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L∞ resolvent bounds for steady Boltzmann's Equation
Indiana University Bloomington, IN 47405, USA |
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.
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.Google Scholar |
show all references
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.Google Scholar |
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