×

From Boltzmann’s equation to the incompressible Navier-Stokes-Fourier system with long-range interactions. (English) Zbl 1257.35140

Summary: We establish a rigorous demonstration of the hydrodynamic convergence of the Boltzmann equation towards a Navier-Stokes-Fourier system under the presence of long-range interactions. This convergence is obtained by letting the Knudsen number tend to zero and has been known to hold, at least formally, for decades. It is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered. However, these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption, which requires that the cross sections be integrable. Indeed, as soon as long-range intermolecular forces are present, non-integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas. In this long-range setting, the Boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case. In fact, the R. J. DiPerna and P. L. Lions renormalized solutions [Ann. Math. (2) 130, No. 2, 321–366 (1989; Zbl 0698.45010)] do not even make sense in this situation and the relevant global solutions to the Boltzmann equation are the so-called renormalized solutions with a defect measure developed by R. Alexandre and C. Villani [Commun. Pure Appl. Math. 55, No. 1, 30–70 (2002; Zbl 1029.82036)]. Our work overcomes the new mathematical difficulties coming from the consideration of long-range interactions by proving the hydrodynamic convergence of the Alexandre-Villani solutions towards the Leray solutions.

MSC:

35Q20 Boltzmann equations
35Q30 Navier-Stokes equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alexandre R., Desvillettes L., Villani C., Wennberg B.: Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152(4), 327–355 (2000) · Zbl 0968.76076
[2] Alexandre R., Morimoto Y., Ukai S., Xu C.-J., Yang T.: The Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential. Anal. Appl. (Singap.) 9(2), 113–134 (2011) · Zbl 1220.35110
[3] Alexandre R., Morimoto Y., Ukai S., Xu C.-J., Yang T.: The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential. J. Funct. Anal. 262(3), 915–1010 (2012) · Zbl 1232.35110
[4] Alexandre R., Villani C.: On the Boltzmann equation for long-range interactions. Comm. Pure Appl. Math. 55(1), 30–70 (2002) · Zbl 1029.82036
[5] Alexandre R., Villani C.: On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(1), 61–95 (2004) · Zbl 1044.83007
[6] Arsénio, D.: On the Boltzmann equation: hydrodynamic limit with long-range interactions and mild solutions. PhD thesis, New York University, New York, 2009
[7] Arsénio, D., Masmoudi, N.: A new approach to velocity averaging lemmas in Besov spaces. J. Math. Pures Appl. (2012) (submitted) · Zbl 1293.35192
[8] Arsénio D., Masmoudi N.: Regularity of renormalized solutions in the Boltzmann equation with long-range interactions. Comm. Pure Appl. Math. 65(4), 508–548 (2012) · Zbl 1234.35172
[9] Bardos C., Golse F., Levermore C.D.: Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46(5), 667–753 (1993) · Zbl 0817.76002
[10] Bardos C., Golse F., Levermore C.D.: Acoustic and Stokes limits for the Boltzmann equation. C. R. Acad. Sci. Paris Sér. I Math. 327(3), 323–328 (1998) · Zbl 0918.35109
[11] Bardos C., Golse F., Levermore C.D.: The acoustic limit for the Boltzmann equation. Arch. Ration. Mech. Anal. 153(3), 177–204 (2000) · Zbl 0973.76075
[12] Bardos C., Golse F., Levermore C.D.: Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Stat. Phys. 63(1–2), 323–344 (1991)
[13] Bézard M.: Régularité L p précisée des moyennes dans les équations de transport. Bull. Soc. Math. France 122(1), 29–76 (1994)
[14] Bouchut, F., Golse, F., Pulvirenti, M.: Kinetic equations and asymptotic theory. In: Series in Applied Mathematics (Paris), vol. 4. Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000. (Edited and with a foreword by Benoît Perthame and Laurent Desvillettes) · Zbl 0979.82048
[15] Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. In: Applied Mathematical Sciences, vol. 106. Springer, New York, 1994 · Zbl 0813.76001
[16] Chemin J.-Y.: Fluides parfaits incompressibles. Astérisque 230, 177 (1995)
[17] DiPerna R.J., Lions P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. (2) 130(2), 321–366 (1989) · Zbl 0698.45010
[18] DiPerna R.J., Lions P.-L., Meyer Y.: L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(3–4), 271–287 (1991)
[19] Golse F., Levermore C.D.: The Stokes–Fourier limit for the Boltzmann equation. C. R. Acad. Sci. Paris Sér. I Math. 333(2), 145–150 (2001) · Zbl 1011.35111
[20] Golse F., Levermore C.D.: Stokes–Fourier and acoustic limits for the Boltzmann equation: convergence proofs. Comm. Pure Appl. Math. 55(3), 336–393 (2002) · Zbl 1044.76055
[21] Golse F., Saint-Raymond L.: Velocity averaging in L 1 for the transport equation. C. R. Math. Acad. Sci. Paris 334(7), 557–562 (2002) · Zbl 1154.35326
[22] Golse F., Saint-Raymond L.: The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155(1), 81–161 (2004) · Zbl 1060.76101
[23] Golse F., Saint-Raymond L.: The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. J. Math. Pures Appl. (9) 91(5), 508–552 (2009) · Zbl 1178.35290
[24] Gressman P.T., Strain R.M.: Global classical solutions of the Boltzmann equation without angular cut-off. J. Am. Math. Soc. 24(3), 771–847 (2011) · Zbl 1248.35140
[25] Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934) · JFM 60.0726.05
[26] Levermore C.D., Masmoudi N.: From the Boltzmann equation to an incompressible Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 196(3), 753–809 (2010) · Zbl 1304.35476
[27] Lions P.-L.: On Boltzmann and Landau equations. Philos. Trans. R. Soc. Lond. Ser. A 346(1679), 191–204 (1994) · Zbl 0809.35137
[28] Lions P.-L., Masmoudi N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. (9) 77(6), 585–627 (1998) · Zbl 0909.35101
[29] Lions, P.-L., Masmoudi, N.: From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II. Arch. Ration. Mech. Anal. 158(3), 173–193, 195–211 (2001) · Zbl 0987.76088
[30] Lions P.-L.: Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire. C. R. Acad. Sci. Paris Sér. I Math. 326(1), 37–41 (1998) · Zbl 0920.35114
[31] Lions P.-L., Masmoudi N.: Une approche locale de la limite incompressible. C. R. Acad. Sci. Paris Sér. I Math. 329(5), 387–392 (1999) · Zbl 0937.35132
[32] Mouhot C.: Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Comm. Partial Diff. Equ. 31(7–9), 1321–1348 (2006) · Zbl 1101.76053
[33] Mouhot C., Strain R.M.: Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J. Math. Pures Appl. (9) 87(5), 515–535 (2007) · Zbl 1388.76338
[34] Saint-Raymond L.: Hydrodynamic limits of the Boltzmann equation. Lecture Notes in Mathematics, vol. 1971. Springer, Berlin, 2009 · Zbl 0224.54045
[35] Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Dynamics, vol. I, pp. 71–305. North-Holland, Amsterdam, 2002 · Zbl 1170.82369
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.