PDF
Abstract
The Boltzmann equation with external potential force exists a unique equilibrium—local Maxwellian. The author constructs the nonlinear stability of the equilibrium when the initial datum is a small perturbation of the local Maxwellian in the whole space ℝ3. Compared with the previous result [Ukai, S., Yang, T. and Zhao, H.-J., Global solutions to the Boltzmann equation with external forces, Anal. Appl. (Singap.), 3, 2005, 157–193], no smallness condition on the Sobolev norm H 1 of the potential is needed in our arguments. The proof is based on the entropy-energy inequality and the L 2 − L ∞ estimates.
Keywords
Boltzmann equation
/
Large potential force
/
Stability
/
Entropy-energy inequality
/
L 2 − L ∞ method
Cite this article
Download citation ▾
Xiuhui Yang.
Stability of the Equilibrium to the Boltzmann Equation with Large Potential Force.
Chinese Annals of Mathematics, Series B, 2018, 39(5): 805-816 DOI:10.1007/s11401-018-0097-1
| [1] |
Duan R.. Stability of the Boltzmann equation with potential forces on torus. Phys. D, 2009, 238(17): 1808-1820
|
| [2] |
Duan R., Ukai S., Yang T., Zhao H.-J.. Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications. Comm. Math. Phys., 2008, 277: 189-236
|
| [3] |
Esposito R., Guo Y., Marra R.. Phase transition in a Vlasov-Boltzmann binary mixture. Commun. Math. Phys., 2010, 296: 1-33
|
| [4] |
Guo Y.. Bounded solutions to the Boltzmann equation. Quart. Appl. Math., 2010, 68: 143-148
|
| [5] |
Guo Y.. Decay and continuity of Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal., 2010, 197: 713-809
|
| [6] |
Guo Y.. The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math., 2002, 55: 1104-1135
|
| [7] |
Kim C.. Boltzmann equation with a large potential in a periodic box. Comm. Part. Diff. Eq., 2014, 39(8): 1393-1423
|
| [8] |
Lei Y.. The non-cutoff Boltzmann equation with potential force in the whole space. Acta Math. Sci. Ser. B (Engl. Ed.), 2014, 34(5): 1519-1539
|
| [9] |
Li F.-C., Yu H.-J.. Global existence of classical solutions to the Boltzmann equation with external force for hard potentials. Int. Math. Res. Not. IMRN, 2008
|
| [10] |
Ukai S., Yang T., Zhao H.-J.. Global solutions to the Boltzmann equation with external forces. Anal. Appl. (Singap.), 2005, 3: 157-193
|
| [11] |
Ukai S., Yang T., Zhao H.-J.. Convergence rate to stationary solutions for Boltzmann equation with external force. Chin. Ann. Math. Ser. B, 2006, 27(4): 363-378
|
| [12] |
Yu H.-J.. Global classical solutions to the Boltzmann equation with external force. Commun. Pure Appl. Anal., 2009, 8: 1647-1668
|
