On the regularity of solutions to the spatially homogeneous Boltzmann equation with polynomially growing collision kernel

Abstract. The paper is devoted to the propagation of smoothness (more precisely L 1 -moments of thederivatives) of the solutions to the spatially homogeneous Boltzmann equation with polynomially growingcollision kernels. 1. IntroductionThe paper is devoted to the spatially homogeneous d -dimensional Boltzmann equation(1.1) @f@t = Q ( f;f ) ; where t ‚ 0, v 2 R d , d ‚ 3, and the collision operator Q is given by the standard formula(1.2) Q ( f;g )( v ) =Z R d dw Z S di 1 wiv; + dnB ( jv iwj;µ )[ f ( v 0 ) g ( w 0 ) if ( v ) g ( w )] : Here n denotes the unit vector in the direction v 0 i v , S di 1 wiv; + = fn 2 S di 1 : ( n;w i v ) ‚ 0 g , dn denotes the Lebesgue measure on S di 1 , µ is the (necessarily acute) angle between w i v and n (or v 0 iv ),(1.3)‰ v 0 = v +( w iv;n ) nw 0 = w i ( w iv;n ) n, ‰ v = v 0 +( w 0 iv 0 ;n ) nw = w 0 i ( w 0 iv 0 ;n ) n and the collision kernel B ( jzj;µ ) is a given measurable non-negative function on R + £ [0 ;…= 2] ofpolynomial growth, i.e.(1.4)Z

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