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Short and long time behavior of the Fokker-Planck equation in a confining potential and applications. (English) Zbl 1120.35016
The author studies the linear Fokker-Planck equation in \(\mathbb{R}^{2d}_{x,v}\), \(d\geq 3\), \[ \partial_t f + v\cdot \partial_x f- \partial_x V\cdot \partial_v f - \gamma \partial_v\cdot(\partial_v +v)f =0, \] where \(f\) is a given external confining potential such that \(V\geq 0\), \(V'' \in W^{\infty,\infty}\), and \(e^{-V}\) is a fast decaying function; moreover \(\gamma\) is a positive constant and \(f\) is a distribution function of particles. The equation admits a unique normalized Maxwellian \(\mathcal{M}\), and the analysis is carried out in the space \(B^2 = \mathcal{M}^{1/2}L^2\). The author proves estimates for the semigroup generated by \(-K = -v\cdot \partial_x f+ \partial_x V\cdot \partial_v f + \gamma \partial_v\cdot(\partial_v +v)f\) and its derivatives as well as estimates determining the rate of convergence of this semigroup to \(\mathcal{M}\) as \(t\to \infty\) (under additional assumption on the spectral gap for the Witten Laplacian associated with \(K\)). These results are used to prove global solvability of the nonlinear mollified Vlasov-Poisson-Fokker-Plank equation and to provide estimates of the rate of decay of the relative entropy associated with the latter equation. Mollification here refers to the fact that the potential is given by a ‘smeared’ solution of the self-consistent Poisson equation.

35B40 Asymptotic behavior of solutions to PDEs
47D06 One-parameter semigroups and linear evolution equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
Full Text: DOI
[1] J.-M. Bismut, G. Lebeau, Le Laplacien hypoelliptique de J.-M. Bismut, Exposé au séminaire X EDP, octobre 2004
[2] Bouchut, F., Existence and uniqueness of a global smooth solution for the vlasov – poisson – fokker – planck system in three dimensions, J. funct. anal., 111, 239-258, (1993) · Zbl 0777.35059
[3] Bouchut, F.; Dolbeault, J., On long time asymptotics of the vlasov – fokker – planck equation and of the vlasov – poisson – fokker – planck system with coulombic and Newtonian potentials, Differential integral equations, 8, 3, 487-514, (1995) · Zbl 0830.35129
[4] Carillo, J.A.; Soler, J.; Vasquez, J.L., Asymptotic behavior and selfsimilarity for the three-dimensional vlasov – poisson – fokker – planck system, J. funct. anal., 141, 99-132, (1996) · Zbl 0873.35066
[5] Coulhon, T.; Saloff-Coste, L.; Varopoulos, N.Th., Analysis and geometry on groups, Cambridge tracts in math., (1992), Cambridge Univ. Press · Zbl 0813.22003
[6] Degond, P., Global existence of smooth solutions for the vlasov – poisson – fokker – planck equation in 1 and 2 space dimensions, Ann. sci. école norm. sup. (4), 19, 519-542, (1986) · Zbl 0619.35087
[7] Desvillettes, L.; Villani, C., On the trend to global equilibrium in spatially inhomogeneous systems. part I: the linear fokker – planck equation, Comm. pure appl. math., 54, 1, 1-42, (2001) · Zbl 1029.82032
[8] Dolbeault, J., Free energy and solutions of the vlasov – poisson – fokker – planck system: external potential and confinement (large time behavior and steady states), J. math. pures appl. (9), 78, 121-157, (1999) · Zbl 1115.82316
[9] Eckmann, J.-P.; Hairer, M., Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Comm. math. phys., 212, 1, 105-164, (2000) · Zbl 1044.82008
[10] Eckmann, J.-P.; Hairer, M., Spectral properties of hypoelliptic operators, Comm. math. phys., 235, 2, 233-253, (2003) · Zbl 1040.35016
[11] Eckmann, J.-P.; Pillet, C.-A.; Rey-Bellet, L., Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat Bath at different temperature, Comm. math. phys., 201, 3, 657-697, (1999) · Zbl 0932.60103
[12] Guo, Y., The Landau equation in a periodic box, Comm. math. phys., 231, 391-434, (2002) · Zbl 1042.76053
[13] Helffer, B.; Nier, F., Hypoellliptic estimates and spectral theory for fokker – planck operators and Witten Laplacians, Lecture notes in math., vol. 1862, (2005), Springer · Zbl 1072.35006
[14] Hérau, F.; Nier, F., Isotropic hypoellipticity and trend to equilibrium for the fokker – planck equation with high degree potential, Arch. ration. mech. anal., 171, 2, 151-218, (2004), Announced in Actes Colloque EDP Forges-les-eaux 2002, 12 p · Zbl 1139.82323
[15] Hérau, F.; Sjöstrand, J.; Stolk, C., Semiclassical analysis for the kramers – fokker – planck equation, Comm. partial differential equations, 30, 4-6, 689-760, (2005) · Zbl 1083.35149
[16] Hörmander, L., Hypoelliptic second order differential equations, Acta math., 119, 147-171, (1967) · Zbl 0156.10701
[17] Kagei, Y., Invariant manifolds and long-time asymptotics for the vlasov – poisson – fokker – planck equation, SIAM J. math. anal., 33, 2, 489-507, (2001) · Zbl 0999.35011
[18] Mouhot, C.; Neumann, L., Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19, 4, 969-998, (2006) · Zbl 1169.82306
[19] O’Dwyer, B.P.; Victory, H.P., On classical solutions of vlasov – poisson – fokker – planck systems, Indiana univ. math. J., 39, 105-157, (1990) · Zbl 0674.60097
[20] Ono, K.; Strauss, W., Regular solutions of the vlasov – poisson – fokker – planck system, Discrete contin. dyn. syst., 6, 4, 751-772, (2000) · Zbl 1034.82058
[21] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Appl. math. sci., (1983), Springer-Verlag Berlin · Zbl 0516.47023
[22] Rein, G.; Weckler, J., Generic global classical solutions of the vlasov – fokker – planck – poisson system in three dimensions, J. differential equations, 99, 59-77, (1992) · Zbl 0810.35090
[23] Soler, J., Asymptotic behavior for the vlasov – poisson – fokker – planck system, Proc. 2nd world congress of nonlinear analysis, Nonlinear anal., 30, 8, 5217-5228, (1997) · Zbl 0895.35082
[24] Talay, D., Approximation of invariant measures of nonlinear Hamiltonian and dissipative stochastic differential equations, (), 139-169
[25] C. Villani, Hypocoercivity, part I, Hörmander operators in Hilbert spaces, in preparation
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