Riv. Mat. Univ. Parma, Vol. 6, No. 1, 2015

Franz Achleitner[1], Anton Arnold[2] and Dominik Stürzer[3],

Large-time behavior in non-symmetric Fokker-Planck equations

Pages: 1-68
Accepted : 6 June 2015
Mathematics Subject Classification (2010): Primary 35Q84, 35H10, 35B20; Secondary 35K10, 35B40, 47D07, 35P99, 47D06.
Keywords: Fokker-Planck equation, hypocoercivity, entropy method, large-time behavior, spectral gap, sharp decay rate, non-local perturbation, spectral analysis, exponential stability.
[1],[2],[3] : Vienna University of Technology, Wiedner Hauptstrasse 8, Vienna, 1040, Austria.

This research was partially supported by the FWF-doctoral school "Dissipation and dispersion in nonlinear partial differential equations" and INDAM - GNFM from Italy. One author (AA) is grateful to J. Schöberl for very helpful discussions.

Abstract: We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First, "hypocoercive" Fokker-Planck equations are degenerate parabolic equations such that the entropy method to study large-time behavior of solutions has to be modified. We review a recent modified entropy method (for non-symmetric Fokker-Planck equations with drift terms that are linear in the position variable). Second, kinetic Fokker-Planck equations with non-quadratic potentials are another example of non-symmetric Fokker-Planck equations. Their drift term is nonlinear in the position variable. In case of potentials with bounded second-order derivatives, the modified entropy method allows to prove exponential convergence of solutions to the steady state. In this application of the modified entropy method symmetric positive definite matrices solving a matrix inequality are needed. We determine all such matrices achieving the optimal decay rate in the modified entropy method. In this way we prove the optimality of previous results. Third, we discuss the spectral properties of Fokker-Planck operators perturbed with convolution operators. For the corresponding Fokker-Planck equation we show existence and uniqueness of a stationary solution. Then, exponential convergence of all solutions towards the stationary solution is proven with a uniform rate.

References

[1] E. Albrecht and F. Vasilescu, Invariant subspaces for some families of unbounded subnormal operators, Glasg. Math. J. 45 (2003), 53–67. [MR1972693]
[2] A. Arnold, E. Carlen and Q. Ju, Large-time behavior of nonsymmetric Fokker-Planck type equations, Commun. Stoch. Anal. 2 (2008), 153–175. [MR2446997]
[3] A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of many-particle systems: an essay on recent research, Monatsh. Math. 142 (2004), 35–43. [MR2065020]
[4] A. Arnold, J. A. Carrillo and C. Manzini, Refined long-time asymptotics for some polymeric fluid flow models, Commun. Math. Sci. 8 (2010), 763–782. [MR2730330]
[5] A. Arnold and J. Erb, Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift, arXiv preprint arXiv:1409.5425 (2014).
[6] A. Arnold, I. M. Gamba, M. P. Gualdani, S. Mischler, C. Mouhot and C. Sparber, The Wigner-Fokker-Planck equation: stationary states and large time behavior, Math. Models Methods Appl. Sci. 22 (2012), 1250034, 31 pp. [MR2974172]
[7] A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations 26 (2001), 43–100. [MR1842428]
[8] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155. [MR2269741]
[9] D. Bakry and M. Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 775–778. [MR0772092]
[10] D. Bakry, and M. Émery, Diffusions hypercontractives, in "Séminaire de probabilités, XIX, 1983/84", vol. 1123 of "Lecture Notes in Math.", Springer, Berlin 1985, pp. 177–206. [MR0889476]
[11] D. Bakry, R. D. Gill and S. A. Molchanov, Lectures on probability theory, vol. 1581 of "Lecture Notes in Mathematics", Springer-Verlag, Berlin 1994. [MR1307412]
[12] F. Baudoin, Bakry-Emery meet Villani, arXiv preprint arXiv:1308.4938 (2013).
[13] F. Bolley and I. Gentil, Phi-entropy inequalities for diffusion semigroups, J. Math. Pures Appl. (9) 93 (2010), 449–473. [MR2609029]
[14] J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 (2001), 1–82. [MR1853037]
[15] M. Costabel and A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z. 265 (2010), 297–320. [MR2609313]
[16] L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54 (2001), 1–42. [MR1787105]
[17] J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc. 367 (2015), 3807–3828. [MR3324910]
[18] K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, vol. 194 of "Graduate Texts in Mathematics", Springer-Verlag, New York 2000. [MR1721989]
[19] F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in $$N \,log_2 \,N$$, SIAM J. Sci. Comput. 28 (2006), 1029–1053 (electronic). [MR2240802]
[20] T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys. 255 (2005), 97–129. [MR123378]
[21] B. Helffer and F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, vol. 1862 of "Lecture Notes in Mathematics", Springer-Verlag, Berlin 2005. [MR2130405 ]
[22] F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), 151–218. [MR2034753]
[23] C. D. Hill, A sharp maximum principle for degenerate elliptic-parabolic equations, Indiana Univ. Math. J. 20 (1970/1971), 213–229. [MR0287175]
[24] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. [MR0222474]
[25] R. A. Horn and C. R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge 1991. [MR1091716]
[26] T. Kato, Perturbation theory for linear operators, vol. 132 of "Die Grundlehren der mathematischen Wissenschaften", Springer-Verlag, New York 1966. [MR0203473]
[27] A. Klar, F. Schneider and O. Tse, Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker–Planck equations, Kinet. Relat. Models 7 (2014), 509–529. [MR3317571]
[28] A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin. 25 (1984), 537–554. [MR0775568]
[29] S. Lee and K.-G. Kang, Numerical analysis of electronic transport characteristics in dielectrics irradiated by ultrashort pulsed laser using the nonlocal Fokker-Planck equation, Numerical Heat Transfer Part A 48 (2005), no. 1, 59–76. DOI:http://dx.doi.org/10.1080/10407780590929838
[30] G. Metafune, $$L^p$$-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30 (2001), 97–124. [MR1882026]
[31] G. Metafune, D. Pallara and E. Priola, Spectrum of Ornstein-Uhlenbeck operators in $$L^p$$ spaces with respect to invariant measures, J. Funct. Anal. 196 (2002), 40–60. [MR1941990]
[32] B. Opic, Necessary and sufficient conditions for imbeddings in weighted Sobolev spaces, Casopis Pest. Mat. 114 (1989), 343–355. [MR1027230]
[33] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of "Applied Mathematical Sciences", Springer-Verlag, New York 1983. [MR0710486]
[34] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, self-adjointness, Academic Press, New York 1975. [MR0493420]
[35] H. Risken, The Fokker-Planck equation. Methods of solution and applications, second ed., vol. 18 of "Springer Series in Synergetics", Springer-Verlag, Berlin 1989. [MR0987631]
[36] D. Stürzer, Spectral analysis and long-time behaviour of a Fokker-Planck equation with a non-local perturbation, PhD thesis, Vienna University of Technology, 2015.
[37] D. Stürzer and A. Arnold, Spectral analysis and long-time behaviour of a Fokker-Planck equation with a non-local perturbation, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25 (2014), 53–89. [MR3180480]
[38] A. E. Taylor and D. C. Lay, Introduction to functional analysis, second ed. John Wiley & Sons, New York-Chichester-Brisbane 1980. [MR0564653]
[39] A. Unterreiter, A. Arnold, P. Markowich and G. Toscani, On generalized Csiszár-Kullback inequalities, Monatsh. Math. 131 (2000), 235–253. [MR1801751]
[40] C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of mathematical fluid dynamics", Vol. I. North-Holland, Amsterdam 2002, pp. 71–305. [MR1942465]
[41] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141 pp. [MR2562709]

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