Partial Regularity in Time for the Landau Equation (with Coulomb Interaction)

François Golse

doi:10.1007/978-3-030-69784-6_13

Partial Regularity in Time for the Landau Equation (with Coulomb Interaction)

François Golse

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

The present paper gives a simplified presentation of a partial regularity result obtained in a joint work with M. P. Gualdani, C. Imbert and A. Vasseur [arXiv:1906.02841 [math.AP]] for the spatially homogeneous Landau equation with Coulomb interaction in three space dimensions. Specifically, we prove that the procedure used in [C. Villani: Arch. Rational Mech. Anal. 143 (1998), 273–307] to construct H-solutions of the Landau equation leads to a class of weak solutions satisfying a truncated relative entropy estimate for values of the distribution function larger than any arbitrary level κ> 0. Using the method introduced in [E. De Giorgi: Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat., 3 (1957), 25–43] for studying the regularity of parabolic equations with bounded diffusion coefficients, we prove that the set of singular times of any such solution has Hausdorff dimension at most 1/2.

Original language	English
Title of host publication	From Particle Systems to Partial Differential Equations - International Conference, Particle Systems and PDEs VI, VII and VIII, 2017-2019
Editors	Cédric Bernardin, François Golse, Patrícia Gonçalves, Valeria Ricci, Ana Jacinta Soares
Publisher	Springer
Pages	283-300
Number of pages	18
ISBN (Print)	9783030697839
DOIs	https://doi.org/10.1007/978-3-030-69784-6_13
Publication status	Published - 1 Jan 2021
Event	International Conference on Particle Systems and PDEs VI, VII and VIII, 2017-2019 - Lisbon, Portugal Duration: 2 Dec 2019 → 6 Dec 2019

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	352
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International Conference on Particle Systems and PDEs VI, VII and VIII, 2017-2019
Country/Territory	Portugal
City	Lisbon
Period	2/12/19 → 6/12/19

Keywords

DeGiorgi-Nash-Moser regularity theory
Hausdorff dimension
Landau equation
Partial regularity

Access to Document

10.1007/978-3-030-69784-6_13

Cite this

Golse, F. (2021). Partial Regularity in Time for the Landau Equation (with Coulomb Interaction). In C. Bernardin, F. Golse, P. Gonçalves, V. Ricci, & A. J. Soares (Eds.), From Particle Systems to Partial Differential Equations - International Conference, Particle Systems and PDEs VI, VII and VIII, 2017-2019 (pp. 283-300). (Springer Proceedings in Mathematics and Statistics; Vol. 352). Springer. https://doi.org/10.1007/978-3-030-69784-6_13

Golse, François. / Partial Regularity in Time for the Landau Equation (with Coulomb Interaction). From Particle Systems to Partial Differential Equations - International Conference, Particle Systems and PDEs VI, VII and VIII, 2017-2019. editor / Cédric Bernardin ; François Golse ; Patrícia Gonçalves ; Valeria Ricci ; Ana Jacinta Soares. Springer, 2021. pp. 283-300 (Springer Proceedings in Mathematics and Statistics).

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title = "Partial Regularity in Time for the Landau Equation (with Coulomb Interaction)",

abstract = "The present paper gives a simplified presentation of a partial regularity result obtained in a joint work with M. P. Gualdani, C. Imbert and A. Vasseur [arXiv:1906.02841 [math.AP]] for the spatially homogeneous Landau equation with Coulomb interaction in three space dimensions. Specifically, we prove that the procedure used in [C. Villani: Arch. Rational Mech. Anal. 143 (1998), 273–307] to construct H-solutions of the Landau equation leads to a class of weak solutions satisfying a truncated relative entropy estimate for values of the distribution function larger than any arbitrary level κ> 0. Using the method introduced in [E. De Giorgi: Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat., 3 (1957), 25–43] for studying the regularity of parabolic equations with bounded diffusion coefficients, we prove that the set of singular times of any such solution has Hausdorff dimension at most 1/2.",

keywords = "DeGiorgi-Nash-Moser regularity theory, Hausdorff dimension, Landau equation, Partial regularity",

author = "Fran{\c c}ois Golse",

note = "Publisher Copyright: {\textcopyright} 2021, Springer Nature Switzerland AG.; International Conference on Particle Systems and PDEs VI, VII and VIII, 2017-2019 ; Conference date: 02-12-2019 Through 06-12-2019",

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language = "English",

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Golse, F 2021, Partial Regularity in Time for the Landau Equation (with Coulomb Interaction). in C Bernardin, F Golse, P Gonçalves, V Ricci & AJ Soares (eds), From Particle Systems to Partial Differential Equations - International Conference, Particle Systems and PDEs VI, VII and VIII, 2017-2019. Springer Proceedings in Mathematics and Statistics, vol. 352, Springer, pp. 283-300, International Conference on Particle Systems and PDEs VI, VII and VIII, 2017-2019, Lisbon, Portugal, 2/12/19. https://doi.org/10.1007/978-3-030-69784-6_13

Partial Regularity in Time for the Landau Equation (with Coulomb Interaction). / Golse, François.
From Particle Systems to Partial Differential Equations - International Conference, Particle Systems and PDEs VI, VII and VIII, 2017-2019. ed. / Cédric Bernardin; François Golse; Patrícia Gonçalves; Valeria Ricci; Ana Jacinta Soares. Springer, 2021. p. 283-300 (Springer Proceedings in Mathematics and Statistics; Vol. 352).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Golse, François

PY - 2021/1/1

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N2 - The present paper gives a simplified presentation of a partial regularity result obtained in a joint work with M. P. Gualdani, C. Imbert and A. Vasseur [arXiv:1906.02841 [math.AP]] for the spatially homogeneous Landau equation with Coulomb interaction in three space dimensions. Specifically, we prove that the procedure used in [C. Villani: Arch. Rational Mech. Anal. 143 (1998), 273–307] to construct H-solutions of the Landau equation leads to a class of weak solutions satisfying a truncated relative entropy estimate for values of the distribution function larger than any arbitrary level κ> 0. Using the method introduced in [E. De Giorgi: Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat., 3 (1957), 25–43] for studying the regularity of parabolic equations with bounded diffusion coefficients, we prove that the set of singular times of any such solution has Hausdorff dimension at most 1/2.

AB - The present paper gives a simplified presentation of a partial regularity result obtained in a joint work with M. P. Gualdani, C. Imbert and A. Vasseur [arXiv:1906.02841 [math.AP]] for the spatially homogeneous Landau equation with Coulomb interaction in three space dimensions. Specifically, we prove that the procedure used in [C. Villani: Arch. Rational Mech. Anal. 143 (1998), 273–307] to construct H-solutions of the Landau equation leads to a class of weak solutions satisfying a truncated relative entropy estimate for values of the distribution function larger than any arbitrary level κ> 0. Using the method introduced in [E. De Giorgi: Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat., 3 (1957), 25–43] for studying the regularity of parabolic equations with bounded diffusion coefficients, we prove that the set of singular times of any such solution has Hausdorff dimension at most 1/2.

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KW - Hausdorff dimension

KW - Landau equation

KW - Partial regularity

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T3 - Springer Proceedings in Mathematics and Statistics

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A2 - Gonçalves, Patrícia

A2 - Ricci, Valeria

A2 - Soares, Ana Jacinta

PB - Springer

T2 - International Conference on Particle Systems and PDEs VI, VII and VIII, 2017-2019

Y2 - 2 December 2019 through 6 December 2019

ER -

Golse F. Partial Regularity in Time for the Landau Equation (with Coulomb Interaction). In Bernardin C, Golse F, Gonçalves P, Ricci V, Soares AJ, editors, From Particle Systems to Partial Differential Equations - International Conference, Particle Systems and PDEs VI, VII and VIII, 2017-2019. Springer. 2021. p. 283-300. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-030-69784-6_13

Partial Regularity in Time for the Landau Equation (with Coulomb Interaction)

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