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doi: 10.3934/krm.2022010
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Local conditional regularity for the Landau equation with Coulomb potential

CMLS, Ecole polytechnique, 91128 Palaiseau cedex, Paris, France

Received  June 2021 Revised  February 2022 Early access March 2022

This paper studies the regularity of Villani solutions of the space homogeneous Landau equation with Coulomb interaction in dimension 3. Specifically, we prove that any such solution belonging to the Lebesgue space $ L_{t}^{\infty}L_{v}^{q} $ with $ q>3 $ in an open cylinder $ (0,S)\times B $, where $ B $ is an open ball of $ \mathbb{R}^{3} $, must have Hölder continuous second order derivatives in the velocity variables, and first order derivative in the time variable locally in any compact subset of that cylinder.

Citation: Immanuel Ben Porat. Local conditional regularity for the Landau equation with Coulomb potential. Kinetic and Related Models, doi: 10.3934/krm.2022010
References:
[1]

R. AlexandreJ. Liao and C. Lin, Some a priori estimates for the homogeneous Landau equation with soft potentials, Kinet. Relat. Models, 8 (2015), 617-650.  doi: 10.3934/krm.2015.8.617.

[2]

A. A. Arsen'ev and N. V. Peskov, The existence of a generalized solution of Landau's equation, (Russian) Z. Vycisl. Mat i Mat. Fiz., 17 (1977), 1096, 1063-1068. 

[3]

L. Desvillettes, Entropy dissipation estimates for the Landau equation, From Particle Systems to Partial Differential Equations III, Springer Proc. Math. Stat., 162 (2016), 121-143.  doi: 10.1007/978-3-319-32144-8_6.

[4]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials part I, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.

[5]

J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/029.

[6]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, 2010. doi: 10.1090/gsm/019.

[7]

G. B. Folland, How to integrate a polynomial over a sphere, Amer. Math. Monthly, 108 (2001), 446-448.  doi: 10.1080/00029890.2001.11919774.

[8]

N. Fournier, Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential, Comm. Math. Phys., 299 (2010), 765-782.  doi: 10.1007/s00220-010-1113-9.

[9]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Nauka, Moscow, 1989.

[10]

F. Golse, M. Gualdani, C. Imbert and A. Vasseur, Partial regularity in time for the space homogeneous Landau equation with Coulomb potential, arXiv: 1906.0284.

[11]

F. GolseC. ImbertC. Mouhot and A. F. Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (2019), 253-295. 

[12]

M. Gualdani and N. Guillen, Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential, Anal. PDE, 9 (2016), 1772-1809.  doi: 10.2140/apde.2016.9.1772.

[13]

M. Gualdani and N. Guillen, On Ap weights and the Landau equation, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 17, 55 pp. doi: 10.1007/s00526-018-1451-6.

[14]

B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle, Arch. Rational Mech. Anal., 75 (1980/81), 51-58.  doi: 10.1007/BF00284620.

[15]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, Providence, R. I. 1968.

[16]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three Dimensional Navier-Stokes Equations, Cambridge studies in advanced mathematics, 2016. doi: 10.1017/CBO9781139095143.

[17]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344.

[18]

L. Silvestre, Upper bounds for parabolic eqyations and the Landau equation, J. Differential Equations, 262 (2017), 3034-3055.  doi: 10.1016/j.jde.2016.11.010.

[19]

A. Vasseur, The De Giorgi Method for Elliptic and Parabolic Equations and Some Applications, Lectures on the analysis of nonlinear partial differential equations. Part 4,195–222, Morningside Lect. Math., 4, Int. Press, Somerville, MA, 2016.

[20]

C. Villani, On a new class of weak solution to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.

[21]

F. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^{p}$, Indiana Univ. Math. J., 29 (1980), 79-102.  doi: 10.1512/iumj.1980.29.29007.

show all references

References:
[1]

R. AlexandreJ. Liao and C. Lin, Some a priori estimates for the homogeneous Landau equation with soft potentials, Kinet. Relat. Models, 8 (2015), 617-650.  doi: 10.3934/krm.2015.8.617.

[2]

A. A. Arsen'ev and N. V. Peskov, The existence of a generalized solution of Landau's equation, (Russian) Z. Vycisl. Mat i Mat. Fiz., 17 (1977), 1096, 1063-1068. 

[3]

L. Desvillettes, Entropy dissipation estimates for the Landau equation, From Particle Systems to Partial Differential Equations III, Springer Proc. Math. Stat., 162 (2016), 121-143.  doi: 10.1007/978-3-319-32144-8_6.

[4]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials part I, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.

[5]

J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/029.

[6]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, 2010. doi: 10.1090/gsm/019.

[7]

G. B. Folland, How to integrate a polynomial over a sphere, Amer. Math. Monthly, 108 (2001), 446-448.  doi: 10.1080/00029890.2001.11919774.

[8]

N. Fournier, Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential, Comm. Math. Phys., 299 (2010), 765-782.  doi: 10.1007/s00220-010-1113-9.

[9]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Nauka, Moscow, 1989.

[10]

F. Golse, M. Gualdani, C. Imbert and A. Vasseur, Partial regularity in time for the space homogeneous Landau equation with Coulomb potential, arXiv: 1906.0284.

[11]

F. GolseC. ImbertC. Mouhot and A. F. Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (2019), 253-295. 

[12]

M. Gualdani and N. Guillen, Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential, Anal. PDE, 9 (2016), 1772-1809.  doi: 10.2140/apde.2016.9.1772.

[13]

M. Gualdani and N. Guillen, On Ap weights and the Landau equation, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 17, 55 pp. doi: 10.1007/s00526-018-1451-6.

[14]

B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle, Arch. Rational Mech. Anal., 75 (1980/81), 51-58.  doi: 10.1007/BF00284620.

[15]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, Providence, R. I. 1968.

[16]

J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three Dimensional Navier-Stokes Equations, Cambridge studies in advanced mathematics, 2016. doi: 10.1017/CBO9781139095143.

[17]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.  doi: 10.1007/BF00253344.

[18]

L. Silvestre, Upper bounds for parabolic eqyations and the Landau equation, J. Differential Equations, 262 (2017), 3034-3055.  doi: 10.1016/j.jde.2016.11.010.

[19]

A. Vasseur, The De Giorgi Method for Elliptic and Parabolic Equations and Some Applications, Lectures on the analysis of nonlinear partial differential equations. Part 4,195–222, Morningside Lect. Math., 4, Int. Press, Somerville, MA, 2016.

[20]

C. Villani, On a new class of weak solution to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.

[21]

F. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^{p}$, Indiana Univ. Math. J., 29 (1980), 79-102.  doi: 10.1512/iumj.1980.29.29007.

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