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Spectral Gap Formation to Kinetic Equations with Soft Potentials in Bounded Domain

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Abstract

It has been unknown in kinetic theory whether the linearized Boltzmann or Landau equation with soft potentials admits a spectral gap in the spatially inhomogeneous setting. Most of existing works indicate a negative answer because the spectrum of two linearized self-adjoint collision operators is accumulated to the origin in case of soft interactions. In the paper we rather prove it in an affirmative way when the space domain is bounded with an inflow boundary condition. The key strategy is to introduce a new Hilbert space with an exponential weight function that involves the inner product of space and velocity variables and also has the strictly positive upper and lower bounds. The action of the transport operator on such space-velocity dependent weight function induces an extra non-degenerate relaxation dissipation in large velocity that can be employed to compensate the degenerate spectral gap and hence give the exponential decay for solutions in contrast with the sub-exponential decay in either the spatially homogeneous case or the case of torus domain. The result reveals a new insight of hypocoercivity for kinetic equations with soft potentials in the specified situation.

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References

  1. Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152(4), 327–355 (2000)

    Article  MATH  Google Scholar 

  2. Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential. J. Funct. Anal. 262(3), 915–1010 (2012)

    Article  MATH  Google Scholar 

  3. Alexandre, R., Hérau, F., Li, W.-X.: Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff. J. Math. Pures Appl. 126, 1–71 (2019)

    Article  MATH  Google Scholar 

  4. Alonso, R., Morimoto, Y., Sun, W., Yang, T.: De Giorgi argument for weighted \(L^2 \cap L^\infty \) solutions to the non-cutoff Boltzmann equation. arXiv:2010.10065

  5. Alonso, R., Morimoto, Y., Sun, W., Yang, T.: Non-cutoff Boltzmann equation with polynomial decay perturbations. Rev. Mat. Iberoam. 37(1), 189–292 (2020)

    Article  MATH  Google Scholar 

  6. Baranger, C., Mouhot, C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Mat. Iberoam. 21, 819–841 (2005)

    Article  MATH  Google Scholar 

  7. Bobylev, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Rev. Sect. C. Math. Phys. Rev. 7, 111–233 (1988)

    MATH  Google Scholar 

  8. Bobylev, A.V., Gamba, I.M., Potapenko, I.: On some properties of the Landau kinetic equation. J. Stat. Phys. 161(6), 1327–1338 (2015)

    Article  MATH  ADS  Google Scholar 

  9. Bobylev, A.V., Gamba, I.M., Zhang, C.: On the rate of relaxation for the Landau kinetic equation and related models. J. Stat. Phys. 168(3), 535–548 (2017)

    Article  MATH  ADS  Google Scholar 

  10. Bouchut, F.: Hypoelliptic regularity in kinetic equations. J. Math. Pures Appl. 81(11), 1135–1159 (2002)

    Article  MATH  Google Scholar 

  11. Caflisch, R.E.: The Boltzmann equation with a soft potential. I. Linear, spatially homogeneous. Commun. Math. Phys. 74(1), 71–95 (1980)

    Article  MATH  ADS  Google Scholar 

  12. Caflisch, R.E.: The Boltzmann equation with a soft potential. II. Nonlinear, spatially periodic. Commun. Math. Phys. 74(2), 97–109 (1980)

    Article  MATH  ADS  Google Scholar 

  13. Carleman, T.: Problèmes mathématiques dans la théorie cinétique des gaz, p. 112. Almqvist & Wiksells Boktryckeri Ab, Uppsala (1957)

    MATH  Google Scholar 

  14. Carrapatoso, K., Mischler, S.: Landau equation for very soft and Coulomb potentials near Maxwellians. Ann. PDE 3(1), 65 (2017)

    Article  MATH  Google Scholar 

  15. Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)

    Book  MATH  Google Scholar 

  16. Wang Chang, C.S., Uhlenbeck, G.E.: On the Propagation of Sound in Monoatomic Gases. Univ. of Michigan Press, Ann Arbor, MI (1952)

    Google Scholar 

  17. Wang Chang, C.S., Uhlenbeck, G.E., de Boer, J.: The heat conductivity and viscosity of polyatomic gases. In: Studies in Statistical Mechanics, vol. II, pp. 241–268. North-Holland, Amsterdam (1964)

  18. Degond, P., Lemou, M.: Dispersion relations for the linearized Fokker–Planck equation. Arch. Ration. Mech. Anal. 138(2), 137–167 (1997)

    Article  MATH  Google Scholar 

  19. Deng, D.-Q.: Dissipation and semigroup on \(H^k_n\): non-cutoff linearized Boltzmann operator with soft potential. SIAM J. Math. Anal. 52(3), 3093–3113 (2020)

    Article  MATH  Google Scholar 

  20. DiPerna, R.J., Lions, P.L.: Global weak solutions of Vlasov–Maxwell systems. Commun. Pure Appl. Math. 42(6), 729–757 (1989)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  22. Duan, R.-J., Huang, F.-M., Wang, Y., Zhang, Z.: Effects of soft interaction and non-isothermal boundary upon long-time dynamics of rarefied gas. Arch. Ration. Mech. Anal. 234(2), 925–1006 (2019)

    Article  MATH  Google Scholar 

  23. Duan, R.-J., Liu, S.-Q., Sakamoto, S., Strain, R.M.: Global mild solutions of the Landau and non-cutoff Boltzmann equations. Commun. Pure Appl. Math. 74(5), 932–1020 (2020)

    Article  MATH  Google Scholar 

  24. Ellis, R.S., Pinsky, M.A.: The first and second fluid approximations to the linearized Boltzmann equation. J. Math. Pures Appl. 54, 125–156 (1975)

    MATH  Google Scholar 

  25. Engel, K.-J., Nagel, R., Campiti, M., Hahn, T.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (1999)

    Google Scholar 

  26. Gohberg, I., Goldberg, S., Kaashoek, M.: Classes of Linear Operators, vol. I (Operator Theory: Advances and Applications) (v. 1). Birkhäuser Verlag, Basel (1990)

  27. Golse, F., Poupaud, F.: Un résultat de compacité pour l’équation de Boltzmann avec potentiel mou. Application au problème du demi-espace. C. R. Acad. Sci. Paris Sér. I Math. 303, 585–586 (1986)

    MATH  Google Scholar 

  28. Grad, H.: Asymptotic Theory of the Boltzmann Equation. II. In: Rarefied Gas Dynamics (Proceedings 3rd International Symposium, Palais de l’UNESCO, Paris, 1962), vol. I, pp. 26–59 Academic Press, New York (1963)

  29. Gressman, P.T., Strain, R.M.: Global classical solutions of the Boltzmann equation without angular cut-off. J. Am. Math. Soc. 24(3), 771–771 (2011)

    Article  MATH  Google Scholar 

  30. Gualdani, M.P., Mischler, S., Mouhot, C.: Factorization of non-symmetric operators and exponential H-theorem. Mémoires de la Société mathématique de France 153, 1–137 (2017)

    Article  MATH  Google Scholar 

  31. Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231(3), 391–434 (2002)

    Article  MATH  ADS  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  34. Guo, Y.: Decay and continuity of the Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal. 197(3), 713–809 (2009)

    Article  MATH  Google Scholar 

  35. Guo, Y.: The Vlasov–Poisson–Landau system in a periodic box. J. Am. Math. Soc. 25(3), 759–812 (2012)

    Article  MATH  Google Scholar 

  36. Guo, Y., Hwang, H.J., Jang, J.W., Ouyang, Z.: The Landau equation with the specular reflection boundary condition. Arch. Ration. Mech. Anal. 236(3), 1389–1454 (2020)

    Article  MATH  Google Scholar 

  37. Liu, S.-Q., Yang, X.-F.: The initial boundary value problem for the Boltzmann equation with soft potential. Arch. Ration. Mech. Anal. 223(1), 463–541 (2016)

    Article  MATH  Google Scholar 

  38. Liu, T.-P., Yang, T., Yu, S.-H.: Energy method for Boltzmann equation. Physica D 188(3–4), 178–192 (2004)

    Article  MATH  ADS  Google Scholar 

  39. Liu, T.-P., Yu, S.-H.: Boltzmann equation: micro–macro decompositions and positivity of shock profiles. Commun. Math. Phys. 246(1), 133–179 (2004)

    Article  MATH  ADS  Google Scholar 

  40. Mokhtar-Kharroubi, M.: Mathematical Topics in Neutron Transport Theory. New Aspects. Series on Advances in Mathematics for Applied Sciences, 46. World Scientific Publishing Co., Singapore (1997)

    Book  MATH  Google Scholar 

  41. Mouhot, C.: Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Commun. Partial Differ. Equ. 31(9), 1321–1348 (2006)

    Article  MATH  Google Scholar 

  42. Mouhot, C., Neumann, L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity 19(4), 969–998 (2006)

    Article  MATH  ADS  Google Scholar 

  43. Mouhot, C., Strain, R.M.: Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J. Math. Pures Appl. 87(5), 515–535 (2007)

    Article  MATH  Google Scholar 

  44. Pao, Y.-P.: Boltzmann collision operator with inverse-power intermolecular potentials, I. Commun. Pure Appl. Math. 27(4), 407–428 (1974)

    Article  MATH  Google Scholar 

  45. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ Press, Princeton (1971)

    Book  Google Scholar 

  46. Strain, R.M., Guo, Y.: Almost exponential decay near Maxwellian. Commun. Partial Differ. Equ. 31(3), 417–429 (2006)

    Article  MATH  Google Scholar 

  47. Strain, R.M., Guo, Y.: Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. 187(2), 287–339 (2007)

    Article  MATH  Google Scholar 

  48. Strain, R.M., Zhu, K.: The Vlasov–Poisson–Landau system in \({\mathbb{R} }^3_x\). Arch. Ration. Mech. Anal. 210(2), 615–671 (2013)

    Article  MATH  Google Scholar 

  49. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Jpn. Acad. 50, 179–184 (1974)

    MATH  Google Scholar 

  50. Ukai, S.: Solutions of the Boltzmann Equation, Patterns and Waves, Studies in Mathematics and Applications, vol. 18, pp. 37–96. North-Holland, Amsterdam (1986)

    Google Scholar 

  51. Ukai, S., Asano, K.: On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. Res. Inst. Math. Sci. 18(2), 477–519 (1982)

    Article  MATH  Google Scholar 

  52. Ukai, S., Yang, T.: Mathematical Theory of Boltzmann Equation. Lecture Notes Series-No. 8. Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, Hong Kong (2006)

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

  54. Yang, T., Yu, H.-J.: Spectrum analysis of some kinetic equations. Arch. Ration. Mech. Anal. 222(2), 731–768 (2016)

    Article  MATH  Google Scholar 

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Acknowledgements

DQD was partially supported by the International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) (YJ20220056) and Direct Grant from BIMSA. RJD was partially supported by the NSFC/RGC Joint Research Scheme (N_CUHK409/19) from RGC in Hong Kong and the Direct Grant (4053452) from CUHK.

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  1. Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Tsinghua University, Beijing, People’s Republic of China

    Dingqun Deng

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s Republic of China

    Renjun Duan

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  1. Dingqun Deng
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Deng, D., Duan, R. Spectral Gap Formation to Kinetic Equations with Soft Potentials in Bounded Domain. Commun. Math. Phys. 397, 1441–1489 (2023). https://doi.org/10.1007/s00220-022-04519-2

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