- Previous Article
- KRM Home
- This Issue
-
Next Article
The stability of contact discontinuity for compressible planar magnetohydrodynamics
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 |
| [1] |
Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1429-1440. doi: 10.3934/dcdss.2020081 |
| [2] |
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic & Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17 |
| [3] |
Xinkuan Chai. The Boltzmann equation near Maxwellian in the whole space. Communications on Pure & Applied Analysis, 2011, 10 (2) : 435-458. doi: 10.3934/cpaa.2011.10.435 |
| [4] |
Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445 |
| [5] |
Roberto Guglielmi. Indirect stabilization of hyperbolic systems through resolvent estimates. Evolution Equations & Control Theory, 2017, 6 (1) : 59-75. doi: 10.3934/eect.2017004 |
| [6] |
Alfredo Lorenzi, Ioan I. Vrabie. An identification problem for a linear evolution equation in a Banach space and applications. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 671-691. doi: 10.3934/dcdss.2011.4.671 |
| [7] |
Yingzhe Fan, Yuanjie Lei. The Boltzmann equation with frictional force for very soft potentials in the whole space. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4303-4329. doi: 10.3934/dcds.2019174 |
| [8] |
Wen Deng. Resolvent estimates for a two-dimensional non-self-adjoint operator. Communications on Pure & Applied Analysis, 2013, 12 (1) : 547-596. doi: 10.3934/cpaa.2013.12.547 |
| [9] |
Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203 |
| [10] |
Niclas Bernhoff. On half-space problems for the weakly non-linear discrete Boltzmann equation. Kinetic & Related Models, 2010, 3 (2) : 195-222. doi: 10.3934/krm.2010.3.195 |
| [11] |
Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115 |
| [12] |
Laurent Gosse. Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension. Kinetic & Related Models, 2012, 5 (2) : 283-323. doi: 10.3934/krm.2012.5.283 |
| [13] |
Robert M. Strain. Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinetic & Related Models, 2012, 5 (3) : 583-613. doi: 10.3934/krm.2012.5.583 |
| [14] |
Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307 |
| [15] |
Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145 |
| [16] |
Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic & Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014 |
| [17] |
Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283 |
| [18] |
Chun-Gil Park. Stability of a linear functional equation in Banach modules. Conference Publications, 2003, 2003 (Special) : 694-700. doi: 10.3934/proc.2003.2003.694 |
| [19] |
Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039 |
| [20] |
Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205 |
2019 Impact Factor: 1.311
Tools
Metrics
Other articles
by authors
[Back to Top]