# American Institute of Mathematical Sciences

August  2022, 15(4): 729-752. doi: 10.3934/krm.2022012

## Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

 1 Kent State University, Kent, OH 44240 2 Indiana University, Bloomington, IN 47405

*Corresponding author: Kevin Zumbrun

In memory of Robert T. Glassey
Preface by Kevin Zumbrun: It is a great privilege to submit a paper to this memorial issue in honor of long-time mentor and colleague Bob Glassey. Bob enlivened the department at Indiana University for many years with his gentle wit and incisive mathematical analysis. Bob is perhaps best known for his work on kinetic equations. What I know about Boltzmann's equation I know mainly from sitting in on his delightful graduate course, which later became his delightful text on the Cauchy problem for kinetic equations. So, this paper concerning Boltzmann's and related kinetic equations seems an appropriate submission, coming as a direct result of Bob's influence as a teacher and researcher. Of course, Bob is equally well known for his seminal work on blowup in nonlinear PDEs. So, it is perhaps also appropriate that our central estimate is a "blow-down" result consisting of the reverse-time version of finite-time blowup for the Riccati equation.

Received  September 2021 Revised  December 2021 Published  August 2022 Early access  April 2022

Fund Project: Research of F.N. was partially supported under NSF grant no. DMS-1600239; research of K.Z. was partially supported under NSF grant no. DMS-0300487

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: Fedor Nazarov, Kevin Zumbrun. Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation. Kinetic and Related Models, 2022, 15 (4) : 729-752. doi: 10.3934/krm.2022012
##### References:
 [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, 2nd 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, 2nd 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.

show all references

##### References:
 [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, 2nd 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, 2nd 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.
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