the total energy balance. The compactness for the total mass density follows from an estimate for the pressure in $L^p$ with $p>1$, the effective viscous flux identity, and uniform bounds related to Feireisl's oscillations defect measure. These bounds rely heavily on the convexity of the free energy and the strong convergence of the relative chemical potentials.

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arXiv2020

Homogenization of stationary Navier--Stokes--Fourier system in domains with tiny 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.

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arXiv2019

Global existence of large data weak solutions for a simplified compressible Oldroyd--B model without stress diffusion

We start with the compressible Oldroyd--B model derived in \cite{Barrett-Lu-Suli} ({\em J. W. Barrett, Y. Lu, E. Süli. 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 dimensional setting with stress diffusion. In the paper, we investigate the case without stress diffusion. We first restrict ourselves to the corotational setting as in \cite{LM00} ({\em P. L. Lions, N. Masmoudi. Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math., Ser. B 21(2) (2000), 131--146.}) We further assume the extra stress tensor is a scalar matrix and we derive a simplified model which takes a similar form as the multi-component compressible Navier--Stokes equations, where, however, the pressure term related to the scalar extra stress tensor has the opposite sign. By employing the techniques developed in \cite{VWY19, Novotny-Pokorny-19}, we can still prove the global-in-time existence of finite energy weak solutions in two or three dimensions, without the presence of stress diffusion.

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arXiv2017

On steady solutions to a model of chemically reacting heat conducting compressible mixture with slip boundary conditions

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 extend the range of parameters for which the existence of weak solutions is known in the case of homogeneous Dirichlet boundary conditions for the velocity.

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arXiv2016

Existence analysis for incompressible fluid model of electrically charged chemically reacting and heat conducting mixtures

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 weak solution. The paper has two key novelties. The first one is that we present a model that is thermodynamically and mechanically consistent and that is able to describe the cross effects in a generality never considered before, i.e., we cover the so-called Soret effect, Dufour effect, Ohm law, Peltier effect, Joul heating, Thompson effect, Seebeck effect and also the generalized Fick law. The second key novelty is that contrary to the previous works on the similar topic, we do not need to deal with the energy equality method and therefore we are able to cover a large range of power-law parameters in the Cauchy-stress. In particular, we cover even the Newtonian case (which is the most used model), for which the existence analysis was missing.

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arXiv2016

Steady solutions to viscous shallow water equations. The case of heavy water

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 located in the construction of solutions. Uniqueness is impossible to obtain, since the gradient of the velocity is of magnitude of the force. The investigation is connected to the corresponding singular limit as Mach number goes to zero and methods for weak solutions to the compressible Navier-Stokes system.

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arXiv2015

Existence of stationary weak solutions for the heat conducting flows

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 $p(\vr,\vt) \sim \vr^γ+ \vr \vt$ and the growth exponent in the heat conductivity, i.e. $κ(\vt) \sim (1+ \vt^m)$, and without any restriction on the size of the data, the main ideas of the construction of weak and variational entropy solutions for the three-dimensional flows with temperature dependent viscosity coefficients are explained. Further, the case when it is possible to prove existence of solutions with bounded density is reviewed. The main changes in the construction of solutions for the two-dimensional flows are mentioned and finally, results for more complex systems are reviewed, where the steady compressible Navier--Stokes--Fourier equations play an important role.

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arXiv2014

On the global existence for a regularized model of viscoelastic non-Newtonian fluid

We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like $μ(\mathbf{D}) \sim |\mathbf{D}|^{p-2}$ ($p>\frac 65$) 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 equations.

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arXiv2014

Heat-conducting, compressible mixtures with multicomponent diffusion: construction of a weak solution

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 several levels of approximation we prove the global-in-time existence of weak solutions on the three-dimensional torus.

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