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Ondřej Kreml

Publications and source records attributed to Ondřej Kreml.

At least 19 recordsLinked to original sources
arXiv2026

Maximal entropy production principle and the Euler system of gas dynamics

Convex integration has revealed that the Euler system of gas dynamics is ill-posed in the class of weak solutions even if the entropy inequality is imposed as an additional constraint. A natural question arises, namely, if a physically relevant solution can be selected by maximizing the entropy production rate. Firstly, we present an example of Riemann initial data in 2-D, for which the standard self-similar solution fails to satisfy the maximal entropy production principle. Hence, maximizing the entropy production rate rules out the 1-D self-similar solution which intuitively seems to be the physically relevant solution in this context. Secondly, we show for a large class of initial data that there exist entropy admissible weak solutions with an arbitrary (non-decreasing) total entropy profile.

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arXiv2025

On dissipative turbulent solutions to the compressible anisotropic Navier-Stokes equations in unbounded domains

Inspired by Abbatiello, Feireisl and Novotný, we prove the global existence of dissipative turbulent solution for the compressible Navier-Stokes equations with anisotropic viscous stress tensor on unbounded domain. Our work complements the result of Bresch and Jabin, where the authors used the new compactness method to prove the existence of a weak solution to the same system in $\mathbb{T}^3$. By virtue of the concept of dissipative turbulent solutions, we are able to relax assumptions on the anisotropic tensor coefficients and the pressure law coefficient. We point out that we establish the existence result on a large class of unbounded domains, which is more conform to geophysical context. We also prove the weak-strong uniqueness property of acquired dissipative turbulent solutions.

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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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arXiv2023

On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams

In this paper, we study a nonlinear fluid-structure interaction problem between a viscoelastic beam and a compressible viscous fluid. The beam is immersed in the fluid which fills a two-dimensional rectangular domain with periodic boundary conditions. Under the effect of periodic forces acting on the beam and the fluid, at least one time-periodic weak solution is constructed which has a bounded energy and a fixed prescribed mass.

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arXiv2020

Measure-valued solutions and weak-strong uniqueness for the incompressible inviscid fluid-rigid body interaction

We consider a coupled system of partial and ordinary differential equations describing the interaction between an isentropic inviscid fluid and a rigid body moving freely inside the fluid. We prove the existence of measure-valued solutions which is generated by the vanishing viscosity limit of incompressible fluid-rigid body interaction system under some physically constitutive relations. Moreover, we show that the measure-value solution coincides with strong solution on the interval of its existence. This relies on the weak-strong uniqueness analysis.

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arXiv2020

On oscillatory solutions to the complete Euler system

The Euler system in fluid dynamics is a model of a compressible inviscid fluid incorporating the three basic physical principles: Conservation of mass, momentum, and energy. We show that the Cauchy problem is basically ill-posed for the $L^\infty$-initial data in the class of weak entropy solutions. As a consequence, there are infinitely many measure-valued solutions for a vast set of initial data. Finally, using the concept of relative energy, we discuss a singular limit problem for the measure-valued solutions, where the Mach and Froude number are proportional to a small parameter.

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arXiv2020

Shocks Make the Riemann Problem for the Full Euler System in Multiple Space Dimensions Ill-posed

The question of (non-)uniqueness of one-dimensional self-similar solutions to the Riemann problem for hyperbolic systems of gas dynamics in sets of multi-dimensional admissible weak solutions was addressed in recent years in several papers culminating in [17] with the proof that the Riemann problem for the isentropic Euler system with a power law pressure is ill-posed if the one-dimensional self-similar solution contains a shock. Natural question then arises whether the same holds also for a more involved system of equations, the full Euler system. After the first step in this direction was made in [1], where the ill-posedness was proved in the case of two shocks appearing in the self-similar solution, we prove in this paper that the presence of just one shock in the self-similar solution implies the same outcome, i.e. the existence of infinitely many admissible weak solutions to the multi-dimensional problem.

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arXiv2019

Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data

We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant $T > 0$. In order to continue the solution after the formation of the discontinuity, we adjust and apply the theory developed by De Lellis and Székelyhidi and we construct infinitely many solutions. We introduce the notion of an admissible generalized fan subsolution to be able to handle data which are not piecewise constant and we reduce the argument to finding a single generalized subsolution.

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arXiv2018

Low stratification of a heat-conducting fluid in time-dependent domain

We study the low Mach number limit of the full Navier-Stokes-Fourier system in the case of low stratification with ill-prepared initial data for the problem stated on moving domain with prescribed motion of the boundary. Similarly as in the case of a fixed domain we recover as a limit the Oberback-Boussinesq system, however we identify one additional term in the temperature equation of the limit system which is related to the motion of the domain and which is not present in the case of a fixed domain. One of the main ingredients in the proof are the properties of the Helmholtz decomposition on moving domains and the dependence of eigenvalues and eigenspaces of the Neumann Laplace operator on time.

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arXiv2018

Local existence of strong solutions and weak-strong uniqueness for the compressible Navier-Stokes system on moving domains

We consider the compressible Navier-Stokes system on time-dependent domains with prescribed motion of the boundary. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong solutions. These results are obtained using a transformation of the problem to a fixed domain and an existence theorem for Navier-Stokes like systems with lower order terms and perturbed boundary conditions. We also show the weak-strong uniqueness principle for slip boundary conditions which remained so far open question.

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arXiv2018

Non-uniqueness of delta shocks and contact discontinuities in the multi-dimensional model of Chaplygin gas

We study the Riemann problem for the isentropic compressible Euler equations in two space dimensions with the pressure law describing the Chaplygin gas. It is well known that there are Riemann initial data for which the 1D Riemann problem does not have a classical $BV$ solution, instead a $δ$-shock appears, which can be viewed as a generalized measure-valued solution with a concentration measure in the density component. We prove that in the case of two space dimensions there exists infinitely many bounded admissible weak solutions starting from the same initial data. Moreover, we show the same property also for a subset of initial data for which the classical 1D Riemann solution consists of two contact discontinuities. As a consequence of the latter result we observe that any criterion based on the principle of maximal dissipation of energy will not pick the classical 1D solution as the physical one.

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arXiv2018

Dissipative measure valued solutions for general conservation laws

In the last years measure-valued solutions started to be considered as a relevant notion of solutions if they satisfy the so-called measure-valued -- strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. This property has been examined for many systems of mathematical physics, including incompressible and compressible Euler system, compressible Navier-Stokes system et al. and there are also some results concerning general hyperbolic systems. Our goal is to provide a unified framework for general systems, that would cover the most interesting cases of systems, and most importantly, we give examples of equations, for which the aspect of measure-valued -- strong uniqueness has not been considered before, like incompressible magentohydrodynamics and shallow water magnetohydrodynamics.

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arXiv2017

On contact discontinuities in multi-dimensional isentropic Euler equations

In this short note we partially extend the recent nonuniqueness results on admissible weak solutions to the Riemann problem for the 2D compressible isentropic Euler equations. We prove nonuniqueness of admissible weak solutions that start from the Riemann initial data allowing a contact discontinuity to emerge.

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arXiv2017

Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations

We study the Riemann problem for the multidimensional compressible isentropic Euler equations. Using the framework developed by Chiodaroli, De Lellis, Kreml and based on the techniques of De Lellis and Székelyhidi, we extend our previous results and prove that whenever the initial Riemann data give rise to a self-similar solution consisting of one admissible shock and one rarefaction wave and are not too far from lying on a simple shock wave, the problem admits also infinitely many admissible weak solutions.

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arXiv2016

Dimension reduction for the full Navier-Stokes-Fourier system

It is well known that the full Navier-Stokes-Fourier system does not possess a strong solution in three dimensions which causes problems in applications. However, when modeling the flow of a fluid in a thin long pipe, the influence of the cross section can be neglected and the flow is basically one-dimensional. This allows us to deal with strong solutions which are more convenient for numerical computations. The goal of this paper is to provide a rigorous justification of this approach. Namely, we prove that any suitable weak solution to the three-dimensional NSF system tends to a strong solution to the one-dimensional system as the thickness of the pipe tends to zero.

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arXiv2015

$\mathcal{A}$-free rigidity and applications to the compressible Euler system

Can every measure-valued solution to the compressible Euler equations be approximated by a sequence of weak solutions? We prove that the answer is negative: Generalizing a well-known rigidity result of Ball and James to a more general situation, we construct an explicit measure-valued solution for the compressible Euler equations which can not be generated by a sequence of distributional solutions. We also give an abstract necessary condition for measure-valued solutions to be generated by weak solutions, relying on work of Fonseca and Müller. This difference between weak and measure-valued solutions in the compressible case is in contrast with the incompressible situation, where every measure-valued solution can be approximated by weak solutions, as shown by Székelyhidi and Wiedemann.

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arXiv2014

Stability of the isentropic Riemann solutions of the full multidimensional Euler system

We consider the complete Euler system describing the time evolution of an inviscid non-isothermal gas. We show that the rarefaction wave solutions of the 1D Riemann problem are stable, in particular unique, in the class of all bounded weak solutions to the associated multi-D problem. This may be seen as a counterpart of the non-uniqueness results of physically admissible solutions emanating from 1D shock waves constructed recently by the method of convex integration.

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