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December  2010, 3(4): 669-683. doi: 10.3934/krm.2010.3.669

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

 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 $\partial_t f + v.\nabla_x f + F.\nabla_v f =0$.
Citation: Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic and 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. [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. [3] M. Bézard, Régularité $L^p$ précisée des moyennes dans les équations de transport, Bull. Soc. Math. France, 122 (1994), 29-76. [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. [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. [6] R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math, 42 (1989), 729-757. [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. [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. [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. [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. [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. [12] P.-E. Jabin, Averaging lemmas and dispersion estimates for kinetic equations, Riv. Mat. Univ. Parma, 8 (2009), 71-138. [13] P.-E. Jabin and L. Vega, A real space method for averaging lemmas, J. de Math. Pures et Appl., 83 (2004), 1309-1351. [14] B. Perthame and P. Souganidis, A limiting case for velocity averaging, Ann. Sci. Ecole Norm. Sup., 31 (1998), 591-598. [15] D. Salort, Weighted dispersion and Strichartz estimates for the Liouville equation in $1D$, Asymptot. Anal., 47 (2006), 85-94. [16] D. Salort, Dispersion and Strichartz estimates for the Liouville equation, J. Differential Equations, 233 (2007), 543-584. [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.

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. [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. [3] M. Bézard, Régularité $L^p$ précisée des moyennes dans les équations de transport, Bull. Soc. Math. France, 122 (1994), 29-76. [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. [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. [6] R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math, 42 (1989), 729-757. [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. [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. [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. [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. [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. [12] P.-E. Jabin, Averaging lemmas and dispersion estimates for kinetic equations, Riv. Mat. Univ. Parma, 8 (2009), 71-138. [13] P.-E. Jabin and L. Vega, A real space method for averaging lemmas, J. de Math. Pures et Appl., 83 (2004), 1309-1351. [14] B. Perthame and P. Souganidis, A limiting case for velocity averaging, Ann. Sci. Ecole Norm. Sup., 31 (1998), 591-598. [15] D. Salort, Weighted dispersion and Strichartz estimates for the Liouville equation in $1D$, Asymptot. Anal., 47 (2006), 85-94. [16] D. Salort, Dispersion and Strichartz estimates for the Liouville equation, J. Differential Equations, 233 (2007), 543-584. [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.
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