Numerical hypocoercivity for the Kolmogorov equation

Porretta, Alessio; Zuazua, Enrique

doi:10.1090/mcom/3157

Numerical hypocoercivity for the Kolmogorov equation
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by Alessio Porretta and Enrique Zuazua PDF

Math. Comp. 86 (2017), 97-119 Request permission

Abstract:

We prove that a finite-difference centered approximation for the Kolmogorov equation in the whole space preserves the decay properties of continuous solutions as $t \to \infty$, independently of the mesh-size parameters. This is a manifestation of the property of numerical hypo-coercivity, and it holds both for semi-discrete and fully discrete approximations. The method of proof is based on the energy methods developed by Herau and Villani, employing well-balanced Lyapunov functionals mixing different energies, suitably weighted and equilibrated by multiplicative powers in time. The decreasing character of this Lyapunov functional leads to the optimal decay of the $L^2$-norms of solutions and partial derivatives, which are of different order because of the anisotropy of the model.

References

Karine Beauchard and Enrique Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 177–227. MR 2754341, DOI 10.1007/s00205-010-0321-y
François Bouchut, Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions, J. Funct. Anal. 111 (1993), no. 1, 239–258. MR 1200643, DOI 10.1006/jfan.1993.1011
Ana Carpio, Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation, Math. Methods Appl. Sci. 21 (1998), no. 11, 985–1014. MR 1635981, DOI 10.1002/(SICI)1099-1476(19980725)21:11<985::AID-MMA919>3.0.CO;2-B
E. L. Foster, J. Lohéac, and M.-B. Tran, A structure preserving scheme for the Kolmogorov equation, preprint 2014 (arXiv:1411.1019v3).
Frédéric Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal. 244 (2007), no. 1, 95–118 (English, with English and French summaries). MR 2294477, DOI 10.1016/j.jfa.2006.11.013
Frédéric Hérau and Francis Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218. MR 2034753, DOI 10.1007/s00205-003-0276-3
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
Liviu I. Ignat, Alejandro Pozo, and Enrique Zuazua, Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws, Math. Comp. 84 (2015), no. 294, 1633–1662. MR 3335886, DOI 10.1090/S0025-5718-2014-02915-3
A. M. Il′in, On a class of ultraparabolic equations, Dokl. Akad. Nauk SSSR 159 (1964), 1214–1217 (Russian). MR 0171084
A. M. Il′in and R. Z. Has′minskiĭ, On the equations of Brownian motion, Teor. Verojatnost. i Primenen. 9 (1964), 466–491 (Russian, with English summary). MR 0168018
A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), no. 1, 116–117 (German). MR 1503147, DOI 10.2307/1968123
J. Lohéac, A splitting method for Kolmogorov-type equations, in preparation.
Aurora Marica and Enrique Zuazua, Propagation of 1D waves in regular discrete heterogeneous media: a Wigner measure approach, Found. Comput. Math. 15 (2015), no. 6, 1571–1636. MR 3413630, DOI 10.1007/s10208-014-9232-x
A. Porretta, A note on the Sobolev and Gagliardo-Nirenberg inequality when $p>N$, in preparation.
Cédric Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141. MR 2562709, DOI 10.1090/S0065-9266-09-00567-5
Cédric Villani, Hypocoercive diffusion operators, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 473–498. MR 2275692
Enrique Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev. 47 (2005), no. 2, 197–243. MR 2179896, DOI 10.1137/S0036144503432862