×

Hypoelliptic estimates for a linear model of the Boltzmann equation without angular cutoff. (English) Zbl 1251.35064

In this paper the authors study the regularity of the following operator \[ P=\partial_t+v\cdot\partial_x+ a(t,x,v)(-\widetilde{\Delta}_v)^\sigma, \quad (t,x,v)\in \mathbb R\times \mathbb R^n\times\mathbb R^n, \] where \(0<\sigma<1\), \((-\widetilde{\Delta}_v)^\sigma\) is the fractional Laplacian, and \(a\) is a smooth function bounded below by a positive constant and whose derivatives of any order are bounded.
By means of microlocal techniques they show that \(P\) satisfies the following hypoelliptic estimate: for any compact subset \(K\subset \mathbb R^{2n+1}\), and for any \(s\geq 0\), there exists \(C=C_{K,s}>0\) such that \[ \|(1+|D_t|^\frac{2\sigma}{2\sigma+1} +|D_x|^\frac{2\sigma}{2\sigma+1} + |D_v|^{2\sigma}) u\|_{s}\leq C\left(\|Pu\|_s+\|u\|_s\right), \tag{1} \] for all smooth functions \(u\) whose support lies in \(K\).
Considering the case when \(a\) is a positive constant, they show that the exponent \({2\sigma}/(2\sigma+1)\) of the time and spatial variables in (1) is optimal.
As a corollary it is shown that if \(u\) is a distribution in the Sobolev space \(H_{-N}(\mathbb R^{2n+1})\) with \(Pu\in H^\infty(\mathbb R^{2n+1})\) then one must have \(u\) in \(C^\infty(\mathbb R^{2n+1})\).

MSC:

35Q20 Boltzmann equations
35H10 Hypoelliptic equations
35B65 Smoothness and regularity of solutions to PDEs
82C40 Kinetic theory of gases in time-dependent statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1080/00411459908205853 · Zbl 0939.35147 · doi:10.1080/00411459908205853
[2] DOI: 10.1051/m2an:2000157 · Zbl 0952.35167 · doi:10.1051/m2an:2000157
[3] DOI: 10.1007/s002050000083 · Zbl 0968.76076 · doi:10.1007/s002050000083
[4] DOI: 10.1016/j.jfa.2008.07.004 · Zbl 1166.35038 · doi:10.1016/j.jfa.2008.07.004
[5] DOI: 10.1007/s00205-010-0290-1 · Zbl 1257.76099 · doi:10.1007/s00205-010-0290-1
[6] Alexandre R., Comm. Math. Phys.
[7] Alexandre R., J. Funct. Anal.
[8] DOI: 10.1142/S0218202505000613 · Zbl 1161.35331 · doi:10.1142/S0218202505000613
[9] DOI: 10.3934/dcds.2009.24.1 · Zbl 1168.35326 · doi:10.3934/dcds.2009.24.1
[10] DOI: 10.1016/S0021-7824(02)01264-3 · Zbl 1045.35093 · doi:10.1016/S0021-7824(02)01264-3
[11] DOI: 10.1007/978-1-4612-1039-9 · doi:10.1007/978-1-4612-1039-9
[12] DOI: 10.1007/978-1-4419-8524-8 · doi:10.1007/978-1-4419-8524-8
[13] DOI: 10.1016/j.jde.2008.05.019 · Zbl 1162.35016 · doi:10.1016/j.jde.2008.05.019
[14] DOI: 10.1080/03605302.2010.507689 · Zbl 1242.35103 · doi:10.1080/03605302.2010.507689
[15] DOI: 10.1007/BF02101556 · Zbl 0827.76081 · doi:10.1007/BF02101556
[16] DOI: 10.1081/PDE-120028847 · Zbl 1103.82020 · doi:10.1081/PDE-120028847
[17] DOI: 10.1090/S0894-0347-2011-00697-8 · Zbl 1248.35140 · doi:10.1090/S0894-0347-2011-00697-8
[18] DOI: 10.1016/j.aim.2011.05.005 · Zbl 1234.35173 · doi:10.1016/j.aim.2011.05.005
[19] DOI: 10.1016/j.matpur.2010.11.003 · Zbl 1221.35107 · doi:10.1016/j.matpur.2010.11.003
[20] Hörmander L., The Analysis of Linear Partial Differential Operators (1985)
[21] DOI: 10.3934/krm.2008.1.453 · Zbl 1158.35332 · doi:10.3934/krm.2008.1.453
[22] DOI: 10.3934/krm.2009.2.647 · Zbl 1194.35089 · doi:10.3934/krm.2009.2.647
[23] Lerner N., Cubo Mat. Educ. 5 pp 213– (2003)
[24] DOI: 10.1007/978-3-540-68268-4_4 · doi:10.1007/978-3-540-68268-4_4
[25] DOI: 10.1007/978-3-7643-8510-1 · doi:10.1007/978-3-7643-8510-1
[26] DOI: 10.1016/S0764-4442(97)82709-7 · Zbl 0920.35114 · doi:10.1016/S0764-4442(97)82709-7
[27] DOI: 10.1007/s11868-010-0008-z · Zbl 1207.35015 · doi:10.1007/s11868-010-0008-z
[28] DOI: 10.3934/dcds.2009.24.187 · Zbl 1169.35315 · doi:10.3934/dcds.2009.24.187
[29] Morimoto Y., J. Math. Kyoto Univ. 47 pp 129– (2007) · Zbl 1146.35027 · doi:10.1215/kjm/1250281072
[30] DOI: 10.1016/j.jde.2009.01.028 · Zbl 1175.35024 · doi:10.1016/j.jde.2009.01.028
[31] DOI: 10.1080/03605300600635004 · Zbl 1101.76053 · doi:10.1080/03605300600635004
[32] DOI: 10.1016/j.matpur.2007.03.003 · Zbl 1388.76338 · doi:10.1016/j.matpur.2007.03.003
[33] DOI: 10.1002/mma.1670130508 · Zbl 0717.35017 · doi:10.1002/mma.1670130508
[34] DOI: 10.1112/S0024610706022952 · Zbl 1106.34060 · doi:10.1112/S0024610706022952
[35] DOI: 10.1007/BF03167864 · Zbl 0597.76072 · doi:10.1007/BF03167864
[36] DOI: 10.1007/s002050050106 · Zbl 0912.45011 · doi:10.1007/s002050050106
[37] DOI: 10.1016/S1874-5792(02)80004-0 · doi:10.1016/S1874-5792(02)80004-0
[38] DOI: 10.1080/03605309408821082 · Zbl 0818.35128 · doi:10.1080/03605309408821082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.