# American Institute of Mathematical Sciences

December  2017, 10(4): 1255-1257. doi: 10.3934/krm.2017048

## L∞ resolvent bounds for steady Boltzmann's Equation

 Indiana University Bloomington, IN 47405, USA

Received  November 2016 Revised  January 2017 Published  March 2017

Fund Project: The author is supported by NSF grant DMS-0300487.

We derive lower bounds on the resolvent operator for the linearized steady Boltzmann equation over weighted $L^\infty$Banach spaces in velocity, comparable to those derived by Pogan & Zumbrun in an analogous weighted $L^2$ Hilbert space setting.These show in particular that the operator norm of the resolvent kernel is unbounded in $L^p(\mathbb{R})$ for all $1<p \leq \infty$, resolving an apparent discrepancy in behavior between the two settings suggested by previous work.

Citation: Kevin Zumbrun. L resolvent bounds for steady Boltzmann's Equation. Kinetic and Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048
##### References:
 [1] H. Grad, Asymptotic theory of the Boltzmann equation. Ⅱ, in Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos. , Palais de l'UNESCO, Paris, 1962), Ⅰ, Academic Press, (1963), 26–59 [2] T.-P. Liu and S.-H. Yu, Invariant manifolds for steady Boltzmann flows and applications, Arch. Rational Mech. Anal., 209 (2013), 869-997.  doi: 10.1007/s00205-013-0640-x. [3] Q. Métivier and K. Zumbrun, Existence and sharp localization in velocity of small-amplitude Boltzmann shocks, Kinet. Relat. Models, 2 (2009), 667-705.  doi: 10.3934/krm.2009.2.667. [4] A. Pogan and K. Zumbrun, Stable manifolds for a class of degenerate evolution equations and exponential decay of kinetic shocks, preprint, https://arxiv.org/pdf/1607.03028.pdf.

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##### References:
 [1] H. Grad, Asymptotic theory of the Boltzmann equation. Ⅱ, in Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos. , Palais de l'UNESCO, Paris, 1962), Ⅰ, Academic Press, (1963), 26–59 [2] T.-P. Liu and S.-H. Yu, Invariant manifolds for steady Boltzmann flows and applications, Arch. Rational Mech. Anal., 209 (2013), 869-997.  doi: 10.1007/s00205-013-0640-x. [3] Q. Métivier and K. Zumbrun, Existence and sharp localization in velocity of small-amplitude Boltzmann shocks, Kinet. Relat. Models, 2 (2009), 667-705.  doi: 10.3934/krm.2009.2.667. [4] A. Pogan and K. Zumbrun, Stable manifolds for a class of degenerate evolution equations and exponential decay of kinetic shocks, preprint, https://arxiv.org/pdf/1607.03028.pdf.
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