Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 719-737.

We prove propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for any value of the coefficient of restitution. The result follows from the uniform in time control of the tails of the Fourier transform of the solution, normalized in order to have constant energy. By standard arguments this implies the convergence of the scaled solution towards the stationary state in Sobolev and L 1 norms in the case of regular initial data as well as the convergence of the original solution to the corresponding self-similar cooling state. In the case of weak inelasticity, similar results have been established by Carlen, Carrillo and Carvalho (2009) in [11] via a precise control of the growth of the Fisher information.

DOI : 10.1016/j.anihpc.2009.11.005
Mots clés : Granular gases, Kinetic models, Boltzmann equation
@article{AIHPC_2010__27_2_719_0,
     author = {Furioli, G. and Pulvirenti, A. and Terraneo, E. and Toscani, G.},
     title = {Convergence to self-similarity for the {Boltzmann} equation for strongly inelastic {Maxwell} molecules},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {719--737},
     publisher = {Elsevier},
     volume = {27},
     number = {2},
     year = {2010},
     doi = {10.1016/j.anihpc.2009.11.005},
     mrnumber = {2595198},
     zbl = {05690778},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.005/}
}
TY  - JOUR
AU  - Furioli, G.
AU  - Pulvirenti, A.
AU  - Terraneo, E.
AU  - Toscani, G.
TI  - Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 719
EP  - 737
VL  - 27
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.005/
DO  - 10.1016/j.anihpc.2009.11.005
LA  - en
ID  - AIHPC_2010__27_2_719_0
ER  - 
%0 Journal Article
%A Furioli, G.
%A Pulvirenti, A.
%A Terraneo, E.
%A Toscani, G.
%T Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 719-737
%V 27
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.005/
%R 10.1016/j.anihpc.2009.11.005
%G en
%F AIHPC_2010__27_2_719_0
Furioli, G.; Pulvirenti, A.; Terraneo, E.; Toscani, G. Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 719-737. doi : 10.1016/j.anihpc.2009.11.005. http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.005/

[1] D. Ben-Avraham, E. Ben-Naim, K. Lindenberg, A. Rosas, Self-similarity in random collision processes, Phys. Rev. E 68 (2003)

[2] M. Bisi, J.A. Carrillo, G. Toscani, Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model, J. Statist. Phys. 124 no. 2–4 (2006), 625-653 | MR | Zbl

[3] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, J.L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Rational Mech. Anal. 191 (2009), 347-385 | MR | Zbl

[4] A.V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Mathematical Physics Reviews, vol. 7, Soviet Sci. Rev. Sect. C Math. Phys. Rev. vol. 7, Harwood Academic Publ., Chur (1988), 111-233 | MR | Zbl

[5] A.V. Bobylev, C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Statist. Phys. 110 no. 1–2 (2003), 333-375 | MR | Zbl

[6] A.V. Bobylev, J.A. Carrillo, I.M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys. 98 no. 3–4 (2000), 743-773 | MR | Zbl

[7] A.V. Bobylev, C. Cercignani, I.M. Gamba, Generalized kinetic Maxwell type models of granular gases, Mathematical Models of Granular Matter, Lecture Notes in Math. vol. 1937, Springer, Berlin (2008), 23-57 | MR | Zbl

[8] A.V. Bobylev, C. Cercignani, G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, J. Statist. Phys. 111 no. 1–2 (2003), 403-417 | MR | Zbl

[9] F. Bolley, J.A. Carrillo, Tanaka theorem for inelastic Maxwell models, Comm. Math. Phys. 276 no. 2 (2007), 287-314 | MR | Zbl

[10] M.J. Càceres, G. Toscani, Kinetic approach to long time behavior of linearized fast diffusion equations, J. Statist. Phys. 128 no. 4 (2007), 883-925 | MR | Zbl

[11] E.A. Carlen, J.A. Carrillo, M. Carvalho, Strong convergence towards homogeneous cooling states for dissipative Maxwell models, Annales IHP Non Linear Analysis 26 no. 5 (2009), 1675-1700 | EuDML | Numdam | MR | Zbl

[12] E.A. Carlen, E. Gabetta, G. Toscani, Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas, Comm. Math. Phys. 199 no. 3 (1999), 521-546 | MR | Zbl

[13] J.A. Carrillo, R. Mccann, C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana 19 (2003), 1-48 | EuDML | MR | Zbl

[14] J.A. Carrillo, R. Mccann, C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal. 179 (2006), 217-263 | MR | Zbl

[15] J.A. Carrillo, G. Toscani, Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J. 49 no. 1 (2000), 113-142 | MR | Zbl

[16] J.A. Carrillo, G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma (7) 6 (2007), 75-198 | MR | Zbl

[17] L. Desvillettes, G. Furioli, E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules, Trans. Amer. Math. Soc. 361 (2009), 1731-1747 | MR | Zbl

[18] G. Furioli, A. Pulvirenti, E. Terraneo, G. Toscani, Strong convergence towards self-similarity for one-dimensional dissipative Maxwell models, J. Funct. Anal. 257 no. 7 (2009), 2291-2324 | MR | Zbl

[19] I.M. Gamba, V. Panferov, C. Villani, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys. 246 no. 3 (2004), 503-541 | MR | Zbl

[20] P.L. Lions, G. Toscani, A strengthened central limit theorem for smooth densities, J. Funct. Anal. 129 no. 1 (1995), 148-167 | MR | Zbl

[21] C. Villani, Topics in Mass Transportation, Grad. Stud. Math. vol. 58 (2003)

[22] C. Villani, Mathematics of granular materials, J. Statist. Phys. 124 no. 2–4 (2006), 781-822 | MR | Zbl

Cité par Sources :