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April  2020, 13(2): 345-371. doi: 10.3934/krm.2020012

Diffusion and kinetic transport with very weak confinement

1. 

CEREMADE (CNRS UMR n° 7534), PSL university, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France

2. 

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

* Corresponding author: Emeric Bouin

Received  January 2019 Revised  September 2019 Published  January 2020

This paper is devoted to Fokker-Planck and linear kinetic equations with very weak confinement corresponding to a potential with an at most logarithmic growth and no integrable stationary state. Our goal is to understand how to measure the decay rates when the diffusion wins over the confinement although the potential diverges at infinity. When there is no confinement potential, it is possible to rely on Fourier analysis and mode-by-mode estimates for the kinetic equations. Here we develop an alternative approach based on moment estimates and Caffarelli-Kohn-Nirenberg inequalities of Nash type for diffusion and kinetic equations.

Citation: Emeric Bouin, Jean Dolbeault, Christian Schmeiser. Diffusion and kinetic transport with very weak confinement. Kinetic and Related Models, 2020, 13 (2) : 345-371. doi: 10.3934/krm.2020012
References:
[1]

E. Aghion, D. A. Kessler and E. Barkai, From non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory, Phys. Rev. Lett., 122 (2019), 010601. doi: 10.1103/PhysRevLett.122.010601.

[2]

D. BakryF. BolleyI. Gentil and P. Maheux, Weighted Nash inequalities, Rev. Mat. Iberoam., 28 (2012), 879-906.  doi: 10.4171/RMI/695.

[3]

D. BakryP. Cattiaux and A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008), 727-759.  doi: 10.1016/j.jfa.2007.11.002.

[4]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, vol. 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[5]

F. Baudoin, Bakry-Émery meet Villani,, J. Funct. Anal., 273 (2017), 2275–2291. doi: 10.1016/j.jfa.2017.06.021.

[6]

J. Ben-Artzi and A. Einav., Weak Poincaré inequalities in the absence of spectral gaps, Annales Henri Poincaré, (Oct. 2019). doi: 10.1007/s00023-019-00858-4.

[7]

E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot and C. Schmeiser, Hypocoercivity without confinement, 2017, https://hal.archives-ouvertes.fr/hal-01575501, To appear in Pure and Applied Analysis.

[8]

E. Bouin, J. Dolbeault and C. Schmeiser, A variational proof of Nash's inequality, https://hal.archives-ouvertes.fr/hal-01940110, To appear in Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.

[9]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259–275, http://www.numdam.org/item/CM_1984__53_3_259_0.

[10]

C. Cao, The kinetic Fokker-Planck equation with weak confinement force, 2018, https://hal.archives-ouvertes.fr/hal-01697058.

[11]

E. A. Carlen and M. Loss, Sharp constant in Nash's inequality, Internat. Math. Res. Notices, 1993 (1993), 213-215.  doi: 10.1155/S1073792893000224.

[12]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, Comptes Rendus Mathématique, 347 (2009), 511–516. doi: 10.1016/j.crma.2009.02.025.

[13]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.

[14]

J. DolbeaultM. Muratori and B. Nazaret, Weighted interpolation inequalities: A perturbation approach, Math. Ann., 369 (2017), 1237-1270.  doi: 10.1007/s00208-016-1480-4.

[15]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[16]

J. B. J. Fourier, Théorie Analytique de la Chaleur, Reprint of the 1822 ed. edition, Cambridge, Cambridge University Press, 2009. doi: 10.1017/CBO9780511693229.

[17]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349–359, https://content.iospress.com/articles/asymptotic-analysis/asy741.

[18]

F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal., 244 (2007), 95-118.  doi: 10.1016/j.jfa.2006.11.013.

[19]

S. Hu and X. Wang, Subexponential decay in kinetic Fokker–Planck equation: Weak hypocoercivity, Bernoulli, 25 (2019), 174-188.  doi: 10.3150/17-BEJ982.

[20]

V. P. Il'in, Some integral inequalities and their applications in the theory of differentiable functions of several variables, Mat. Sb. (N.S.), 54(96) (1961), 331–380, http://mi.mathnet.ru/msb4741.

[21]

O. Kavian and S. Mischler, The Fokker-Planck equation with subcritical confinement force, 2015, https://hal.archives-ouvertes.fr/hal-01241680.

[22]

J. D. Morgan III, Schrödinger operators whose potentials have separated singularities, J. Operator Theory, 1 (1979), 109-115. 

[23]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.

[24]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.

[25]

A. Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand., 8 (1960), 143-153. 

[26]

M. Röckner and F.-Y. Wang, Weak Poincaré inequalities and $\mathrm L^2$-convergence rates of Markov semigroups, J. Funct. Anal., 185 (2001), 564-603.  doi: 10.1006/jfan.2001.3776.

[27]

B. Simon, Semiclassical analysis of low lying eigenvalues. Ⅰ. Nondegenerate minima: Asymptotic expansions, Ann. Inst. H. Poincaré Sect. A (N.S.), 38 (1983), 295-308. 

[28]

J. L. Vázquez, The mathematical theories of diffusion: Nonlinear and fractional diffusion, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, vol. 2186 of Lecture Notes in Math., Springer, Cham, 2017,205–278.

[29]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5.

[30]

F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245.  doi: 10.1006/jfan.1999.3516.

[31]

F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288–310. doi: 10.1006/jfan.2002.3968.

[32]

F.-Y. Wang, Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures, Sci. China Math., 53 (2010), 895-904.  doi: 10.1007/s11425-010-0019-5.

show all references

References:
[1]

E. Aghion, D. A. Kessler and E. Barkai, From non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory, Phys. Rev. Lett., 122 (2019), 010601. doi: 10.1103/PhysRevLett.122.010601.

[2]

D. BakryF. BolleyI. Gentil and P. Maheux, Weighted Nash inequalities, Rev. Mat. Iberoam., 28 (2012), 879-906.  doi: 10.4171/RMI/695.

[3]

D. BakryP. Cattiaux and A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008), 727-759.  doi: 10.1016/j.jfa.2007.11.002.

[4]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, vol. 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[5]

F. Baudoin, Bakry-Émery meet Villani,, J. Funct. Anal., 273 (2017), 2275–2291. doi: 10.1016/j.jfa.2017.06.021.

[6]

J. Ben-Artzi and A. Einav., Weak Poincaré inequalities in the absence of spectral gaps, Annales Henri Poincaré, (Oct. 2019). doi: 10.1007/s00023-019-00858-4.

[7]

E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot and C. Schmeiser, Hypocoercivity without confinement, 2017, https://hal.archives-ouvertes.fr/hal-01575501, To appear in Pure and Applied Analysis.

[8]

E. Bouin, J. Dolbeault and C. Schmeiser, A variational proof of Nash's inequality, https://hal.archives-ouvertes.fr/hal-01940110, To appear in Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.

[9]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259–275, http://www.numdam.org/item/CM_1984__53_3_259_0.

[10]

C. Cao, The kinetic Fokker-Planck equation with weak confinement force, 2018, https://hal.archives-ouvertes.fr/hal-01697058.

[11]

E. A. Carlen and M. Loss, Sharp constant in Nash's inequality, Internat. Math. Res. Notices, 1993 (1993), 213-215.  doi: 10.1155/S1073792893000224.

[12]

J. Dolbeault, C. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, Comptes Rendus Mathématique, 347 (2009), 511–516. doi: 10.1016/j.crma.2009.02.025.

[13]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.

[14]

J. DolbeaultM. Muratori and B. Nazaret, Weighted interpolation inequalities: A perturbation approach, Math. Ann., 369 (2017), 1237-1270.  doi: 10.1007/s00208-016-1480-4.

[15]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[16]

J. B. J. Fourier, Théorie Analytique de la Chaleur, Reprint of the 1822 ed. edition, Cambridge, Cambridge University Press, 2009. doi: 10.1017/CBO9780511693229.

[17]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349–359, https://content.iospress.com/articles/asymptotic-analysis/asy741.

[18]

F. Hérau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal., 244 (2007), 95-118.  doi: 10.1016/j.jfa.2006.11.013.

[19]

S. Hu and X. Wang, Subexponential decay in kinetic Fokker–Planck equation: Weak hypocoercivity, Bernoulli, 25 (2019), 174-188.  doi: 10.3150/17-BEJ982.

[20]

V. P. Il'in, Some integral inequalities and their applications in the theory of differentiable functions of several variables, Mat. Sb. (N.S.), 54(96) (1961), 331–380, http://mi.mathnet.ru/msb4741.

[21]

O. Kavian and S. Mischler, The Fokker-Planck equation with subcritical confinement force, 2015, https://hal.archives-ouvertes.fr/hal-01241680.

[22]

J. D. Morgan III, Schrödinger operators whose potentials have separated singularities, J. Operator Theory, 1 (1979), 109-115. 

[23]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.

[24]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.

[25]

A. Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand., 8 (1960), 143-153. 

[26]

M. Röckner and F.-Y. Wang, Weak Poincaré inequalities and $\mathrm L^2$-convergence rates of Markov semigroups, J. Funct. Anal., 185 (2001), 564-603.  doi: 10.1006/jfan.2001.3776.

[27]

B. Simon, Semiclassical analysis of low lying eigenvalues. Ⅰ. Nondegenerate minima: Asymptotic expansions, Ann. Inst. H. Poincaré Sect. A (N.S.), 38 (1983), 295-308. 

[28]

J. L. Vázquez, The mathematical theories of diffusion: Nonlinear and fractional diffusion, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions, vol. 2186 of Lecture Notes in Math., Springer, Cham, 2017,205–278.

[29]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5.

[30]

F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245.  doi: 10.1006/jfan.1999.3516.

[31]

F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288–310. doi: 10.1006/jfan.2002.3968.

[32]

F.-Y. Wang, Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures, Sci. China Math., 53 (2010), 895-904.  doi: 10.1007/s11425-010-0019-5.

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