Exponential Convergence to Equilibrium for the Homogeneous Boltzmann Equation for Hard Potentials Without Cut-Off

This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the so-called (non cut-off with a moderate angular singularity and non mollified) hard potentials. We prove an exponential in time convergence towards the equilibrium, improving results of Villani (Commun Math Phys 234(3): 455–490, 2003) where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a $$L^1$$L1 space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani et al. (http://hal.archives-ouvertes.fr/ccsd-00495786, 2013). We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.

[1]  Cl'ement Mouhot,et al.  Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials , 2005, math/0607532.

[2]  T. Goudon On boltzmann equations and fokker—planck asymptotics: Influence of grazing collisions , 1997 .

[3]  Leif Arkeryd,et al.  Intermolecular forces of infinite range and the Boltzmann equation , 1981 .

[4]  A. Bobylev,et al.  Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems , 1997 .

[5]  S. Mischler,et al.  Spectral analysis of semigroups and growth-fragmentation equations , 2013, 1310.7773.

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

[7]  Isabelle Tristani,et al.  Fractional Fokker-Planck equation , 2013, 1312.1652.

[8]  Stéphane Mischler,et al.  On the spatially homogeneous Boltzmann equation , 1999 .

[9]  G. B. The Dynamical Theory of Gases , 1916, Nature.

[10]  Bernt Wennberg,et al.  The Povzner inequality and moments in the Boltzmann equation , 1996 .

[11]  Cédric Villani,et al.  Cercignani's Conjecture is Sometimes True and Always Almost True , 2003 .

[12]  B. Wennberg Entropy dissipation and moment production for the Boltzmann equation , 1997 .

[13]  Young-Ping Pao,et al.  Boltzmann collision operator with inverse-power intermolecular potentials , 2011 .

[14]  C. Mouhot,et al.  Quantitative Lower Bounds for the Full Boltzmann Equation, Part I: Periodic Boundary Conditions , 2005, math/0607541.

[15]  T. Carleman,et al.  Sur la théorie de l'équation intégrodifférentielle de Boltzmann , 1933 .

[16]  Cl'ement Mouhot,et al.  Explicit Coercivity Estimates for the Linearized Boltzmann and Landau Operators , 2006, math/0607538.

[17]  Irene M. Gamba,et al.  Moment Inequalities and High-Energy Tails for Boltzmann Equations with Inelastic Interactions , 2004 .

[18]  Tosio Kato Perturbation theory for linear operators , 1966 .

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

[20]  C. Cercignani The Boltzmann equation and its applications , 1988 .

[21]  T. Elmroth Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range , 1983 .

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

[23]  Laurent Desvillettes,et al.  Some applications of the method of moments for the homogeneous Boltzmann and Kac equations , 1993 .

[24]  N. Fournier Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition , 2012, 1203.0130.

[25]  Cl'ement Mouhot,et al.  Stability, Convergence to Self-Similarity and Elastic Limit for the Boltzmann Equation for Inelastic Hard Spheres , 2007, math/0701449.

[26]  Cl'ement Mouhot Rate of Convergence to Equilibrium for the Spatially Homogeneous Boltzmann Equation with Hard Potentials , 2006 .

[27]  Maria Pia Gualdani,et al.  Factorization for non-symmetric operators and exponential H-theorem , 2010, 1006.5523.

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

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

[30]  Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions , 2006, math/0606307.

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

[32]  Bernt Wennberg,et al.  A maxwellian lower bound for solutions to the Boltzmann equation , 1997 .

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

[34]  Giuseppe Toscani,et al.  Sharp Entropy Dissipation Bounds and Explicit Rate of Trend to Equilibrium for the Spatially Homogeneous Boltzmann Equation , 1999 .

[35]  Lingbing He,et al.  Smoothing Estimates for Boltzmann Equation with Full-Range Interactions: Spatially Homogeneous Case , 2011 .

[36]  V. Bally,et al.  Regularization properties of the 2D homogeneous Boltzmann equation without cutoff , 2009, 0911.2614.

[37]  Cl'ement Mouhot,et al.  Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff , 2006, math/0607495.

[38]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[39]  C. Mouhot,et al.  On the Well-Posedness of the Spatially Homogeneous Boltzmann Equation with a Moderate Angular Singularity , 2007, math/0703283.