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June  2022, 15(3): 467-516. doi: 10.3934/krm.2022003

Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition

Hongjie Dong , Yan Guo and  Timur Yastrzhembskiy

Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA

Received  November 2021 Published  June 2022 Early access  January 2022

Fund Project: H. Dong’s research is supported in part by the Simons Foundation, grant no. 709545, a Simons fellowship, grant no. 007638, and the NSF under agreement DMS-2055244.
Y. Guo's research is supported in part by NSF DMS-grant 2106650

We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the $ S_p $ estimate of [7], we prove regularity in the kinetic Sobolev spaces $ S_p $ and anisotropic Hölder spaces for such weak solutions. Such $ S_p $ regularity leads to the uniqueness of weak solutions.

Keywords: Kinetic Kolmogorov-Fokker-Planck equations, specular boundary condition, Landau equation, plasma physics, $ S_p $ estimates.
Mathematics Subject Classification: Primary: 35Q84, 35K70, 35H10, 35B45, 34A12; Secondary: 35Q70.
Citation: Hongjie Dong, Yan Guo, Timur Yastrzhembskiy. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition. Kinetic and Related Models, 2022, 15 (3) : 467-516. doi: 10.3934/krm.2022003
References:
[1]

D. Albritton, S. Armstrong, J.-C. Mourrat and M. Novack, Variational methods for the kinetic Fokker-Planck equation, arXiv: 1902.04037.

[2]

R. Beals and V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), 370-405.  doi: 10.1016/0022-247X(87)90252-6.

[3]

M. Bramanti, M.C. Cerutti, M. Manfredini, $L^p$ estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. Appl., 200 (1996), no. 2,332–354.

[4]

M. Bramanti, G. Cupini, E. Lanconelli, E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients., Math. Nachr., 286 (2013), no. 11-12, 1087–1101.

[5]

H. Dong, Timur Yastrzhembskiy, Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in divergence form, in preparation.

[6]

H. Dong, Y. Guo and Z. Ouyang, The Vlasov-Poisson-Landau system with the specular reflection boundary condition, arXiv: 2010.05314.

[7]

H. Dong and T. Yastrzhembskiy, Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in nondivergence form, arXiv: 2107.08568, to appear in Arch. Ration. Mech. Anal..

[8]

R. Duan, S. Liu, S. Sakamoto and R. M. Strain, Global mild solutions of the Landau and non-cutoff Boltzmann equations, Comm. Pure Appl. Math., 74 (2021), 932-1020.  doi: 10.1002/cpa.21920.

[9]

Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 245-263.  doi: 10.1007/BF02096997.

[10]

Y. Guo, Singular solutions of the Vlasov-Maxwell system on a half Line, Arch. Rational Mech. Anal., 131 (1995), 241-304.  doi: 10.1007/BF00382888.

[11]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.

[12]

Y. Guo, H. J. Hwang, J. W. Jang and Z. Ouyang, The Landau equation with the specular reflection boundary condition, Arch. Ration. Mech. Anal., 236 (2020), 1389-1454.  doi: 10.1007/s00205-020-01496-5.

[13]

H. J. Hwang, J. Jang and J. Jung, The Fokker-Planck equation with absorbing boundary conditions in bounded domains, SIAM J. Math. Anal., 50 (2018), 2194-2232.  doi: 10.1137/16M1109928.

[14]

H. J. Hwang, J. Jang and J. J. L. Velázquez, The Fokker-Planck equation with absorbing boundary conditions, Arch. Ration. Mech. Anal., 214 (2014), 183-233.  doi: 10.1007/s00205-014-0758-5.

[15]

H. J. Hwang, J. Jang and J. J. L. Velázquez, Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions, Arch. Ration. Mech. Anal., 231 (2019), 1309-1400.  doi: 10.1007/s00205-018-1299-0.

[16]

J. Kim, Y. Guo and H. J. Hwang, An $L^2$ to $L^{\infty}$ framework for the Landau equation, Peking Math. J., 3 (2020), 131-202.  doi: 10.1007/s42543-019-00018-x.

[17]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, 96. American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096.

[18]

N. V. Krylov, On a representation of fully nonlinear elliptic operators in terms of pure second order derivatives and its applications, J. Math. Sci. (N.Y.), 177 (2011), 1-26.  doi: 10.1007/s10958-011-0445-0.

[19]

M. Litsgård and K. Nyström, The Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients, J. Funct. Anal., 281 (2021), Paper No. 109226, 39 pp. doi: 10.1016/j.jfa.2021.109226.

[20]

T. S. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Physics, 31 (1953), 253-259. 

[21]

L. Niebel and R. Zacher, Kinetic maximal $L^p$-regularity with temporal weights and application to quasilinear kinetic diffusion equations., J. Differential Equations, 307 (2022), 29-82. 

[22]

S. Polidoro and M. A. Ragusa, Sobolev-Morrey spaces related to an ultraparabolic equation, Manuscripta Math., 96 (1998), 371-392.  doi: 10.1007/s002290050072.

show all references

References:
[1]

D. Albritton, S. Armstrong, J.-C. Mourrat and M. Novack, Variational methods for the kinetic Fokker-Planck equation, arXiv: 1902.04037.

[2]

R. Beals and V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), 370-405.  doi: 10.1016/0022-247X(87)90252-6.

[3]

M. Bramanti, M.C. Cerutti, M. Manfredini, $L^p$ estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. Appl., 200 (1996), no. 2,332–354.

[4]

M. Bramanti, G. Cupini, E. Lanconelli, E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients., Math. Nachr., 286 (2013), no. 11-12, 1087–1101.

[5]

H. Dong, Timur Yastrzhembskiy, Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in divergence form, in preparation.

[6]

H. Dong, Y. Guo and Z. Ouyang, The Vlasov-Poisson-Landau system with the specular reflection boundary condition, arXiv: 2010.05314.

[7]

H. Dong and T. Yastrzhembskiy, Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in nondivergence form, arXiv: 2107.08568, to appear in Arch. Ration. Mech. Anal..

[8]

R. Duan, S. Liu, S. Sakamoto and R. M. Strain, Global mild solutions of the Landau and non-cutoff Boltzmann equations, Comm. Pure Appl. Math., 74 (2021), 932-1020.  doi: 10.1002/cpa.21920.

[9]

Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 245-263.  doi: 10.1007/BF02096997.

[10]

Y. Guo, Singular solutions of the Vlasov-Maxwell system on a half Line, Arch. Rational Mech. Anal., 131 (1995), 241-304.  doi: 10.1007/BF00382888.

[11]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.

[12]

Y. Guo, H. J. Hwang, J. W. Jang and Z. Ouyang, The Landau equation with the specular reflection boundary condition, Arch. Ration. Mech. Anal., 236 (2020), 1389-1454.  doi: 10.1007/s00205-020-01496-5.

[13]

H. J. Hwang, J. Jang and J. Jung, The Fokker-Planck equation with absorbing boundary conditions in bounded domains, SIAM J. Math. Anal., 50 (2018), 2194-2232.  doi: 10.1137/16M1109928.

[14]

H. J. Hwang, J. Jang and J. J. L. Velázquez, The Fokker-Planck equation with absorbing boundary conditions, Arch. Ration. Mech. Anal., 214 (2014), 183-233.  doi: 10.1007/s00205-014-0758-5.

[15]

H. J. Hwang, J. Jang and J. J. L. Velázquez, Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions, Arch. Ration. Mech. Anal., 231 (2019), 1309-1400.  doi: 10.1007/s00205-018-1299-0.

[16]

J. Kim, Y. Guo and H. J. Hwang, An $L^2$ to $L^{\infty}$ framework for the Landau equation, Peking Math. J., 3 (2020), 131-202.  doi: 10.1007/s42543-019-00018-x.

[17]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, 96. American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096.

[18]

N. V. Krylov, On a representation of fully nonlinear elliptic operators in terms of pure second order derivatives and its applications, J. Math. Sci. (N.Y.), 177 (2011), 1-26.  doi: 10.1007/s10958-011-0445-0.

[19]

M. Litsgård and K. Nyström, The Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients, J. Funct. Anal., 281 (2021), Paper No. 109226, 39 pp. doi: 10.1016/j.jfa.2021.109226.

[20]

T. S. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Physics, 31 (1953), 253-259. 

[21]

L. Niebel and R. Zacher, Kinetic maximal $L^p$-regularity with temporal weights and application to quasilinear kinetic diffusion equations., J. Differential Equations, 307 (2022), 29-82. 

[22]

S. Polidoro and M. A. Ragusa, Sobolev-Morrey spaces related to an ultraparabolic equation, Manuscripta Math., 96 (1998), 371-392.  doi: 10.1007/s002290050072.

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