The Gelfand–Shilov smoothing effect for the radially symmetric homogeneous Landau equation with Shubin initial datum

Li, Hao-Guang

Comptes Rendus. Mathématique, Tome 356 (2018) no. 6, pp. 613-625.

Dans cet article, nous étudions le problème de Cauchy associé à l'équation de Landau spatialement homogène à symétrie radiale et sans troncature angulaire avec des molécules de Maxwell, tandis que la donnée initiale appartient à un espace de Shubin d'indice négatif, qui peut être caractérisé à partir de la décomposition spectrale de l'oscillateur harmonique quantique. En utilisant cette décomposition spectrale, nous construisons une solution faible avec une donnée initiale dans un espace de Shubin, puis nous prouvons l'unicité et un effet de régularisation de Gelfand–Shilov de la solution à ce problème de Cauchy.

In this paper, we study the Cauchy problem associated with the radially symmetric spatially homogeneous non-cutoff Landau equation with Maxwellian molecules, while the initial datum belongs to negative-index Shubin space, which can be characterized by spectral decomposition of the harmonic oscillators. Based on this spectral decomposition, we construct the weak solution with Shubin's class initial datum, and then we prove the uniqueness and the Gelfand–Shilov smoothing effect of the solution to this Cauchy problem.

[1] Glangetas, L.; Najeme, M. Analytical regularizing effect for the radial homogeneous Boltzmann equation, Kinet. Relat. Models, Volume 6 (2013), pp. 407-427

[2] Glangetas, L.; Li, H.-G.; Xu, C.-J. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation, Kinet. Relat. Models, Volume 9 (2016), pp. 299-371

[3] Gramchev, T.; Pilipović, S.; Rodino, L., Birkhäuser, Basel, Switzerland (2009), pp. 15-31

[4] Lekrine, N.; Xu, C.-J. Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac equation, Kinet. Relat. Models, Volume 2 (2009), pp. 647-666

[5] Lerner, N.; Morimoto, Y.; Pravda-Starov, K.; Xu, C.-J. Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators, Kinet. Relat. Models, Volume 6 (2013), pp. 625-648

[6] Lerner, N.; Morimoto, Y.; Pravda-Starov, K.; Xu, C.-J. Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff, J. Differ. Equ., Volume 256 (2014), pp. 797-831

[7] Li, H.-G.; Xu, C.-J. Cauchy problem for the spatially homogeneous Landau equation with Shubin class initial datum and Gelfand–Shilov smoothing effect | arXiv

[8] Morimoto, Y.; Ukai, S. Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff, J. Pseud.-Differ. Oper. Appl., Volume 1 (2010), pp. 139-159

[9] Morimoto, Y.; Ukai, S.; Xu, C.-J.; Yang, T. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete Contin. Dyn. Syst., Volume 24 (2009), pp. 187-212

[10] Morimoto, Y.; Pravda-Starov, K.; Xu, C.-J. A remark on the ultra-analytic smoothing properties of the spatially homogeneous Boltzmann equation, Kinet. Relat. Models, Volume 6 (2013), pp. 715-727

[11] Sansone, G. Orthogonal Functions, Pure and Applied Mathematics, vol. IX, 1959

[12] Shubin, M. Pseudodifferential Operators and Spectral Theory, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987

[13] Villani, C. On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., Volume 143 (1998), pp. 273-307

[14] Villani, C. On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., Volume 8 (1998), pp. 957-983

[15] Villani, C. A review of mathematical topics in collisional kinetic theory (Friedlander, S.; Serre, D., eds.), Handbook of Mathematical Fluid Dynamics, vol. 1, 2002, pp. 71-305

[16] Zhang, T.-F.; Yin, Z. Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff, J. Differ. Equ., Volume 253 (2012), pp. 1172-1190