Hydrodynamic Limits : Some Improvements of the Relative Entropy Method
Saint-Raymond, Laure
Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009), p. 705-744 / Harvested from Numdam
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@article{AIHPC_2009__26_3_705_0,
     author = {Saint-Raymond, Laure},
     title = {Hydrodynamic Limits : Some Improvements of the Relative Entropy Method},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {26},
     year = {2009},
     pages = {705-744},
     doi = {10.1016/j.anihpc.2008.01.001},
     mrnumber = {2526399},
     zbl = {1170.35500},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2009__26_3_705_0}
}

Saint-Raymond, Laure. Hydrodynamic Limits : Some Improvements of the Relative Entropy Method. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) pp. 705-744. doi : 10.1016/j.anihpc.2008.01.001. http://gdmltest.u-ga.fr/item/AIHPC_2009__26_3_705_0/

[1] Aoki K., Sone Y., Steady Gas Flows Past Bodies at Small Knudsen Numbers - Boltzmann and Hydrodynamic Systems, Trans. Theory Stat. Phys. 16 (1987) 189-199. | Zbl 0622.76082

[2] Bardos C., Golse F., Levermore C. D., Fluid Dynamic Limits of the Boltzmann Equation I, J. Stat. Phys. 63 (1991) 323-344. | MR 1115587

[3] Bardos C., Golse F., Levermore C. D., Fluid Dynamic Limits of the Boltzmann Equation II: Convergence Proofs, Comm. Pure Appl. Math. 46 (1993) 667-753. | MR 1213991 | Zbl 0817.76002

[4] Biryuk A., Craig W., Panferov V., Strong Solutions of the Boltzmann Equation in One Spatial Dimension, C. R. Acad. Sci. Paris 342 (2006) 843-848. | MR 2224633 | Zbl 1096.35001

[5] Bouchut F., Golse F., Pulvirenti M., Desvillettes L., Perthame B. (Eds.), Kinetic Equations and Asymptotic Theory, Editions scientifiques et médicales Elsevier, Paris, 2000. | MR 2065070

[6] Caflisch R., The Boltzmann Equation With a Soft Potential. I. Linear, Spatially-Homogeneous, Commun. Math. Phys. 74 (1980) 71-95. | MR 575897 | Zbl 0434.76065

[7] Cercignani C., Global Weak Solutions of the Boltzmann Equation, J. Stat. Phys. 118 (2005) 333-342. | MR 2122558 | Zbl 1097.82022

[8] Chapman S., Cowling T. G., The Mathematical Theory of Non-Uniform Gases: an Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases, Cambridge University Press, New York, 1960. | JFM 65.1541.01 | MR 116537 | Zbl 0726.76084

[9] Chemin J.-Y., Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 14, The Clarendon Press, Oxford University Press, New York, 1998. | MR 1688875 | Zbl 0927.76002

[10] Chemin J.-Y., Desjardins B., Gallagher I., Grenier E., Mathematical Geophysics. an Introduction to Rotating Fluids and the Navier-Stokes Equations, Oxford Lecture Series in Mathematics and its Applications, vol. 32, The Clarendon Press, Oxford University Press, Oxford, 2006. | MR 2228849 | Zbl pre05029231

[11] Demasi A., Esposito R., Lebowitz J., Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation, Comm Pure Appl. Math. 42 (1990) 1189-1214. | MR 1029125 | Zbl 0689.76024

[12] Desjardins B., Grenier E., Low Mach Number Limit of Viscous Compressible Flows in the Whole Space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999) 2271-2279. | MR 1702718 | Zbl 0934.76080

[13] Di Perna R., Lions P.-L., On the Cauchy Problem for the Boltzmann Equation: Global Existence and Weak Stability Results, Ann. of Math. 130 (1990) 321-366. | MR 1014927 | Zbl 0698.45010

[14] Gallagher I., Saint-Raymond L., On the Influence of the Earth's Rotation on Geophysical Flows, in: Handbook of Mathematical Fluid Dynamics, vol. 4, Elsevier, 2007.

[15] F. Golse, L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for hard potentials, submitted for publication.

[16] H. Grad, Asymptotic theory of the Boltzmann equation. II, in: Rarefied Gas Dynamics, vol. I, Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962, 1963, pp. 26-59. | MR 156656

[17] Grenier E., Quelques Limites Singulières Oscillantes, Séminaire sur les Equations aux Dérivées Partielles, vol. 21, Ecole Polytech., Palaiseau, 1995. | Numdam | MR 1362569 | Zbl 0873.35068

[18] Grenier E., On the Nonlinear Instability of Euler and Prandtl Equations, Comm. Pure Appl. Math. 53 (2000) 1067-1091. | MR 1761409 | Zbl 1048.35081

[19] Guo Y., The Vlasov-Poisson-Boltzmann System Near Maxwellians, Comm. Pure Appl. Math. 55 (2002) 1104-1135. | MR 1908664 | Zbl 1027.82035

[20] Guo Y., The Boltzmann Equation in the Whole Space, Indiana Univ. Math. J. 53 (2004) 1081-1094. | MR 2095473 | Zbl 1065.35090

[21] Hilbert D., Begründung Der Kinetischen Gastheorie, Math. Ann. 72 (1912) 562-577. | JFM 43.1055.03 | MR 1511713

[22] Lions P.-L., Conditions at Infinity for Boltzmann's Equation, Comm. Partial Differential Equations 19 (1994) 335-367. | MR 1257008 | Zbl 0799.35210

[23] Lions P.-L., Masmoudi N., From Boltzmann Equation to the Navier-Stokes and Euler Equations I, Arch. Ration Mech. Anal. 158 (2001) 173-193. | MR 1842343 | Zbl 0987.76088

[24] Lions P.-L., Masmoudi N., Une Approche Locale De La Limite Incompressible, C. R. Acad. Sci. Paris 329 (1999) 387-392. | MR 1710123 | Zbl 0937.35132

[25] Masmoudi N., Ekman Layers of Rotating Fluids: the Case of General Initial Data, Commun. Pure Appl. Math. 53 (2000) 432-483. | MR 1733696 | Zbl 1047.76124

[26] S. Mischler, Kinetic equations with Maxwell boundary condition, Preprint.

[27] Saint-Raymond L., Du Modèle BGK De L'équation De Boltzmann Aux Équations D'Euler Des Fluides Incompressibles, Bull. Sci. Math. 126 (2002) 493-506. | MR 1931626 | Zbl 1023.76042

[28] Saint-Raymond L., Convergence of Solutions to the Boltzmann Equation in the Incompressible Euler Limit, Arch. Ration. Mech. Anal. 166 (2003) 47-80. | MR 1952079 | Zbl 1016.76071

[29] L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Lectures at SISSA, Trieste, Lecture Notes in Mathematics, Preprint. | Zbl 1171.82002

[30] Schochet S., Fast Singular Limits of Hyperbolic PDEs, J. Differential Equations 114 (1994) 476-512. | MR 1303036 | Zbl 0838.35071

[31] Yau H. T., Relative Entropy and Hydrodynamics of Ginzburg-Landau Models, Lett. Math. Phys. 22 (1991) 63-80. | MR 1121850 | Zbl 0725.60120