2017
DOI: 10.1016/j.jfa.2016.08.017
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The Boltzmann equation with weakly inhomogeneous data in bounded domain

Abstract: This paper is concerned with the Boltzmann equation with specular reflection boundary condition. We construct a unique global solution and obtain its large time asymptotic behavior in the case that the initial data is close enough to a radially symmetric homogeneous datum. The result extends the case of Cauchy problem considered by Arkeryd-Esposito-Pulvirenti [Comm. Math. Phys. 111(3): 393-407 (1987)] to the specular reflection boundary value problem in bounded domain.

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Cited by 9 publications
(6 citation statements)
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“…In fact, for the Boltzmann equation with cutoff hard potentials in the torus or even in a general bounded domain, Guo [49] developed a mathematical theory of global existence of small-amplitude L I x;v solutions by an L 2 -L I interplay method. Since then, there have been extensive generalizations and further developments of the Boltzmann global existence theory in the angular cutoff setting, for instance, [10,13,26,31,50,51,57,58,63,68]. Particularly, motivated by [26,49], Nishimura [68]…”
Section: Motivation Of the Current Workmentioning
confidence: 99%
See 1 more Smart Citation
“…In fact, for the Boltzmann equation with cutoff hard potentials in the torus or even in a general bounded domain, Guo [49] developed a mathematical theory of global existence of small-amplitude L I x;v solutions by an L 2 -L I interplay method. Since then, there have been extensive generalizations and further developments of the Boltzmann global existence theory in the angular cutoff setting, for instance, [10,13,26,31,50,51,57,58,63,68]. Particularly, motivated by [26,49], Nishimura [68]…”
Section: Motivation Of the Current Workmentioning
confidence: 99%
“…Duan-Wang [31] showed the global well-posedness for a class of large-amplitude initial data. Guo-Liu [51] constructed the global existence for the Boltzmann equation with specular reflection boundary condition around the polynomial initial data differing from a global equilibrium Maxwellian. Esposito-Guo-Kim-Marra [33] and Duan-Liu [27] studied the hydrodynamic limits of the Boltzmann equation in bounded domains based on an improved L 2 -L I method.…”
Section: Related Literaturementioning
confidence: 99%
“…Further progress on high-order Sobolev regularity of solutions was recently made in [30]; see also references therein. For other related works on the study of the IBVP on the nonlinear Boltzmann equation, we would mention [9] for the general Maxwell boundary condition, [31] for the global existence of solutions with weakly inhomogeneous data in the case of specular reflection, [33] for the specular boundary condition in convex domains with C 3 smoothness, and [36] for a direct extension of [29] from hard potentials to soft potentials.…”
Section: )mentioning
confidence: 99%
“…To treat the difficulty, we resort to the L ∞ -L 2 method developed recently by Guo [25]; see also [19,20,27]. One of the key points when applying this approach is the decay of the operator K for large velocity.…”
mentioning
confidence: 99%