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Spectrum structure and decay rate estimates on the Landau equation with Coulomb potential

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Abstract

In this paper we first deduce the estimates on the linearized Landau operator with Coulomb potential and then analyze its spectrum structure by using semigroup theory and linear operator perturbation theory. Based on these estimates, we give the precise time decay rate estimates on the semigroup generated by the linearized Landau operator so that the optimal time decay rates of the nonlinear Landau equation follow. In addition, we present a similar result for the non-angular cutoff Boltzmann equation with soft potentials.

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References

  1. Alexandre R, Morimoto Y, Ukai S, et al. The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions. Arch Ration Mech Anal, 2011, 202: 599–661

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandre R, Morimoto Y, Ukai S, et al. The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential. J Funct Anal, 2012, 262: 915–1010

    Article  MathSciNet  MATH  Google Scholar 

  3. Bardos C, Ukai S. The classical incompressible Navier-Stokes limit of the Boltzmann equation. Math Models Methods Appl Sci, 1991, 1: 235–257

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouchut F. Hypoelliptic regularity in kinetic equations. J Math Pures Appl (9), 2002, 81: 1135–1159

    Article  MathSciNet  MATH  Google Scholar 

  5. Caflisch R. The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous. Comm Math Phys, 1980, 74: 71–95

    Article  MathSciNet  MATH  Google Scholar 

  6. Caflisch R. The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic. Comm Math Phys, 1980, 74: 97–109

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Duan R J, Yu H J. The Vlasov-Poisson-Landau system near a local Maxwellian. Adv Math, 2020, 362: 106956

    Article  MathSciNet  MATH  Google Scholar 

  10. Ellis R, Pinsky M. The first and second fluid approximations to the linearized Boltzmann equation. J Math Pures Appl (9), 1975, 54: 125–156

    MathSciNet  MATH  Google Scholar 

  11. Glassey R T. The Cauchy Problem in Kinetic Theory. Philadelphia: SIAM, 1996

    Book  MATH  Google Scholar 

  12. Gressman T, Strain R M. Global classical solutions of the Boltzmann equation without angular cut-off. J Amer Math Soc, 2011, 24: 771–847

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo Y. The Landau equation in a periodic box. Comm Math Phys, 2002, 231: 391–434

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo Y. Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm Pure Appl Math, 2006, 59: 626–687

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo Y. The Vlasov-Poisson-Landau system in a periodic box. J Amer Math Soc, 2012, 25: 759–812

    Article  MathSciNet  MATH  Google Scholar 

  16. Hsiao L, Yu H J. On the Cauchy problem of the Boltzmann and Landau equations with soft potentials. Quart Appl Math, 2007, 65: 281–315

    Article  MathSciNet  MATH  Google Scholar 

  17. Kato T. Perturbation Theory for Linear Operators. New York: Springer-Verlag, 1976

    MATH  Google Scholar 

  18. Kawashima S. The Boltzmann equation and thirteen moments. North-Holl Math Stud, 1987, 148: 157–172

    Article  MathSciNet  MATH  Google Scholar 

  19. Kawashima S, Matsumura A, Nishida T. On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation. Comm Math Phys, 1979, 70: 97–124

    Article  MathSciNet  MATH  Google Scholar 

  20. Lei Y J, Wan L, Zhao H J. The Vlasov-Poisson-Landau system with a uniform ionic background and algebraic decay initial perturbation. Bull Inst Math Acad Sin (NS), 2015, 10: 311–347

    MathSciNet  MATH  Google Scholar 

  21. Li H L, Yang T, Zhong M Y. Spectrum structure and behaviors of the Vlasov-Maxwell-Boltzmann systems. SIAM J Math Anal, 2016, 48: 595–669

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu T P, Yu S H. Solving Boltzmann equation, part I: Green’s function. Bull Inst Math Acad Sin (NS), 2011, 6: 115–234

    MathSciNet  MATH  Google Scholar 

  24. Mouhot C, Strain R M. Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J Math Pures Appl (9), 2007, 87: 515–553

    Article  MathSciNet  MATH  Google Scholar 

  25. Nicolaenko B. Dispersion laws for plane wave propagation. In: The Boltzmann Equation. New York: Courant Institute of Mathematical Sciences, 1971, 125–173

    Google Scholar 

  26. Nishida T. Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Comm Math Phys, 1978, 61: 119–148

    Article  MathSciNet  MATH  Google Scholar 

  27. Reed M, Simon B. Methods of Modern Mathematical Physics. IV: Analysis of Operators. New York: Academic Press, 1978

    MATH  Google Scholar 

  28. Seeley R T. Extension of C∞ functions defined in a half space. Proc Amer Math Soc, 1964, 15: 625–626

    MathSciNet  MATH  Google Scholar 

  29. Strain R M. Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinet Relat Models, 2012, 5: 583–613

    Article  MathSciNet  MATH  Google Scholar 

  30. Strain R M, Guo Y. Almost exponential decay near Maxwellian. Comm Partial Differential Equations, 2006, 31: 417–429

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  34. Ukai S. Les solutions globales de l’equation de Boltzmann dans l’espace tout entier et dans le demi-espace. C R Acad Sci Paris Sér I Math, 1976, 282: A317–A320

    MathSciNet  MATH  Google Scholar 

  35. Ukai S, Asano K. On the Cauchy problem of the Boltzmann equation with a soft potential. Publ Res Inst Math Sci, 1982, 18: 57–99

    Article  MathSciNet  MATH  Google Scholar 

  36. Ukai S, Asano K. The Euler limit and initial layer of the nonlinear Boltzmann equation. Hokkaido Math J, 1983, 3: 311–332

    MathSciNet  MATH  Google Scholar 

  37. Ukai S, Yang T. Mathematical theory of Boltzmann equation. http://www.cityu.edu.hk/rcms/publications/ln8.pdf, 2006

  38. Villani C. A review of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Dynamics, vol. 1. Amsterdam: North-Holland, 2002, 71–74

    Chapter  MATH  Google Scholar 

  39. Yang T, Yu H J. Optimal convergence rates of Landau equation with external forcing in the whole space. Acta Math Sci Ser B Engl Ed, 2009, 29: 1035–1562

    MathSciNet  MATH  Google Scholar 

  40. Yang T, Yu H J. Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space. J Differential Equations, 2010, 248: 1518–1560

    Article  MathSciNet  MATH  Google Scholar 

  41. Yang T, Yu H J. Spectrum analysis of some kinetic equations. Arch Ration Mech Anal, 2016, 222: 731–768

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author was supported by the Research Grants Council of the Hong Kong Special Administrative Region of the People’s Republic of China (Grant No. SRF2021-1S01) and National Natural Science Foundation of China (Grant No. 11971200). The second author was supported by National Natural Science Foundation of China (Grant No. 11871229) and the Project Supported by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme 2017. The authors thank the anonymous referees for their helpful comments on this paper.

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Hong Kong, China

    Tong Yang

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Hongjun Yu

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  1. Tong Yang
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  2. Hongjun Yu
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Correspondence to Hongjun Yu.

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Yang, T., Yu, H. Spectrum structure and decay rate estimates on the Landau equation with Coulomb potential. Sci. China Math. 66, 37–78 (2023). https://doi.org/10.1007/s11425-020-1901-4

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