American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity
  • KRM Home
  • This Issue
  • Next Article
    Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials
December  2010, 3(4): 669-683. doi: 10.3934/krm.2010.3.669

$L^1$ averaging lemma for transport equations with Lipschitz force fields

Daniel Han-Kwan 1,

1. 

École Normale Suprieure, Dpartement de Mathmatiques et Applications, 75230 Paris Cedex 05, France

Received  April 2010 Revised  September 2010 Published  October 2010

The purpose of this note is to extend the $L^1$ averaging lemma of Golse and Saint-Raymond [10] to the case of a kinetic transport equation with a force field $F(x)\in W^{1,\infty}$. To this end, we will prove a local in time mixing property for the transport equation $\partial_t f + v.\nabla_x f + F.\nabla_v f =0$.
Keywords: $L^1$ averaging lemma, External force field, Local in time mixing estimate..
Mathematics Subject Classification: Primary: 35Q83; Secondary: 82C7.
Citation: Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic & Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669
References:
[1]

V.I. Agoshkov, Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation, Dokl. Akad. Nauk SSSR, 276 (1984), 1289-1293.  Google Scholar

[2]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  Google Scholar

[3]

M. Bézard, Régularité $L^p$ précisée des moyennes dans les équations de transport, Bull. Soc. Math. France, 122 (1994), 29-76.  Google Scholar

[4]

F. Castella and B. Perthame, Estimations de Strichartz pour les équations de transport cinétique, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 535-540.  Google Scholar

[5]

R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 130 (1989), 321-366.  Google Scholar

[6]

R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math, 42 (1989), 729-757.  Google Scholar

[7]

R. J. DiPerna, P.-L. Lions and Y. Meyer, $L^p$ regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 271-287.  Google Scholar

[8]

F. Golse, B. Perthame and R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 341-344.  Google Scholar

[9]

F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.  Google Scholar

[10]

F. Golse and L. Saint-Raymond, Velocity averaging in $L^1$ for the transport equation, C. R. Acad. Sci. Paris Sér. I Math., 334 (2002), 557-562.  Google Scholar

[11]

F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  Google Scholar

[12]

P.-E. Jabin, Averaging lemmas and dispersion estimates for kinetic equations, Riv. Mat. Univ. Parma, 8 (2009), 71-138.  Google Scholar

[13]

P.-E. Jabin and L. Vega, A real space method for averaging lemmas, J. de Math. Pures et Appl., 83 (2004), 1309-1351.  Google Scholar

[14]

B. Perthame and P. Souganidis, A limiting case for velocity averaging, Ann. Sci. Ecole Norm. Sup., 31 (1998), 591-598.  Google Scholar

[15]

D. Salort, Weighted dispersion and Strichartz estimates for the Liouville equation in $1D$, Asymptot. Anal., 47 (2006), 85-94.  Google Scholar

[16]

D. Salort, Dispersion and Strichartz estimates for the Liouville equation, J. Differential Equations, 233 (2007), 543-584.  Google Scholar

[17]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Models Methods Appl. Sci., 19 (2009), 199-228.  Google Scholar

show all references

References:
[1]

V.I. Agoshkov, Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation, Dokl. Akad. Nauk SSSR, 276 (1984), 1289-1293.  Google Scholar

[2]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  Google Scholar

[3]

M. Bézard, Régularité $L^p$ précisée des moyennes dans les équations de transport, Bull. Soc. Math. France, 122 (1994), 29-76.  Google Scholar

[4]

F. Castella and B. Perthame, Estimations de Strichartz pour les équations de transport cinétique, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 535-540.  Google Scholar

[5]

R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 130 (1989), 321-366.  Google Scholar

[6]

R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math, 42 (1989), 729-757.  Google Scholar

[7]

R. J. DiPerna, P.-L. Lions and Y. Meyer, $L^p$ regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 271-287.  Google Scholar

[8]

F. Golse, B. Perthame and R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 341-344.  Google Scholar

[9]

F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125.  Google Scholar

[10]

F. Golse and L. Saint-Raymond, Velocity averaging in $L^1$ for the transport equation, C. R. Acad. Sci. Paris Sér. I Math., 334 (2002), 557-562.  Google Scholar

[11]

F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  Google Scholar

[12]

P.-E. Jabin, Averaging lemmas and dispersion estimates for kinetic equations, Riv. Mat. Univ. Parma, 8 (2009), 71-138.  Google Scholar

[13]

P.-E. Jabin and L. Vega, A real space method for averaging lemmas, J. de Math. Pures et Appl., 83 (2004), 1309-1351.  Google Scholar

[14]

B. Perthame and P. Souganidis, A limiting case for velocity averaging, Ann. Sci. Ecole Norm. Sup., 31 (1998), 591-598.  Google Scholar

[15]

D. Salort, Weighted dispersion and Strichartz estimates for the Liouville equation in $1D$, Asymptot. Anal., 47 (2006), 85-94.  Google Scholar

[16]

D. Salort, Dispersion and Strichartz estimates for the Liouville equation, J. Differential Equations, 233 (2007), 543-584.  Google Scholar

[17]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Models Methods Appl. Sci., 19 (2009), 199-228.  Google Scholar

[1]

Shaofei Wu, Mingqing Wang, Maozhu Jin, Yuntao Zou, Lijun Song. Uniform $L^1$ stability of the inelastic Boltzmann equation with large external force for hard potentials. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1005-1013. doi: 10.3934/dcdss.2019068
[2]

Huaiyu Jian, Hongjie Ju, Wei Sun. Traveling fronts of curve flow with external force field. Communications on Pure & Applied Analysis, 2010, 9 (4) : 975-986. doi: 10.3934/cpaa.2010.9.975
[3]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997
[4]

Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic & Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499
[5]

Bernard Ducomet. Asymptotics for 1D flows with time-dependent external fields. Conference Publications, 2007, 2007 (Special) : 323-333. doi: 10.3934/proc.2007.2007.323
[6]

Fei Meng, Xiao-Ping Yang. Elastic limit and vanishing external force for granular systems. Kinetic & Related Models, 2019, 12 (1) : 159-176. doi: 10.3934/krm.2019007
[7]

Hongjun Yu. Global classical solutions to the Boltzmann equation with external force. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1647-1668. doi: 10.3934/cpaa.2009.8.1647
[8]

Lan Wen. A uniform $C^1$ connecting lemma. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 257-265. doi: 10.3934/dcds.2002.8.257
[9]

Zhigang Wu, Wenjun Wang. Uniform stability of the Boltzmann equation with an external force near vacuum. Communications on Pure & Applied Analysis, 2015, 14 (3) : 811-823. doi: 10.3934/cpaa.2015.14.811
[10]

T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665
[11]

Rafael López, Óscar Perdomo. Constant-speed ramps for a central force field. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3447-3464. doi: 10.3934/dcds.2021003
[12]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175
[13]

Jacky Cresson. The transfer lemma for Graff tori and Arnold diffusion time. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 787-800. doi: 10.3934/dcds.2001.7.787
[14]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763
[15]

Arnaud Goullet, Ian Glasgow, Nadine Aubry. Dynamics of microfluidic mixing using time pulsing. Conference Publications, 2005, 2005 (Special) : 327-336. doi: 10.3934/proc.2005.2005.327
[16]

Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253
[17]

T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265
[18]

Karine Beauchard, Morgan Morancey. Local controllability of 1D Schrödinger equations with bilinear control and minimal time. Mathematical Control & Related Fields, 2014, 4 (2) : 125-160. doi: 10.3934/mcrf.2014.4.125
[19]

T. Tachim Medjo. Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1281-1305. doi: 10.3934/cpaa.2011.10.1281
[20]

T. Tachim Medjo. Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1119-1140. doi: 10.3934/cpaa.2014.13.1119

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences