Abstract
Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space above its condensed phase. The present paper studies the existence and uniqueness of a uniformly, exponentially decaying solution in the vicinity of the Maxwellian equilibrium with zero bulk velocity, with the same temperature as that of the condensed phase, and whose pressure is the saturating vapor pressure at the temperature of the interface. This problem has been studied numerically by Y. Sone, K. Aoki and their collaborators—see section 2 of (Bardos et al., J Stat Phys 124:275–300, 2006) for a detailed presentation of these works. More recently Liu and Yu (Arch Ration Mech Anal 209:869–997, 2013) have proposed a mathematical strategy to handle problems of this type. In this paper, we describe an alternative approach to one of their results obtained in collaboration with Bernhoff (Arch Ration Mech Anal 240:51–98, 2021).
In memory of Basil Nicolaenko (1942–2007)
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Notes
- 1.
The possibility of extending the Nicolaenko-Thurber theory to the case |u| < < 1 was mentioned to me by Prof. Nicolaenko in the late 1990s during one of my visits to his department at Arizona State University in Phoenix.
- 2.
During the meeting Prof. Schmeiser kindly reminded me that a somewhat reminiscent penalization of the linearized collision integral had been used in the paper [11], which predates the introduction of the penalization method in [32]. See the definition of the operator denoted M in formula (3.39) of [11], and Proposition 3.3 on p. 171 in the same reference. However, the idea of penalizing the collision integral is used quite differently in [11] and [32]. That the penalization method of [32] escaped the notice of the authors of the first fundamental contribution [4] to the theory of the half-space problem for the Boltzmann equation, who were obviously aware of its importance in the shock profile problem treated in [11], says a lot about the originality and depth of the ideas in [32].
- 3.
One should pay attention to the fact that the appropriate function space used in this argument is an anisotropic, or mixed Lebesgue space of the form \(L^2_\xi (L^\infty _z)\), and not \(L^\infty _z(L^2_\xi )\). That \(L^2_\xi (L^\infty _z)\) is the function space of interest for this type of problem has been known for a long time—for instance it was already used in [14].
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Acknowledgements
I wish to thank Profs. Sone and Aoki who spent a lot of time teaching me their theory of the half-space problem with condensation/evaporation during several visits in Kyoto. I am also very much indebted to the late Prof. Nicolaenko for explaining to me his approach to the generalized eigenvalue problem described in Sect. 5—along with so many other things in mathematics. The question reported in the last paragraph of this paper had been a long standing program of ours, which we unfortunately did not have the time to complete. My collaboration on this problem with N. Bernhoff started from our joint interest in the topics discussed in [7, 8], to which I was introduced by Prof. Bobylev. Finally, I would like to thank Prof. Aoki for his friendly interest and support in our joint work [6].
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Golse, F. (2021). The Half-Space Problem for the Boltzmann Equation with Phase Transition at the Boundary. In: Salvarani, F. (eds) Recent Advances in Kinetic Equations and Applications. Springer INdAM Series, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-030-82946-9_8
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