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

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

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.

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. DuanS. LiuS. 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. GuoH. J. HwangJ. 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. HwangJ. 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. HwangJ. 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. HwangJ. 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. KimY. 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. DuanS. LiuS. 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. GuoH. J. HwangJ. 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. HwangJ. 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. HwangJ. 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. HwangJ. 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. KimY. 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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