Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation

Abstract

The Boltzmann equation without Grad's angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and Maxwellian type decay in velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C ∞ regularity for positive time.2000 Mathematics Subject Classification. 35A05, 35B65, 35D10, 35H20, 76P05, 84C40.

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

“…taking the weakly collisional limit of [28,29]. Weakly collisional limits of non-cutoff Boltzmann equations would also be interesting (see [1] for hypoelliptic smoothing effects).…”

Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff

“…Over the time, this point of view transformed into the following widespread heuristic conjecture on the diffusive behavior of the Boltzmann operator as a flat fractional Laplacian [1,2,3,24,25,28]:…”

Well-Posedness of Spatially Homogeneous Boltzmann Equation with Full-Range Interaction

“…Thus we start with the global weak solutions established in [24]. Then by using the commutator estimates in [2], we obtain the propagation of the smoothness for the approximated solutions when the initial data are in some Sobolev space.…”

“…Until recently, Alexandre-Morimoto-Ukai-Xu-Yang [2] introduced the pseudo-differential operator and harmonic analysis to study the upper and lower bounds and commutator estimates for the collision operator with modified kinetic factor (v) = v γ . Very recently, Chen-He [9] established the new estimates for both upper and lower bounds for the collision operator with generalized kinetic factor (v) = ( 2 + |v| 2 ) γ /2 .…”

“…Owing to Proposition 2.9 in [2], one has that if 0 < s < 1 2 , then for any K > 0, there exists a constant C(K , ) > 0 such that…”