Skip to main content
SpringerLink
Log in

The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

Boundary effects are central to the dynamics of the dilute particles governed by the Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for the Boltzmann equation with a soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate \({L^\infty}\) analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in \({L^\infty}\) space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori \({L^\infty}\) estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the \({L^2-L^\infty}\) theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted \({L^\infty}\) theory so that we could deal with the greater difficulties stemming from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove that the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in \({L^\infty}\) space with the aid of the L 2 theory and a bootstrap argument. These methods, in the latter case, can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Arkeryd L., Cercignani C.: A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal. Arch. Rational Mech. Anal. 125(3), 271–287 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Arkeryd L., Maslova N.: On diffuse reflection at the boundary for the Boltzmann equation and related equations. J. Stat. Phys. 77(5–6), 1051–1077 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Briant, M., Guo, Y.: Asymptotic stability of the Boltzmann equation with Maxwell boundary conditions. arXiv:1511.01305

  4. Cercignani C.: On the initial-boundary value problem for the Boltzmann equation. Arch. Ration. Mech. Anal. 116(4), 307–315 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cercignani C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  6. Chapman, S., Colwing, T. G.: The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge Math. Lib., Cambridge University Press, Cambridge, 1990

  7. Desvillettes L.: Convergence to equilibrium in large time for Boltzmann and B.G.K. equations. Arch. Rational Mech. Anal. 110(1), 73–91 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Desvillettes L., Villani C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. DiPerna R.J., Lions P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130(2), 321–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. DiPerna R.J., Lions P.-L.: Global weak solution of Vlasov-Maxwell systems. Commun. Pure Appl. Math. 42, 729–757 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duan R.-J.: Global smooth dynamics of a fully ionized plasma with long-range collisions. Ann. Inst. H. Poincar Anal. Non Linaire 31(4), 751–778 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Duan R.-J., Liu S.-Q.: Stability of the rarefaction wave of the Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal. 47(5), 3585–3647 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duan R.-J., Liu S.-Q., Yang T., Zhao H.-J.: Stability of the nonrelativistic Vlasov–Maxwell–Boltzmann system for angular non-cutoff potentials. Kinet. Relat. Models 6(1), 159–204 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Duan R.-J., Yang T., Zhao H.-J.: The Vlasov-Poisson-Boltzmann system in the whole space: the hard potential case. J. Differ. Equ. 252(12), 6356–6386 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Duan R.-J., Yang T., Zhao H.-J.: The Vlasov–Poisson–Boltzmann system for soft potentials. Math. Models Methods Appl. Sci. 23(6), 979–1028 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Esposito R., Guo Y., Marra R.: Phase transition in a Vlasov–Boltzmann binary mixture. Commun. Math. Phys. 296(1), 1–33 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Esposito R., Guo Y., Kim C., Marra R.: Non-isothermal boundary in the Boltzmann theory and Fourier law. Commun. Math. Phys. 323(1), 177–239 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Esposito, R., Guo, Y., Kim, C., Marra, R.: Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit. arXiv:1502.05324

  19. Esposito R., Lebowitz J. L., Marra R.: Hydrodynamic limit of the stationary Boltzmann equation in a slab. Commun. Math. Phys. 160, 49–80 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Esposito R., Lebowitz J. L., Marra R.: The Navier–Stokes limit of stationary solutions of the nonlinear Boltzmann equation. J. Stat. Phys. 78, 389–412 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Golse F., Perthame B., Sulem C.: On a boundary layer problem for the nonlinear Boltzmann equation Arch. Ration. Mech. Anal. 103, 81–96 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Grad, H.: Principles of the kinetic theory of gases. In: Handbuch der Physik, vol. XII, pp. 205–294. Springer, Berlin, 1958

  23. Grad, H.: Asymptotic theory of the Boltzmann equation. II. 1963 Rarefied Gas Dynamics. Proceedings of the 3rd international Symposium, pp. 26–59, Paris, 1962

  24. Guo Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169(4), 305–353 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guo Y.: The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53(4), 1081–1094 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Guo Y.: Boltzmann diffusive limit beyond the Navier–Stokes approximation. Commun. Pure. Appl. Math. 55(9), 0626–0687 (2006)

    Article  MathSciNet  Google Scholar 

  27. Guo Y.: Decay and continuity of the Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal. 197(3), 713–809 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Guo, Y., Kim, C., Tonon, D., Trescases, A.: Regularity of the Boltzmann equation in convex domains, to appear in Invent. Math. doi:10.1007/s00222-016-0670-8

  29. Guo Y., Kim C., Tonon D., Trescases A.: BV-regularity of the Boltzmann equation in non-convex domains. Arch. Ration. Mech. Anal. 220(3), 1045–1093 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hamdache K.: Initial boundary value problems for Boltzmann equation: global existence of week solutions. Arch. Ration. Mech. Anal. 119(4), 309–353 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kim C.: Formation and propagation of discontinuity for Boltzmann equation in non-convex domains. Commun. Math. Phys. 308(3), 641–701 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Kuo H.-W., Liu T.-P., Tsai L.-C.: Equilibrating effects of boundary and collision in rarefied gases. Commun. Math. Phys. 328(2), 421–480 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Liu T.-P., Yu S.-H.: Invariant manifolds for steady Boltzmann flows and applications. Arch. Ration. Mech. Anal. 209(3), 869–997 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu T.-P., Yu S.-H.: The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation. Commun. Pure Appl. Math. 57(12), 1543–1608 (2004)

    Article  MATH  Google Scholar 

  35. Liu T.-P., Yu S.-H.: Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation. Commun. Pure Appl. Math. 60(3), 295–356 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu T.-P., Yu S.-H.: Boltzmann equation, boundary effects. Discr. Contin. Dyn. Syst. 24(1), 145–157 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Masmoudi N., Saint-Raymond L.: From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Commun. Pure Appl. Math. 56(9), 1263–1293 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mischler S.: On the initial boundary value problem for the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 210(2), 447–466 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Shizuta Y., Asano K.: Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Jpn Acad. Ser. A Math. Sci. 53(1), 3–5 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sone, Y.: Molecular Gas Dynamics. Theory, Techniques, and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Inc., Boston, MA, 2007

  41. Strain R. M.: Asymptotic stability of the relativistic Boltzmann equation for the soft potentials. Commun. Math. Phys. 300(2), 529–597 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Strain R.M., Guo Y.: Almost exponential decay near Maxwellian. Commun. Partial Differ. Equ. 31, 417–429 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Strain R.M., Guo Y.: Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. 187(2), 287–339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ukai S.: Solutions of the Boltzmann equations In: pattern and Waves–Qualitative Analysis of Nonlinear Differential Equations. Stud. Math. Appl. 18, 37–96 (1986)

    MathSciNet  Google Scholar 

  45. Ukai S., Yang T.: The Boltzmann equation in the space \({L^2\cap L^\infty_\beta}\): global and time-periodic solutions. Anal. Appl. (Singap.) 4, 263–310 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  46. Vidav I.: Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl. 30, 264–279 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  47. Villani, C.: Hypocoercivity, Mem. Am. Math. Soc. 202(950), iv+141 (2009)

  48. Yang T., Zhao H.-J.: A half-space problem for the Boltzmann equation with specular reflection boundary condition. Commun. Math. Phys. 255(3), 683–726 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Yu S.-H.: Stochastic formulation for the initial-boundary value problems of the Boltzmann equation. Arch. Ration. Mech. Anal. 192(2), 217–274 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Jinan University, Guangzhou, 510632, People’s Republic of China

    Shuangqian Liu

  2. School of Mathematical Science, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Xiongfeng Yang

Authors
  1. Shuangqian Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiongfeng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuangqian Liu.

Additional information

Communicated by L. Saint-Raymond

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, S., Yang, X. The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential. Arch Rational Mech Anal 223, 463–541 (2017). https://doi.org/10.1007/s00205-016-1038-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-016-1038-3

Search

Navigation