# American Institute of Mathematical Sciences

December  2015, 8(4): 617-650. doi: 10.3934/krm.2015.8.617

## Some a priori estimates for the homogeneous Landau equation with soft potentials

 1 Arts et Metiers Paris Tech, Paris 75013, France 2 School of Science, East China University of Science and Technology, Shanghai, 200237 3 Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China

Received  May 2014 Revised  June 2015 Published  July 2015

This paper is devoted to some a priori estimates for the homogeneous Landau equation with soft potentials. Using coercivity properties of the Landau operator for soft potentials, we prove that the global in time a priori estimates of weak solutions in $L^2$ space hold true for moderately soft potential cases $\gamma \in[-2, 0)$ without any smallness assumption on the initial data. For very soft potential cases $\gamma \in[-3, -2)$, which cover in particular the Coulomb case $\gamma=-3$, we get local in time estimates of weak solutions in $L^{2}$.
In the proofs of these estimates, global ones for the special case $\gamma=-2$ and local ones for very soft potential cases $\gamma \in[-3, -2)$, the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality.
Citation: Radjesvarane Alexandre, Jie Liao, Chunjin Lin. Some a priori estimates for the homogeneous Landau equation with soft potentials. Kinetic & Related Models, 2015, 8 (4) : 617-650. doi: 10.3934/krm.2015.8.617
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##### References:
 [1] Ricardo J. Alonso, Véronique Bagland, Bertrand Lods. Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations. Kinetic & Related Models, 2019, 12 (5) : 1163-1183. doi: 10.3934/krm.2019044 [2] Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483 [3] Michele Coti Zelati, Piotr Kalita. Smooth attractors for weak solutions of the SQG equation with critical dissipation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1857-1873. doi: 10.3934/dcdsb.2017110 [4] Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic & Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405 [5] Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019032 [6] Yingzhe Fan, Yuanjie Lei. The Boltzmann equation with frictional force for very soft potentials in the whole space. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4303-4329. doi: 10.3934/dcds.2019174 [7] Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic & Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645 [8] Mingchun Wang, Jiankai Xu, Huoxiong Wu. On Positive solutions of integral equations with the weighted Bessel potentials. Communications on Pure & Applied Analysis, 2019, 18 (2) : 625-641. doi: 10.3934/cpaa.2019031 [9] Xueke Pu, Boling Guo, Jingjun Zhang. Global weak solutions to the 1-D fractional Landau-Lifshitz equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 199-207. doi: 10.3934/dcdsb.2010.14.199 [10] Kay Kirkpatrick. Rigorous derivation of the Landau equation in the weak coupling limit. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1895-1916. doi: 10.3934/cpaa.2009.8.1895 [11] Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293 [12] Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731 [13] Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801 [14] Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic & Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691 [15] Lijuan Wang, Weike Wang. Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2835-2854. doi: 10.3934/cpaa.2019127 [16] Jiankai Xu, Song Jiang, Huoxiong Wu. Some properties of positive solutions for an integral system with the double weighted Riesz potentials. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2117-2134. doi: 10.3934/cpaa.2016030 [17] Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential. Kinetic & Related Models, 2011, 4 (4) : 919-934. doi: 10.3934/krm.2011.4.919 [18] N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647 [19] Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867 [20] Xianbo Sun, Pei Yu. Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 965-987. doi: 10.3934/dcdsb.2018341

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