Global in time estimates for the spatially homogeneous Landau equation with soft potentials

Abstract This paper deals with some global in time a priori estimates of the spatially homogeneous Landau equation for soft potentials γ ∈ [ − 2 , 0 ) . For the first result, we obtain the estimate of weak solutions in L t α L v 3 − e for α = 2 ( 3 − e ) 3 ( 2 − e ) and 0 e 1 , which is an improvement over estimates by Fournier and Guerin [10] . For the second result, we have the estimate of weak solutions in L t ∞ L v p , p > 1 , which extends part of results by Fournier and Guerin [10] and Alexandre, Liao and Lin [1] . As an application, we deduce some global well-posedness results for γ ∈ [ − 2 , 0 ) . Our estimates include the case γ = − 2 , which is the key point in this paper.

[1]  N. V. Peskov,et al.  On the existence of a generalized solution of Landau's equation☆ , 1977 .

[2]  Cédric Villani,et al.  On the spatially homogeneous landau equation for hard potentials part i : existence, uniqueness and smoothness , 2000 .

[3]  C. Villani.  Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory , 2002 .

[4]  Cédric Villani,et al.  ON THE SPATIALLY HOMOGENEOUS LANDAU EQUATION FOR MAXWELLIAN MOLECULES , 1998 .

[5]  W. Beckner.  Weighted inequalities and Stein-Weiss potentials , 2006, math/0607309.

[6]  Nicolas Fournier,et al.  Well-posedness of the spatially homogeneous Landau equation for soft potentials , 2008, 0806.3379.

[7]  Cédric Villani,et al.  On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations , 1998 .

[8]  Yan Guo,et al.  The Landau Equation in a Periodic Box , 2002 .

[9]  Radjesvarane Alexandre,et al.  Some A priori estimates for the homogeneous Landau equation with soft potentials , 2013, 1302.1814.

[10]  N. Fournier.  Uniqueness of Bounded Solutions for the Homogeneous Landau Equation with a Coulomb Potential , 2009, 0909.0647.

[11]  Cédric Villani,et al.  On the spatially homogeneous landau equation for hard potentials part ii : h-theorem and applications , 2000 .

[12]  William Beckner,et al.  Pitt’s inequality with sharp convolution estimates , 2006, math/0612486.

[13]  W. Beckner.  Pitt's inequality and the fractional Laplacian: Sharp error estimates , 2007, math/0701939.

[14]  H. Guerin.  Solving Landau equation for some soft potentials through a probabilistic approach , 2003 .

[15]  L. Desvillettes,et al.  Smoothing Effects for Classical Solutions of the Full Landau Equation , 2009 .

[16]  Marion Kee,et al.  Analysis , 2004, Machine Translation.