On weak solutions of steady Navier-Stokes equations for monatomic gas

Jan Březina; Antonín Novotný

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 4, page 611-632
  • ISSN: 0010-2628

Abstract

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We use L ∞ estimates for the inverse Laplacian of the pressure introduced by Plotnikov, Sokolowski and Frehse, Goj, Steinhauer together with the nonlinear potential theory due to Adams, Hedberg, to get a priori estimates and to prove existence of weak solutions to steady isentropic Navier-Stokes equations with the adiabatic constant γ > 1 3 ( 1 + 13 ) ≈ 1 . 53 for the flows powered by volume non-potential forces and with γ > 1 8 ( 3 + 41 ) ≈ 1 . 175 for the flows powered by potential forces and arbitrary non-volume forces. According to our knowledge, it is the first result that treats in three dimensions existence of weak solutions in the physically relevant case γ ≤ 5 3 with arbitrary large external data. The solutions are constructed in a rectangular domain with periodic boundary conditions.

How to cite

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Březina, Jan, and Novotný, Antonín. "On weak solutions of steady Navier-Stokes equations for monatomic gas." Commentationes Mathematicae Universitatis Carolinae 49.4 (2008): 611-632. <http://eudml.org/doc/250496>.

@article{Březina2008,
abstract = {We use $L^\infty $ estimates for the inverse Laplacian of the pressure introduced by Plotnikov, Sokolowski and Frehse, Goj, Steinhauer together with the nonlinear potential theory due to Adams, Hedberg, to get a priori estimates and to prove existence of weak solutions to steady isentropic Navier-Stokes equations with the adiabatic constant $\gamma >\{1\over 3\}(1+\sqrt\{13\})\approx 1.53$ for the flows powered by volume non-potential forces and with $\gamma >\{1\over 8\}(3+\sqrt\{41\}) \approx 1.175$ for the flows powered by potential forces and arbitrary non-volume forces. According to our knowledge, it is the first result that treats in three dimensions existence of weak solutions in the physically relevant case $\gamma \le \{5\over 3\}$ with arbitrary large external data. The solutions are constructed in a rectangular domain with periodic boundary conditions.},
author = {Březina, Jan, Novotný, Antonín},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {steady compressible Navier-Stokes equations; periodic domain; isentropic flow; existence of the weak solution; potential theory; steady compressible Navier-Stokes equations; periodic domain; isentropic flow; existence of a weak solution; nonlinear potential theory},
language = {eng},
number = {4},
pages = {611-632},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On weak solutions of steady Navier-Stokes equations for monatomic gas},
url = {http://eudml.org/doc/250496},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Březina, Jan
AU - Novotný, Antonín
TI - On weak solutions of steady Navier-Stokes equations for monatomic gas
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 4
SP - 611
EP - 632
AB - We use $L^\infty $ estimates for the inverse Laplacian of the pressure introduced by Plotnikov, Sokolowski and Frehse, Goj, Steinhauer together with the nonlinear potential theory due to Adams, Hedberg, to get a priori estimates and to prove existence of weak solutions to steady isentropic Navier-Stokes equations with the adiabatic constant $\gamma >{1\over 3}(1+\sqrt{13})\approx 1.53$ for the flows powered by volume non-potential forces and with $\gamma >{1\over 8}(3+\sqrt{41}) \approx 1.175$ for the flows powered by potential forces and arbitrary non-volume forces. According to our knowledge, it is the first result that treats in three dimensions existence of weak solutions in the physically relevant case $\gamma \le {5\over 3}$ with arbitrary large external data. The solutions are constructed in a rectangular domain with periodic boundary conditions.
LA - eng
KW - steady compressible Navier-Stokes equations; periodic domain; isentropic flow; existence of the weak solution; potential theory; steady compressible Navier-Stokes equations; periodic domain; isentropic flow; existence of a weak solution; nonlinear potential theory
UR - http://eudml.org/doc/250496
ER -

References

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