On the entropy method for the hydrodynamical limit of some kinetic theory models. (English) Zbl 1145.82340
Summary: This paper develops an asymptotic theory for an initial value problem for the Boltzmann equation to obtain macroscopic hydrodynamical models from the underlying description delivered by the kinetic theory. The diffusive macroscopic limit is obtained using relative entropy methods. The method is technically applied to some specific models of the kinetic theory.
MSC:
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
Software:
ChemotaxisReferences:
| [1] | Bardos, C.; Golse, F.; Levermore, D., Fluid dynamic limits of kinetic equations II: Convergence proofs for the Boltzmann equation, Commun. Pure Appl. Math., 46, 667-753 (1993) · Zbl 0817.76002 |
| [2] | Degond, P.; Goudon, T.; Poupaud, F., Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49, 3, 1175-1198 (2000) · Zbl 0971.82035 |
| [3] | Dogbe, C., Diffusion approximation for a Knudsen gas in a thin domain with reflexive chaotic law, Bull. Austral. Math. Soc., 64, 497-514 (2001) · Zbl 1108.82321 |
| [4] | Bellomo, N.; Bellouquid, A., From a class of kinetic models to the macroscopic equations for multicellular systems in biology, Discrete Contin. Dyn. Syst. Ser. B, Biol., \(4, n^∘1, 59-80 (2004)\) · Zbl 1044.92021 |
| [5] | Chalub, F. A.; Markowich, P.; Perthame, B.; Schmeiser, C., Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142, 1-2, 123-141 (2004) · Zbl 1052.92005 |
| [6] | Chalub, F.; Dolak-Struss, Y.; Markowich, P.; Oeltz, D.; Schmeiser, C.; Soref, A., Model hierarchies for cell aggregation by chemotaxis, Math. Models Methods Appl. Sci., 16, 1173-1198 (2006) · Zbl 1094.92009 |
| [7] | Bardos, C.; Santos, C.; Sentis, R., Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284, A, 617-649 (1984) · Zbl 0508.60067 |
| [8] | Bonilla, L. L.; Soler, J. S., High field limit for the Vlasov-Poisson-Fokker-Plank system: A comparison of different perturbation methods, Math. Models Methods Appl. Sci., 11, 8, 1457-1681 (2001) · Zbl 1012.82023 |
| [9] | Filbet, F.; Laurençot, P.; Perthame, B., Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50, 189-207 (2005) · Zbl 1080.92014 |
| [10] | Bellomo, N.; Bellouquid, A.; Nieto, J.; Soler, J., Multicellular growing systems: Hyperbolic limits towards macroscopic description, Math. Models Methods Appl. Sci., 17 (2007) · Zbl 1135.92009 |
| [11] | Yau, H.-T., Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys., 22, 1, 63-80 (1991) · Zbl 0725.60120 |
| [12] | G. Bolliat, C.M. Dafermos, P.D. Lax, T.P. Liu, Recent Mathematical Methods in Nonlinear Wave Propagation, in: Lecture Notes in Mathematics, vol. 1640, Springer-Verlag, Berlin; G. Bolliat, C.M. Dafermos, P.D. Lax, T.P. Liu, Recent Mathematical Methods in Nonlinear Wave Propagation, in: Lecture Notes in Mathematics, vol. 1640, Springer-Verlag, Berlin · Zbl 0853.00040 |
| [13] | Arnold, A.; Markowich, P.; Toscani, G.; Unterreiter, A., On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26, 43-100 (2001) · Zbl 0982.35113 |
| [14] | Carrillo, J. A.; Toscani, G., Asymptotic \(L^1\)-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49, 113-141 (2000) · Zbl 0963.35098 |
| [15] | Brenier, Y., Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25, 737-754 (2000) · Zbl 0970.35110 |
| [16] | Cáceres, M. J.; Carrillo, J.-A.; Goudon, T., Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles, Comm. Partial Differential Equations, 28, 5, 969-989 (2003) · Zbl 1045.35094 |
| [17] | Cercignani, C.; Illner, R.; Pulvirenti, M., (The Mathematical Theory of Dilute Gases. The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106 (1994), Springer-Verlag: Springer-Verlag New York) · Zbl 0813.76001 |
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