Skip to main content
Top

2024 | OriginalPaper | Chapter

Hydrodynamic Limit from the Boltzmann Equation in a Slightly Compressible Regime

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We discuss the hydrodynamic limit for the Boltzmann equation under the diffusive scaling with initial and/or boundary non homogeneous conditions for density and temperature with gradients of order 1.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Arkeryd, L., Esposito, R., Marra, R., Nouri, A.: Stability of the laminar solution of the Boltzmann equation for the Benard problem. Bull. Acad. Sin. 3, 51–97 (2008)MathSciNet Arkeryd, L., Esposito, R., Marra, R., Nouri, A.: Stability of the laminar solution of the Boltzmann equation for the Benard problem. Bull. Acad. Sin. 3, 51–97 (2008)MathSciNet
2.
go back to reference Arkeryd, L., Esposito, R., Marra, R., Nouri, A.: Stability for Rayleigh-Benard convective solutions of the Boltzmann equation. Arch. Rat. Mech. 198, 125–187 (2010)MathSciNetCrossRef Arkeryd, L., Esposito, R., Marra, R., Nouri, A.: Stability for Rayleigh-Benard convective solutions of the Boltzmann equation. Arch. Rat. Mech. 198, 125–187 (2010)MathSciNetCrossRef
3.
go back to reference Arkeryd, L., Esposito, R., Marra, R., Nouri, A.: Ghost effect by curvature in planar Couette flow. Kinet. Relat. Model. 4, 109–138 (2011)MathSciNetCrossRef Arkeryd, L., Esposito, R., Marra, R., Nouri, A.: Ghost effect by curvature in planar Couette flow. Kinet. Relat. Model. 4, 109–138 (2011)MathSciNetCrossRef
4.
go back to reference Arkeryd, L., Nouri, A.: Asymptotic techniques for kinetic problems of Boltzmann type. In: Proceedings of the 3rd edition of the summer school in “Methods and Models of kinetic theory", Riv. Mat. Univ. Parma 7, 1–74 (2007) Arkeryd, L., Nouri, A.: Asymptotic techniques for kinetic problems of Boltzmann type. In: Proceedings of the 3rd edition of the summer school in “Methods and Models of kinetic theory", Riv. Mat. Univ. Parma 7, 1–74 (2007)
5.
6.
go back to reference Brull, S.: The stationary Boltzmann equation for a two-component gas in the slab. Math. Methods Appl. Sci. 31, 153–178 (2008)MathSciNetCrossRef Brull, S.: The stationary Boltzmann equation for a two-component gas in the slab. Math. Methods Appl. Sci. 31, 153–178 (2008)MathSciNetCrossRef
7.
go back to reference Brull, S.: Problem of evaporation-condensation for a two component gas in the slab. Kinet. Relat. Models 1, 185–221 (2008)MathSciNetCrossRef Brull, S.: Problem of evaporation-condensation for a two component gas in the slab. Kinet. Relat. Models 1, 185–221 (2008)MathSciNetCrossRef
8.
go back to reference Caflisch, R.E.: The fluid dynamic limit of the nonlinear Boltzmann equation. Commun. Pure Appl. Math. 33, 651–666 (1980)MathSciNetCrossRef Caflisch, R.E.: The fluid dynamic limit of the nonlinear Boltzmann equation. Commun. Pure Appl. Math. 33, 651–666 (1980)MathSciNetCrossRef
9.
go back to reference De Masi, A., Esposito, R., Lebowitz, J.L.: Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Comm. Pure Appl. Math. 42, 1189–1214 (1989)MathSciNetCrossRef De Masi, A., Esposito, R., Lebowitz, J.L.: Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Comm. Pure Appl. Math. 42, 1189–1214 (1989)MathSciNetCrossRef
10.
go back to reference Esposito, R., Lebowitz, J.L., Marra, R.: Hydrodynamic limit of the stationary Boltzmann equation in a slab. Comm. Math. Phys. 160, 49–80 (1994). The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation. J. Stat. Phys. 78, 389–412 (1995) Esposito, R., Lebowitz, J.L., Marra, R.: Hydrodynamic limit of the stationary Boltzmann equation in a slab. Comm. Math. Phys. 160, 49–80 (1994). The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation. J. Stat. Phys. 78, 389–412 (1995)
11.
go back to reference Esposito, R., Lebowitz, J.L., Marra, R.: Solutions to the Boltzmann equation in the Boussinesq regime. J. Stat. Phys. 90, 1129-1178 (1998). Fluid Dynamics, vol. 3. In: Serre, D., Friedlander, F. (eds.). Elsevier (2004) Esposito, R., Lebowitz, J.L., Marra, R.: Solutions to the Boltzmann equation in the Boussinesq regime. J. Stat. Phys. 90, 1129-1178 (1998). Fluid Dynamics, vol. 3. In: Serre, D., Friedlander, F. (eds.). Elsevier (2004)
12.
go back to reference Esposito, R., Garrido, P.L., Lebowitz, J.L., Marra, R.: Diffusive limit for a Boltzmann-like equation with non-conserved momentum. Nonlinearity 32, 4834–4852 (2019)MathSciNetCrossRef Esposito, R., Garrido, P.L., Lebowitz, J.L., Marra, R.: Diffusive limit for a Boltzmann-like equation with non-conserved momentum. Nonlinearity 32, 4834–4852 (2019)MathSciNetCrossRef
13.
go back to reference Garrido, P.L., Lebowitz, J.L.: Diffusion equations from kinetic models with non-conserved momentum. Nonlinearity 32, 5441–5462 (2018)MathSciNetCrossRef Garrido, P.L., Lebowitz, J.L.: Diffusion equations from kinetic models with non-conserved momentum. Nonlinearity 32, 5441–5462 (2018)MathSciNetCrossRef
14.
go back to reference Esposito, R., Guo, Y., Kim, C., Marra, R.: Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann. PDE 4, 1–119 (2018)MathSciNetCrossRef Esposito, R., Guo, Y., Kim, C., Marra, R.: Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann. PDE 4, 1–119 (2018)MathSciNetCrossRef
15.
go back to reference Esposito, R., Guo, Y., Marra, R., et al.: Ghost effect from Boltzmann theory: expansion with remainder. Vietnam J. Math. 52, 883–914 (2024) Esposito, R., Guo, Y., Marra, R., et al.: Ghost effect from Boltzmann theory: expansion with remainder. Vietnam J. Math. 52, 883–914 (2024)
17.
18.
go back to reference Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Inven. Math. 155, 81–161 (2004)MathSciNetCrossRef Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Inven. Math. 155, 81–161 (2004)MathSciNetCrossRef
19.
go back to reference Golse, F., Saint-Raymond, L.: The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. 91, 508–552 (2009)MathSciNetCrossRef Golse, F., Saint-Raymond, L.: The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. 91, 508–552 (2009)MathSciNetCrossRef
20.
go back to reference Guo, Y.: Boltzmann diffusive limit beyond the Navier-Stokes approximation. Commun. Pure Appl. Math. 59, 626–687 (2006)MathSciNetCrossRef Guo, Y.: Boltzmann diffusive limit beyond the Navier-Stokes approximation. Commun. Pure Appl. Math. 59, 626–687 (2006)MathSciNetCrossRef
21.
go back to reference Guo, Y., Jang, N., Jang, J.: Acoustic limit for the Boltzmann equation in optimal scaling. Commun. Pure Appl. Math. 63, 337–361 (2010)MathSciNetCrossRef Guo, Y., Jang, N., Jang, J.: Acoustic limit for the Boltzmann equation in optimal scaling. Commun. Pure Appl. Math. 63, 337–361 (2010)MathSciNetCrossRef
23.
go back to reference Kogan, M.N., Galkin, V.S., Fridlender, O.G.: Stresses produced in gases by temperature and concentration inhomogeneities. New types of free convection. Sov. Phys. Usp. 19(5), 420–428 (1976) Kogan, M.N., Galkin, V.S., Fridlender, O.G.: Stresses produced in gases by temperature and concentration inhomogeneities. New types of free convection. Sov. Phys. Usp. 19(5), 420–428 (1976)
24.
go back to reference Saint-Raymond, L.: Hydrodynamic Limits of the Boltzmann Equation. Springer-Verlag, Berlin (2009)CrossRef Saint-Raymond, L.: Hydrodynamic Limits of the Boltzmann Equation. Springer-Verlag, Berlin (2009)CrossRef
25.
go back to reference Sone, Y.: Thermal creep in rarefied gas. J. Phys. Soc. Jpn 21, 1836–1837 (1966)CrossRef Sone, Y.: Thermal creep in rarefied gas. J. Phys. Soc. Jpn 21, 1836–1837 (1966)CrossRef
26.
go back to reference Sone, Y.: Flow induced by thermal stress in rarefied gas. Phys. Fluids 15, 1418–1423 (1972)CrossRef Sone, Y.: Flow induced by thermal stress in rarefied gas. Phys. Fluids 15, 1418–1423 (1972)CrossRef
27.
go back to reference Sone, Y.: Kinetic Theory and Fluid Dynamics. Birkhäuser, Boston (2002); Molecular Gas Dynamics, Theory, Techniques, and Applications. World Scientific, Birkhäuser, Boston (2007) Sone, Y.: Kinetic Theory and Fluid Dynamics. Birkhäuser, Boston (2002); Molecular Gas Dynamics, Theory, Techniques, and Applications. World Scientific, Birkhäuser, Boston (2007)
Metadata
Title
Hydrodynamic Limit from the Boltzmann Equation in a Slightly Compressible Regime
Author
Rossana Marra
Copyright Year
2024
DOI
https://doi.org/10.1007/978-3-031-65195-3_9

Premium Partner