The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential

Abstract

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable. By defining and analyzing a non-isotropy norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces. Contents2000 Mathematics Subject Classification. 35A05, 35B65, 35D10, 35H20, 76P05, 84C40.

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

“…The more interesting thing is that the estimates obtained including the lower and upper bounds are uniform with respect to the parameter . Finally, we refer to [3][4][5]16,17] for the progress on the well-posedness of the inhomogeneous Boltzmann equation when the initial data are near Maxwellian equilibrium.…”

Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator

“…The unraveling of these special features of the non-cutoff Boltzmann operator have led them to conjecture that this collision operator behaves and induces smoothing effects as a fractional Laplacian. The following coercive estimate was later proven in [5] (see also [4,9,18,19])…”

A Revisiting of the $$L^2$$ L 2 -Stability Theory of the Boltzmann Equation Near Global Maxwellians

“…Adopting the terminology given in [2][3][4], we use s = 1 n ∈ (0, 1) instead of n. Then, the angular part b γ (cos θ) satisfies b γ (cos θ) ∼ K θ −(2+2s) , as θ → 0 + for some constant K > 0, which gives a non-integrable singularity at θ = 0.…”

“…The Boltzmann equation (1) with angular cutoff potentials near a global Maxwellian has been extensively studied (see [5,[15][16][17]23,24] and references therein) since the seminal work of Ukai [22]. Recently, Alexandre-Morimoto-Ukai-Xu-Yang [2][3][4] and Gressman-Strain [11] extended the classical theory of Ukai to the Boltzmann equation without angular cutoff using the sophisticated methods from the theory of pseudo-differential operators and harmonic analysis. More precisely, they established the precise coercivity estimate [2][3][4]11,[19][20][21] of the linear collision operator and the trilinear type estimate for the nonlinear collision operator in some nonisotropic norms.…”

“…Recently, Alexandre-Morimoto-Ukai-Xu-Yang [2][3][4] and Gressman-Strain [11] extended the classical theory of Ukai to the Boltzmann equation without angular cutoff using the sophisticated methods from the theory of pseudo-differential operators and harmonic analysis. More precisely, they established the precise coercivity estimate [2][3][4]11,[19][20][21] of the linear collision operator and the trilinear type estimate for the nonlinear collision operator in some nonisotropic norms. These key estimates led to the global existence theory of the Cauchy problem for the Boltzmann equation in a non cutoff setting.…”