We establish an instantaneous smoothing property for decaying solutions on the half-line $ (0, +\infty) $ of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of $ H^1 $ stable manifolds of such equations, showing that $ L^2_{loc} $ solutions that remain sufficiently small in $ L^\infty $ (i) decay exponentially, and (ii) are $ C^\infty $ for $ t>0 $, hence lie eventually in the $ H^1 $ stable manifold constructed by Pogan and Zumbrun.
Citation: |
[1] | G. Boillat and T. Ruggeri, On the shock structure problem for hyperbolic system of balance laws and convex entropy, Contin. Mech. Thermodyn., 10 (1998), 285-292. doi: 10.1007/s001610050094. |
[2] | J. B. Conway, A Course in Functional Analysis, 2^{nd} edition, Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990. |
[3] | J. Diestel and J. J. Uhl, Vector Measures, Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977. |
[4] | L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. |
[5] | Y. Latushkin and A. Pogan, The dichotomy theorem for evolution bi-families, J. Diff. Eq., 245 (2008), 2267-2306. doi: 10.1016/j.jde.2008.01.023. |
[6] | G. 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. |
[7] | A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. |
[8] | A. Pogan and K. Zumbrun, Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks,, Kinet. Relat. Models, 12 (2019), 1-36. doi: 10.3934/krm.2019001. |
[9] | A. Pogan and K. Zumbrun, Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks,, J. Diff. Eq., 264 (2018), 6752-6808. doi: 10.1016/j.jde.2018.01.049. |
[10] | M. Reed and B. Simon, Methods of Mathematical Physics, 2 edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. |
[11] | W. Rudin, Functional Analysis, 2^{nd} edition, International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. |
[12] | K. Zumbrun, Invariant manifolds for a class of degenerate evolution equations and structure of kinetic shock layers,, Springer Proc. Math. Stat., 237 (2018), 691-714. |