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Invariant Manifolds for Steady Boltzmann Flows and Applications

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Abstract

We develop a theory of invariant manifolds for the steady Boltzmann equation and apply it to the study of boundary layers and nonlinear waves. The steady Boltzmann equation is an infinite dimensional differential equation, so the standard center manifold theory for differential equations based on spectral information does not apply here. Instead, we employ a time-asymptotic approach using the pointwise information of Green’s function for the construction of the linear invariant manifolds. At the resonance cases when the Mach number at the far field is around one of the critical values of −1, 0 or 1, the truly nonlinear theory arises. In such a case, there are wave patterns combining the fast decaying Knudsen-type and slow varying fluid-like waves. The key Knudsen manifolds consisting of only Knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical Mach numbers. Around Mach number ± 1, the fluidlike waves are compressive and expansive waves; and around the Mach number 0, they are linear thermal layers. The quantitative analysis of the fluid-like waves is done using the reduction of dimensions to the center manifolds.Two-scale nonlinear dynamics based on those on the Knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns. There are striking bifurcations in the transition of evaporation to condensation and in the transition of the Milne’s problem with a subsonic far field to one with a supersonic far field. The analysis of these wave patterns allows us to understand the Sone Diagram for the study of the complete condensation boundary value problem. The monotonicity of the Boltzmann shock profiles, a problem that initially motivated the present study, is shown as a consequence of the quantitative analysis of the nonlinear fluid-like waves.

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References

  1. Bardos C., Caflisch R.E., Nicolaenko B.: The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas. Commun. Pure Appl. Math. 49, 323–352 (1986)

    Article  MathSciNet  Google Scholar 

  2. Bhatnagar P.L., Gross E.P., Krook M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)

    Article  ADS  MATH  Google Scholar 

  3. Boltzmann, L.: Vorlesungen über Gastheorie, 2 Vols., Barth, Leipzig, 1896–1898. English translation by Bush, S.G.: Lectures on Gas Theory. University of California Press, California, 1964

  4. Caflisch R.E., Nicolaenko B.: Shock profile solutions of the Boltzmann equation. Commun. Math. Phys. 86, 161–194 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Carleman T.: Sur La Théorie de l’Équation Intégrodifférentielle de Boltzmann. Acta Math. 60, 91–142 (1933)

    Article  MathSciNet  Google Scholar 

  6. Carleman, T.: Problémes mathématiques dans la théorie cinétique des gaz. (French) Publ. Sci. Inst. Mittag-Leffler. 2 Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957

  7. Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, New York, 1994

  8. Sone Y.: Molecular Gas Dynamics: Theory, Techniques, and Applications. Birkhäuser, Boston (2007)

    Book  Google Scholar 

  9. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 325. Springer, Berlin, 2005

  10. CoronF.; Golse F., Sulem C.: A classification of well-posed kinetic layer problems. Commun. Pure Appl. Math. 41, 409–435 (1988)

    Article  Google Scholar 

  11. Golse F: Analysis of the boundary layer equation in the kinetic theory of gases. Bull. Inst. Math. Acad. Sin. (N.S.) 3, 211–242 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Golse F., Poupaud F.: Steady solutions of the linearized Boltzmann equation in a half-space. Math. Methods Appl. Sci. 11, 483–502 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Grad, H.: Asymptotic theory of the Boltzmann equation. In: Laurmann, J.A. (ed.) Rarefied Gas Dynamics, vol. 1, pp. 26–59. Academic Press, New York, 1963

  14. Grad H.: Asymptotic theory of the Boltzmann equation. Phys Fluids 6, 147–181 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Grad, H.: Singular and nonuniform limits of solutions of the Boltzmann equation. Transport Theory (Proc. Sympos. Appl. Math., New York, 1967), SIAM-AMS Proc., vol. I, pp. 269–308. American Mathematical Society, Providence, 1969

  16. Kuo H.-W., Liu T.-P., Noh S.-E.: Mixture lemma. Bull. Inst. Math. Acad. Sin. (N.S.) 5, 1–10 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Lax P.D.: Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math. 10, 537–566 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu T.-P.: Pointwise convergence to shock waves for viscous conservation laws. Commun. Pure Appl. Math. 50, 1113–1182 (1997)

    Article  MATH  Google Scholar 

  19. Liu T.-P., Yu S.-H.: Propagation of a steady shock layer in the presence of a boundary. Arch Ration. Mech. Anal. 139(1), 57–82 (1997)

    Article  MATH  Google Scholar 

  20. Liu T.-P., Yu S.-H.: Boltzmann equation: micro–macro decompositions and positivity of shock profiles. Commun Math. Phys. 246, 133–179 (2004)

    Article  ADS  MATH  Google Scholar 

  21. Liu T.-P., Yu S.-H.: The Green’s function and large-time behavior of solutions for one-dimensional Boltzmann equation. Commun. Pure Appl. Math. 57, 1543–1608 (2004)

    Article  MATH  Google Scholar 

  22. Liu T.-P., Yu S.-H.: Green’s function of Boltzmann equation, 3-D waves,. Bull Inst. Math. Acad. Sin. (N.S.) 1, 1–78 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Liu T.-P., Yu S.-H.: Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation. Commun. Pure Appl. Math. 60, 295–356 (2007)

    Article  MATH  Google Scholar 

  24. Liu T.-P., Yu S.-H.: Solving the Boltzmann equation, Part I: Green’s function. Bull. Inst. Math. Acad. Sin. (N.S.) 6, 115–243 (2006)

    Google Scholar 

  25. Sone Y.: Kinetic theory analysis of linearized Rayleigh problem. J. Phys. Soc. Japan 19, 1463–1473 (1964)

    Article  ADS  Google Scholar 

  26. Sone Y.: Effect of sudden change of wall temperature in a rarefied gas. J. Phys. Soc. Japan 20, 222–229 (1965)

    Article  ADS  Google Scholar 

  27. Sone Y.: Thermal creep in rarefied gas. J. Phys. Soc. Japan 21, 1836–1837 (1966)

    Article  ADS  Google Scholar 

  28. Sone, Y.: Asymptotic theory of flow of rarefied gas over a smooth boundary I. In: Trilling, L., Wachman, H.Y. (eds.) Rarefied Gas Dynamics, vol. I, pp. 243–253. Academic Press, New York, 1969

  29. Sone, Y.: Asymptotic theory of flow of rarefied gas over a smooth boundary II. In: Dini, D. (ed.) Rarefied Gas Dynamics, vol. II, pp. 737–749. Editrice Tecnico Scientifica, Pisa, 1971

  30. Sone Y.: Kinetic theory of evaporation and condensation Linear and nonlinear problems. J. Phys. Soc. Japan 45, 315–320 (1978)

    Article  ADS  Google Scholar 

  31. Sone Y.: Kinetic theoretical studies of the half-space problem of evaporation and condensation. Transp. Theory Stat. Phys. 29, 227–260 (2000)

    Article  MATH  Google Scholar 

  32. Sone Y., Onishi Y.: Kinetic theory of evaporation and condensation: Hydrodynamic equation and slip boundary condition. J. Phys. Soc. Japan 44, 1981–1994 (1978)

    Article  ADS  Google Scholar 

  33. Sone, Y., Aoki, K, Yamashita, K.: A study of unsteady strong condensation on a plane condensed phase with special interest in formation of steady profile. In: Boffi, V., Cercignani, C. (eds.) Rarefied Gas Dynamics, vol. II, pp. 323–333. Teubner, Stuttgart, 1986

  34. Sone Y., Golse F., Ohwada T., Doi T.: Analytical study of transonic flows of a gas condensing onto its plane condensed phase on the basis of kinetic theory. Eur J. Mech. B/Fluids 17, 277–306 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sone, Y.: Kinetic Theory and Fluid Dynamics. Birkhäuser, Boston, 2002

  36. Sone, Y.: Molecular Gas Dynamics: Theory, Techniques, and Applications. Birkhäuser, Boston, 2007

  37. Tsai L.-C.: Viscous shock propagation with boundary effect. Bull. Inst. Math. Acad. Sin. (N.S.) 6, 1–26 (2011)

    MathSciNet  MATH  Google Scholar 

  38. Ukai S., Yang T., Yu S.-H.: Nonlinear boundary layers of the Boltzmann equation: I. Existence. Commun. Math. Phys. 236, 373–393 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Welander P.: On the temperature jump in a rarefied gas. Ark. Fys. 7, 507–553 (1954)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Academia Sinica, Taipei, Taiwan

    Tai-Ping Liu

  2. Department of Mathematics, Stanford University, Stanford, USA

    Tai-Ping Liu

  3. Department of Mathematics, National University of Singapore, Singapore, Singapore

    Shih-Hsien Yu

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  1. Tai-Ping Liu
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  2. Shih-Hsien Yu
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Liu, TP., Yu, SH. Invariant Manifolds for Steady Boltzmann Flows and Applications. Arch Rational Mech Anal 209, 869–997 (2013). https://doi.org/10.1007/s00205-013-0640-x

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  • DOI: https://doi.org/10.1007/s00205-013-0640-x

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