A Priori Estimates for Solutions to Landau Equation Under Prodi–Serrin Like Criteria

Ricardo J. Alonso, Véronique Bagland, Laurent Desvillettes, Bertrand Lods

Type: Article

Publication Date: 2024-05-13

Citations: 3

DOI: https://doi.org/10.1007/s00205-024-01992-y

Abstract

Abstract In this paper, we introduce Prodi–Serrin like criteria which enable us to provide a priori estimates for the solutions to the spatially homogeneous Landau equation for all classical soft potentials and dimensions $$d \geqq 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≧</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . The physical case of Coulomb interaction in dimension $$d=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> is included in our analysis; this generalizes the work of Silvestre (J Differ Equ 262:3034–3055, 2017). Our approach is quantitative and does not require a preliminary knowledge of elaborate tools for nonlinear parabolic equations.

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