Milan Pokorný
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Milan Pokorný has a total of 42 co-authors. Here are the top co-authors by number of shared publications:
- Piotr Boguslaw Mucha
Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland
13 shared publications - Ondřej Kreml
Czech Academy of Sciences
6 shared publications - Tomasz Piasecki
Institute of Applied Mathematics and Mechanics University of Warsaw Banacha 2 02-097 Warsaw Poland
6 shared publications - Antonín Novotný
University of Toulon, IMATH, EA 2134, La Garde, France
5 shared publications - Emil Skříšovský
Charles University, Faculty of Mathematics and Physics, Mathematical Inst. of Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
3 shared publications - Ewelina Zatorska
Institute of Applied Mathematics and Mechanics University of Warsaw ul Banacha 2 02-097 Warszawa Poland
3 shared publications
Latest Publications
- Open Access
- 1 November 2025
- Published by IOP Publishing in Physica Scripta
- Vol. 100 (11) , 115236
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 ( ϑ ), ξ ( ϑ ) ∼ (1 + ϑ ) α for 0 ≤ α ≤ 1. This extends the known theory for α = 1 from [1] to the values α ∈ (0, 1) and improves significantly the results for α = 0 from [2]. 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 > 3/2 for the pressure law p ( ϱ , ϑ ) ∼ ϱ γ + ϱϑ , where ϱ is the volume density of the gas. 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.
- 1 November 2025
- Published by IOP Publishing in Physica Scripta
- Vol. 100 (11) , 115236
Open AccessSteady 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 ( ϑ ), ξ ( ϑ ) ∼ (1 + ϑ ) α for 0 ≤ α ≤ 1. This extends the known theory for α = 1 from [1] to the values α ∈ (0, 1) and improves significantly the results for α = 0 from [2]. 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 > 3/2 for the pressure law p ( ϱ , ϑ ) ∼ ϱ γ + ϱϑ , where ϱ is the volume density of the gas. 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.
- Open Access
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.Open AccessWeak 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. - Open AccessOn a few counterexamples to solvability of the $div$ equation in domains with external cuspsThis 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.Open AccessOn a few counterexamples to solvability of the $div$ equation in domains with external cuspsThis 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.
- Open Access
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.Open AccessSteady 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. - Open AccessHomogenization 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.Open AccessHomogenization 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.
- 29 January 2023
- Published by Wiley in Mathematical Methods in the Applied Sciences
- Vol. 46 (8) , 9736-9750
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, modeling the behavior of a compressible non‐Newtonian fluid for small Reynolds number. We construct the weak solutions despite the lack of the 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 functions, which allowed us to adjust the method from Feireisl et al. (2015) to more relaxed conditions on the velocity.
- 29 January 2023
- Published by Wiley in Mathematical Methods in the Applied Sciences
- Vol. 46 (8) , 9736-9750
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, modeling the behavior of a compressible non‐Newtonian fluid for small Reynolds number. We construct the weak solutions despite the lack of the 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 functions, which allowed us to adjust the method from Feireisl et al. (2015) to more relaxed conditions on the velocity.
- Open Access
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.Open AccessWeak 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. - 1 December 2022
- Published by Springer Nature
- p. 335-350
Steady Compressible Navier–Stokes–Fourier Equations with Dirichlet Boundary Condition for the Temperature
- 1 December 2022
- Published by Springer Nature
- p. 335-350
Steady Compressible Navier–Stokes–Fourier Equations with Dirichlet Boundary Condition for the Temperature
- 22 July 2022
- Published by Springer Nature in Journal of Mathematical Fluid Mechanics
- Vol. 24 (3) , 1-27
Two-Phase Compressible/Incompressible Navier–Stokes System with Inflow-Outflow Boundary Conditions
- 22 July 2022
- Published by Springer Nature in Journal of Mathematical Fluid Mechanics
- Vol. 24 (3) , 1-27
Two-Phase Compressible/Incompressible Navier–Stokes System with Inflow-Outflow Boundary Conditions