Entropy Dissipation Estimates for the Landau Equation: General Cross Sections

We present here an extension to the case of general cross sections of the lower bound obtained in Desvillettes (J Funct Anal 269:1359–1403, 2015, [14]) on the entropy dissipation of Landau’s collision kernel (in the case of soft potentials, including the Coulomb potential). We also simplify somewhat the proof of the lower bound proposed in Desvillettes (J Funct Anal 269:1359–1403, 2015, [14]).

[1]  C. Villani Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation wi , 1999 .

[2]  Radjesvarane Alexandre,et al.  Entropy Dissipation and Long-Range Interactions , 2000 .

[3]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases : notes added in 1951 , 1951 .

[4]  L. Desvillettes Entropy dissipation rate and convergence in kinetic equations , 1989 .

[5]  Giuseppe Toscani,et al.  On the Trend to Equilibrium for Some Dissipative Systems with Slowly Increasing a Priori Bounds , 2000 .

[6]  C. Villani,et al.  Celebrating Cercignani's conjecture for the Boltzmann equation , 2010, 1009.4006.

[7]  Philip T. Gressman,et al.  Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production , 2010, 1007.1276.

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

[9]  N. N. Bogolyubov,et al.  Problems of a Dynamical Theory in Statistical Physics , 1959 .

[10]  Radjesvarane Alexandre Sur le taux de dissipation d'entropie sans troncature angulaire , 1998 .

[11]  Philip T. Gressman,et al.  Global classical solutions of the Boltzmann equation without angular cut-off , 2009, 0912.0888.

[12]  Lingbing He,et al.  Estimates for the Large Time Behavior of the Landau Equation in the Coulomb Case , 2015, Archive for Rational Mechanics and Analysis.

[13]  Eric A. Carlen,et al.  Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation , 1992 .

[14]  Radjesvarane Alexandre,et al.  On the Landau approximation in plasma physics , 2004 .

[15]  Laurent Desvillettes,et al.  Entropy dissipation estimates for the Landau equation in the Coulomb case and applications , 2014, 1408.6025.

[16]  Philip T. Gressman,et al.  Global classical solutions of the Boltzmann equation with long-range interactions , 2010, Proceedings of the National Academy of Sciences.

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

[18]  Laurent Desvillettes,et al.  On asymptotics of the Boltzmann equation when the collisions become grazing , 1992 .

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

[20]  Chao-Jiang Xu,et al.  Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff , 2010 .

[21]  Kung-Chien Wu,et al.  Global in time estimates for the spatially homogeneous Landau equation with soft potentials , 2013, 1306.1220.

[22]  Eric A. Carlen,et al.  Entropy production estimates for Boltzmann equations with physically realistic collision kernels , 1994 .

[23]  Chao-Jiang Xu,et al.  Smoothing effect of weak solutions for the spatially homogeneous Boltzmann Equation without angular cutoff , 2011, 1104.5648.

[24]  M. Pulvirenti,et al.  From Particle Systems to the Landau Equation: A Consistency Result , 2012, 1207.5987.