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

Ondřej Kreml, Tomasz Piasecki, Milan Pokorný, Emil Skříšovský

TL;DR

This work studies the steady compressible Navier–Stokes–Fourier system with general temperature-dependent viscosities, establishing existence results via Bogovskii-type density estimates and ballistic-energy frameworks. By combining energy and entropy (or ballistic energy) inequalities with density estimates and compactness arguments, the authors prove the existence of renormalized variational entropy solutions under heat-flux boundary conditions and, for Dirichlet temperature data, ballistic-energy variants, subject to precise constraints on the exponents $\alpha$, $m$, and $\gamma$. The analysis hinges on the effective viscous flux identity and oscillation defect measures to obtain strong convergence of the density, enabling passage to the limit in nonlinear pressure and energy terms. The results extend prior works (notably for $\alpha=1$ and $\alpha=0$) and set the stage for a second paper that treats a broader range of pressure estimates and larger $\gamma$ regimes, including $\gamma>1$ cases. Overall, the paper advances the existence theory for steady NSF–Fourier systems with temperature-dependent viscosities and multiple boundary conditions, via a robust density-estimation framework based on Bogovskii operators.

Abstract

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.

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

TL;DR

This work studies the steady compressible Navier–Stokes–Fourier system with general temperature-dependent viscosities, establishing existence results via Bogovskii-type density estimates and ballistic-energy frameworks. By combining energy and entropy (or ballistic energy) inequalities with density estimates and compactness arguments, the authors prove the existence of renormalized variational entropy solutions under heat-flux boundary conditions and, for Dirichlet temperature data, ballistic-energy variants, subject to precise constraints on the exponents , , and . The analysis hinges on the effective viscous flux identity and oscillation defect measures to obtain strong convergence of the density, enabling passage to the limit in nonlinear pressure and energy terms. The results extend prior works (notably for and ) and set the stage for a second paper that treats a broader range of pressure estimates and larger regimes, including cases. Overall, the paper advances the existence theory for steady NSF–Fourier systems with temperature-dependent viscosities and multiple boundary conditions, via a robust density-estimation framework based on Bogovskii operators.

Abstract

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 , for . This extends the known theory for from and improves significantly the results for . 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 for the pressure law . 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 only is based on different type of pressure estimates and will be the content of the subsequent paper.
Paper Structure (15 sections, 10 theorems, 214 equations)

This paper contains 15 sections, 10 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Constitutive relations. Main results
  3. Constitutive relations
  4. Weak solution
  5. Main results
  6. Preliminaries
  7. Proof for the heat flux boundary condition
  8. Energy and entropy estimates
  9. Estimates of the density based on Bogovskii operator
  10. Limit passage I
  11. Limit passage II: strong convergence of the density
  12. Proof for the Dirichlet boundary conditions for the temperature
  13. Energy and entropy estimates
  14. Density estimates by Bogovskii operator
  15. Limit passage for the sequence of solutions

Key Result

Theorem 1

Let $\Omega \subset \mathbb{R}^3$ be a $C^2$ bounded domain. Let $M>0$ in tot_mass, $L>0$, $\vartheta_0 \in L^1(\partial \Omega)$, $\vartheta_0 \geq T_0 >0$ a.e. at $\partial \Omega$ in temp_heatflux and ${\bf f}\in L^\infty(\Omega)$. Let Then there exists a renormalized variational entropy solution to problem NSF_1, vel_Dir (or vel_Nav with $\sigma=0$), temp_heatflux and tot_mass in the sense of

Theorems & Definitions (13)

  • Definition 1: Heat flux boundary condition for the temperature
  • Definition 2: Dirichlet condition for the temperature
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 3: Korn inequality I
  • Lemma 4: Korn inequality II
  • Lemma 5: Commutators I
  • Lemma 6: Commutators II
  • Lemma 7: Friedrich commutator lemma
  • ...and 3 more