# American Institute of Mathematical Sciences

August  2021, 14(4): 599-638. doi: 10.3934/krm.2021017

## Incompressible Navier-Stokes-Fourier limit from the Landau equation

 Université de Nantes, Laboratoire de Mathematiques Jean Leray, 2, rue de la Houssinière, BP 92208 F-44322 Nantes Cedex 3, France

Received  December 2020 Revised  March 2021 Published  August 2021 Early access  May 2021

In this work, we provide a result on the derivation of the incompressible Navier-Stokes-Fourier system from the Landau equation for hard, Maxwellian and moderately soft potentials. To this end, we first investigate the Cauchy theory associated to the rescaled Landau equation for small initial data. Our approach is based on proving estimates of some adapted Sobolev norms of the solution that are uniform in the Knudsen number. These uniform estimates also allow us to obtain a result of weak convergence towards the fluid limit system.

Citation: Mohamad Rachid. Incompressible Navier-Stokes-Fourier limit from the Landau equation. Kinetic and Related Models, 2021, 14 (4) : 599-638. doi: 10.3934/krm.2021017
##### References:
 [1] R. J. Alonso, B. Lods and I. Tristani, Fluid dynamic limit of boltzmann equation for granular hard–spheres in a nearly elastic regime, arXiv: 2008.05173, 2020. [2] D. Arsénio and L. Saint-Raymond, From the Vlasov-Maxwell-Boltzmann System to Incompressible Viscous Electro-Magneto-Hydrodynamics. Vol. 1, EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2019. doi: 10.4171/193. [3] C. Bardos, F. Golse and C. D. Levermore, The acoustic limit for the Boltzmann equation, Arch. Ration. Mech. Anal., 153 (2000), 177-204.  doi: 10.1007/s002050000080. [4] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅰ: Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608. [5] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183., Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0. [6] M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, J. Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022. [7] M. Briant, S. Merino-Aceituno and C. Mouhot, From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight, Anal. Appl. (Singap.), 17 (2019), 85-116.  doi: 10.1142/S021953051850015X. [8] K. Carrapatoso, I. Tristani and K.-C. Wu, Cauchy problem and exponential stability for the inhomogeneous Landau equation, Arch. Ration. Mech. Anal., 221 (2016), 363-418.  doi: 10.1007/s00205-015-0963-x. [9] P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in $1$ and $2$ space dimensions, Ann. Sci. École Norm. Sup., 19 (1986), 519-542.  doi: 10.24033/asens.1516. [10] 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.  doi: 10.2307/1971423. [11] R. S. Ellis and M. A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl., 54 (1975), 125-156. [12] F. Golse and C. D. Levermore, Stokes-Fourier and acoustic limits for the Boltzmann equation, Comm. Pure Appl. Math., 55 (2002), 336-393.  doi: 10.1002/cpa.3011. [13] F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81–161. doi: 10.1007/s00222-003-0316-5. [14] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization of non-symmetric operators and exponential $H$-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137. doi: 10.24033/msmf.461. [15] Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9. [16] Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.  doi: 10.1512/iumj.2004.53.2574. [17] Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Commun. Pure Appl. Math., 59 (2006), 626–687. doi: 10.1002/cpa.20121. [18] N. Jiang, C. D. Levermore and N. Masmoudi, Remarks on the acoustic limit for the Boltzmann equation, Comm. Partial Differential Equations, 35 (2010), 1590-1609.  doi: 10.1080/03605302.2010.496096. [19] N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain Ⅰ, Comm. Pure Appl. Math., 70 (2017), 90-171.  doi: 10.1002/cpa.21631. [20] N. Jiang, C.-J. Xu and H. Zhao, Incompressible Navier-Stokes-Fourier limit from the Boltzmann equation: Classical solutions, Indiana Univ. Math. J., 67 (2018), 1817–1855. doi: 10.1512/iumj.2018.67.5940. [21] J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. [22] P.-L. Lions and N. Masmoudi, Une approche locale de la limite incompressible, C. R. Acad. Sci., Paris, Sér. I, Math., 329 (1999), 387–392. doi: 10.1016/S0764-4442(00)88611-5. [23] P.-L. Lions and N. Masmoudi, From Boltzmann equation to Navier-Stokes and Euler equations Ⅰ, Arch. Ration. Mech. Anal., 158 (2001), 173-193.  doi: 10.1007/s002050100143. [24] P.-L. Lions and N. Masmoudi, From Boltzmann equation to Navier-Stokes and Euler equations Ⅱ, Arch. Ration. Mech. Anal., 158 (2001), 195-211. [25] A. J. Majda, A. L. Bertozzi and A. Ogawa, Vorticity and incompressible flow. Cambridge texts in applied mathematics, Appl. Mech. Rev., 55 (2002), B77–B78. [26] N. Masmoudi and L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, Comm. Pure Appl. Math., 56 (2003), 1263-1293.  doi: 10.1002/cpa.10095. [27] S. Mischler, Kinetic equations with Maxwell boundary conditions, Ann. Sci. Éc. Norm. Supér., 43 (2010), 719-760.  doi: 10.24033/asens.2132. [28] C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, J. Math. Pures Appl., 87 (2007), 515-535.  doi: 10.1016/j.matpur.2007.03.003. [29] L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, vol. 1971 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-92847-8. [30] H. Wang and K.-C. Wu, Solving linearized Landau equation pointwisely, arXiv preprint, 2017. arXiv: 1709.00839.

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##### References:
 [1] R. J. Alonso, B. Lods and I. Tristani, Fluid dynamic limit of boltzmann equation for granular hard–spheres in a nearly elastic regime, arXiv: 2008.05173, 2020. [2] D. Arsénio and L. Saint-Raymond, From the Vlasov-Maxwell-Boltzmann System to Incompressible Viscous Electro-Magneto-Hydrodynamics. Vol. 1, EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2019. doi: 10.4171/193. [3] C. Bardos, F. Golse and C. D. Levermore, The acoustic limit for the Boltzmann equation, Arch. Ration. Mech. Anal., 153 (2000), 177-204.  doi: 10.1007/s002050000080. [4] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅰ: Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608. [5] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183., Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0. [6] M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, J. Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022. [7] M. Briant, S. Merino-Aceituno and C. Mouhot, From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight, Anal. Appl. (Singap.), 17 (2019), 85-116.  doi: 10.1142/S021953051850015X. [8] K. Carrapatoso, I. Tristani and K.-C. Wu, Cauchy problem and exponential stability for the inhomogeneous Landau equation, Arch. Ration. Mech. Anal., 221 (2016), 363-418.  doi: 10.1007/s00205-015-0963-x. [9] P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in $1$ and $2$ space dimensions, Ann. Sci. École Norm. Sup., 19 (1986), 519-542.  doi: 10.24033/asens.1516. [10] 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.  doi: 10.2307/1971423. [11] R. S. Ellis and M. A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl., 54 (1975), 125-156. [12] F. Golse and C. D. Levermore, Stokes-Fourier and acoustic limits for the Boltzmann equation, Comm. Pure Appl. Math., 55 (2002), 336-393.  doi: 10.1002/cpa.3011. [13] F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81–161. doi: 10.1007/s00222-003-0316-5. [14] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization of non-symmetric operators and exponential $H$-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137. doi: 10.24033/msmf.461. [15] Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9. [16] Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.  doi: 10.1512/iumj.2004.53.2574. [17] Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Commun. Pure Appl. Math., 59 (2006), 626–687. doi: 10.1002/cpa.20121. [18] N. Jiang, C. D. Levermore and N. Masmoudi, Remarks on the acoustic limit for the Boltzmann equation, Comm. Partial Differential Equations, 35 (2010), 1590-1609.  doi: 10.1080/03605302.2010.496096. [19] N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain Ⅰ, Comm. Pure Appl. Math., 70 (2017), 90-171.  doi: 10.1002/cpa.21631. [20] N. Jiang, C.-J. Xu and H. Zhao, Incompressible Navier-Stokes-Fourier limit from the Boltzmann equation: Classical solutions, Indiana Univ. Math. J., 67 (2018), 1817–1855. doi: 10.1512/iumj.2018.67.5940. [21] J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. [22] P.-L. Lions and N. Masmoudi, Une approche locale de la limite incompressible, C. R. Acad. Sci., Paris, Sér. I, Math., 329 (1999), 387–392. doi: 10.1016/S0764-4442(00)88611-5. [23] P.-L. Lions and N. Masmoudi, From Boltzmann equation to Navier-Stokes and Euler equations Ⅰ, Arch. Ration. Mech. Anal., 158 (2001), 173-193.  doi: 10.1007/s002050100143. [24] P.-L. Lions and N. Masmoudi, From Boltzmann equation to Navier-Stokes and Euler equations Ⅱ, Arch. Ration. Mech. Anal., 158 (2001), 195-211. [25] A. J. Majda, A. L. Bertozzi and A. Ogawa, Vorticity and incompressible flow. Cambridge texts in applied mathematics, Appl. Mech. Rev., 55 (2002), B77–B78. [26] N. Masmoudi and L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, Comm. Pure Appl. Math., 56 (2003), 1263-1293.  doi: 10.1002/cpa.10095. [27] S. Mischler, Kinetic equations with Maxwell boundary conditions, Ann. Sci. Éc. Norm. Supér., 43 (2010), 719-760.  doi: 10.24033/asens.2132. [28] C. Mouhot and R. M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, J. Math. Pures Appl., 87 (2007), 515-535.  doi: 10.1016/j.matpur.2007.03.003. [29] L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, vol. 1971 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-92847-8. [30] H. Wang and K.-C. Wu, Solving linearized Landau equation pointwisely, arXiv preprint, 2017. arXiv: 1709.00839.
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