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  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 & Related Models, 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. Google Scholar

[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.  Google Scholar

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C. BardosF. Golse and C. D. Levermore, The acoustic limit for the Boltzmann equation, Arch. Ration. Mech. Anal., 153 (2000), 177-204.  doi: 10.1007/s002050000080.  Google Scholar

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C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅰ: Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. BriantS. 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.  Google Scholar

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K. CarrapatosoI. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.  Google Scholar

[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.  Google Scholar

[17]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Commun. Pure Appl. Math., 59 (2006), 626–687. doi: 10.1002/cpa.20121.  Google Scholar

[18]

N. JiangC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[27]

S. Mischler, Kinetic equations with Maxwell boundary conditions, Ann. Sci. Éc. Norm. Supér., 43 (2010), 719-760.  doi: 10.24033/asens.2132.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

H. Wang and K.-C. Wu, Solving linearized Landau equation pointwisely, arXiv preprint, 2017. arXiv: 1709.00839. Google Scholar

show all references

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. Google Scholar

[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.  Google Scholar

[3]

C. BardosF. Golse and C. D. Levermore, The acoustic limit for the Boltzmann equation, Arch. Ration. Mech. Anal., 153 (2000), 177-204.  doi: 10.1007/s002050000080.  Google Scholar

[4]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅰ: Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. BriantS. 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.  Google Scholar

[8]

K. CarrapatosoI. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.  Google Scholar

[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.  Google Scholar

[17]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Commun. Pure Appl. Math., 59 (2006), 626–687. doi: 10.1002/cpa.20121.  Google Scholar

[18]

N. JiangC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[27]

S. Mischler, Kinetic equations with Maxwell boundary conditions, Ann. Sci. Éc. Norm. Supér., 43 (2010), 719-760.  doi: 10.24033/asens.2132.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

H. Wang and K.-C. Wu, Solving linearized Landau equation pointwisely, arXiv preprint, 2017. arXiv: 1709.00839. Google Scholar

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