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Global solutions in the critical Sobolev space for the Landau equation

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Abstract

The Landau equation is studied for hard potential with −2 ≤ γ ≤ 1. Under a perturbation setting, a unique global solution of the Cauchy problem to the Landau equation is established in a critical Sobolev space \(H_x^dL_v^2(d > {3 \over 2})\), which extends the results of [11] in the torus domain to the whole space \(\mathbb{R}_x^3\). Here we utilize the pseudo-differential calculus to derive our desired result.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Alexandre R, Morimoto Y, Ukai S, et al. Regularizing effect and local existence for non-cutoff Boltzmann equation. Arch Ration Mech Anal, 2010, 198: 39–123

    Article  MathSciNet  Google Scholar 

  3. Alexandre R, Morimoto Y, Ukai S, et al. Local existence with mild regularity for the Boltzmann equation. Kinet Relat Models, 2013, 6: 1011–1041

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Carrapatoso K. Exponential convergence to equalilibrium for the homogeneous Landau equation with hard potentials. Bull Sci Math, 2015, 139: 777–805

    Article  MathSciNet  Google Scholar 

  6. Cao H M, Li W X, Xu C J. Analytic smoothing effect of the spatially inhomogeneous Landau equations for hard potentials. J Math Pures Appl, 2023, 176: 138–182

    Article  MathSciNet  Google Scholar 

  7. Cao H M, Li H G, Xu C J, Xu J. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinet Relat Models, 2019, 12: 829–884

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Desvillettes L, Villani C. On the spatially homogeneous Landau equation for hard potentials I. Existence, uniqueness and smoothness. Comm Partial Differential Equations, 2000, 25: 179–259

    Article  MathSciNet  Google Scholar 

  10. Desvillettes L, Villani C. On the spatially homogeneous Landau equation for hard potentials II. H-theorem and applications. Comm Partial Differential Equations, 2000, 25: 261–298

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Duan R J, He L B, Yang T, Zhou Y L. Solutions to the non-cutoff Boltzmann equation in the grazing limit. arXiv: 2105.13606

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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  Google Scholar 

  17. Kim C, Guo Y, Hwang H J. A L2 to L∞ approach for the Landau equation. Peking Math J, 2020, 3: 131–202

    Article  MathSciNet  Google Scholar 

  18. Li H G, Xu C J. Cauchy problem for the spatially homogeneous Landau equation with Shubin class initial datum and Gelfand-Shilov smoothing effect. Siam J Math Anal, 2019, 51: 532–564

    Article  MathSciNet  Google Scholar 

  19. Deng D Q. Global existence of non-cutoff Boltzmann equation in weighted Sobolev space. J Stat Phys, 2022, 188: Art 25

  20. Villani C. On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equation. Arch Rational Mech Anal, 1998, 143: 273–307

    Article  MathSciNet  Google Scholar 

  21. Villani C. On the Cauchy problem for Landau equation: Sequential stability, global existence. Adv Differential Equations, 1996, 1: 793–816

    Article  MathSciNet  Google Scholar 

  22. Wang H, Fang Z D. Global existence and time decay of the non-cutoff Boltzmann equation with hard potential. Nonlinear Anal: Real World Appl, 2021, 63: 103416

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Hao Wang

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  1. Hao Wang
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Correspondence to Hao Wang.

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Conflict of Interest The author declares no conflict of interest.

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This research was supported by the NSFC (12301284).

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Wang, H. Global solutions in the critical Sobolev space for the Landau equation. Acta Math Sci 44, 1347–1372 (2024). https://doi.org/10.1007/s10473-024-0410-x

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  • DOI: https://doi.org/10.1007/s10473-024-0410-x

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