Commutator method for averaging lemmas. (English) Zbl 1501.35109
Summary: We introduce a commutator method with multiplier to prove averaging lemmas, the regularizing effect for the velocity average of solutions for kinetic equations. Our method requires only elementary techniques in Fourier analysis and highlights a new range of assumptions that are sufficient for the velocity average to be in \smash{\(L^2([0,T],H^{1/2}_x)\)}. Our result provides a direct proof (without interpolation) and improves the regularizing result for the measure-valued solutions to scalar conservation laws in space dimension 1.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35L65 | Hyperbolic conservation laws |
| 35Q49 | Transport equations |
| 35Q83 | Vlasov equations |
| 42B37 | Harmonic analysis and PDEs |
Keywords:
velocity-averaging lemma; kinetic transport equation; dispersion; singular integral; multiplier; scalar conservation law; measure-valued solutionReferences:
| [1] | ; Agoshkov, V. I., Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation, Dokl. Akad. Nauk SSSR, 276, 6, 1289 (1984) · Zbl 0599.35009 |
| [2] | 10.2140/paa.2021.3.319 · Zbl 1478.35062 · doi:10.2140/paa.2021.3.319 |
| [3] | 10.1016/j.matpur.2013.06.012 · Zbl 1293.35192 · doi:10.1016/j.matpur.2013.06.012 |
| [4] | 10.2140/apde.2019.12.333 · Zbl 1404.35076 · doi:10.2140/apde.2019.12.333 |
| [5] | 10.1016/j.jfa.2011.07.020 · Zbl 1231.42023 · doi:10.1016/j.jfa.2011.07.020 |
| [6] | 10.24033/bsmf.2222 · Zbl 0798.35025 · doi:10.24033/bsmf.2222 |
| [7] | 10.1016/S0021-7824(02)01264-3 · Zbl 1045.35093 · doi:10.1016/S0021-7824(02)01264-3 |
| [8] | 10.1016/S0294-1449(03)00024-6 · Zbl 1041.35019 · doi:10.1016/S0294-1449(03)00024-6 |
| [9] | 10.1090/S0894-0347-00-00359-3 · Zbl 1001.35079 · doi:10.1090/S0894-0347-00-00359-3 |
| [10] | 10.1007/BF00752112 · Zbl 0616.35055 · doi:10.1007/BF00752112 |
| [11] | 10.1002/cpa.3160420603 · Zbl 0698.35128 · doi:10.1002/cpa.3160420603 |
| [12] | 10.2307/1971423 · Zbl 0698.45010 · doi:10.2307/1971423 |
| [13] | 10.1007/BF01393835 · Zbl 0696.34049 · doi:10.1007/BF01393835 |
| [14] | 10.1016/S0294-1449(16)30264-5 · Zbl 0763.35014 · doi:10.1016/S0294-1449(16)30264-5 |
| [15] | 10.1215/kjm/1250519013 · Zbl 0807.35026 · doi:10.1215/kjm/1250519013 |
| [16] | ; Escobedo, Miguel; Mischler, Stéphane; Valle, Manuel A., Homogeneous Boltzmann equation in quantum relativistic kinetic theory. Elec. J. Differential Eq. Monogr., 4 (2003) · Zbl 1103.82022 |
| [17] | 10.1017/S0962492916000088 · Zbl 1382.76001 · doi:10.1017/S0962492916000088 |
| [18] | 10.1007/s10208-015-9299-z · Zbl 1371.65080 · doi:10.1007/s10208-015-9299-z |
| [19] | 10.1006/jdeq.1998.3595 · Zbl 0927.35048 · doi:10.1006/jdeq.1998.3595 |
| [20] | 10.1007/978-3-642-54271-8_1 · Zbl 1319.35155 · doi:10.1007/978-3-642-54271-8\_1 |
| [21] | 10.4171/RMI/765 · Zbl 1288.35343 · doi:10.4171/RMI/765 |
| [22] | 10.3233/ASY-1992-6202 · Zbl 0784.35084 · doi:10.3233/ASY-1992-6202 |
| [23] | 10.1016/S1631-073X(02)02302-6 · Zbl 1154.35326 · doi:10.1016/S1631-073X(02)02302-6 |
| [24] | ; Golse, François; Saint-Raymond, Laure, Hydrodynamic limits for the Boltzmann equation, Riv. Mat. Univ. Parma (7), 4**, 1 (2005) · Zbl 1109.35112 |
| [25] | 10.1016/0022-1236(88)90051-1 · Zbl 0652.47031 · doi:10.1016/0022-1236(88)90051-1 |
| [26] | 10.3934/krm.2010.3.669 · Zbl 1209.35139 · doi:10.3934/krm.2010.3.669 |
| [27] | 10.1007/BF02392081 · Zbl 0156.10701 · doi:10.1007/BF02392081 |
| [28] | 10.1002/cpa.3005 · Zbl 1124.35312 · doi:10.1002/cpa.3005 |
| [29] | 10.1051/cocv:2002033 · Zbl 1065.35185 · doi:10.1051/cocv:2002033 |
| [30] | 10.1016/j.crma.2003.09.004 · Zbl 1030.35005 · doi:10.1016/j.crma.2003.09.004 |
| [31] | 10.1016/j.matpur.2004.03.004 · Zbl 1082.35043 · doi:10.1016/j.matpur.2004.03.004 |
| [32] | 10.2969/jmsj/05010179 · Zbl 0917.35129 · doi:10.2969/jmsj/05010179 |
| [33] | ; Kruzhkov, S. N., First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81(123), 2, 228 (1970) · Zbl 0202.11203 |
| [34] | 10.2307/2152725 · Zbl 0820.35094 · doi:10.2307/2152725 |
| [35] | 10.1007/BF02102014 · Zbl 0799.35151 · doi:10.1007/BF02102014 |
| [36] | 10.1137/050630763 · Zbl 1206.82133 · doi:10.1137/050630763 |
| [37] | ; Oleinik, O. A., Discontinuous solutions of non-linear differential equations, Usp. Mat. Nauk (N.S.), 12, 3(75), 3 (1957) · Zbl 0131.31803 |
| [38] | ; Perthame, Benoît, Kinetic formulation of conservation laws. Oxford Lect. Ser. Math. Appl., 21 (2002) · Zbl 1030.35002 |
| [39] | 10.1007/978-3-540-92847-8 · Zbl 1171.82002 · doi:10.1007/978-3-540-92847-8 |
| [40] | 10.1081/PDE-120005841 · Zbl 1010.35015 · doi:10.1081/PDE-120005841 |
| [41] | ; Stein, Elias M., Singular integrals and differentiability properties of functions. Princeton Math. Ser., 30 (1970) · Zbl 0207.13501 |
| [42] | 10.1002/cpa.20180 · Zbl 1131.35004 · doi:10.1002/cpa.20180 |
| [43] | 10.1137/S0036141000380760 · Zbl 1067.35021 · doi:10.1137/S0036141000380760 |
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.
