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Global classical solutions of the Boltzmann equation without angular cut-off. (English) Zbl 1248.35140

Summary: This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, \( r^{-(p-1)}\) with \( p>2\), for initial perturbations of the Maxwellian equilibrium states, as announced in an earlier paper by the authors. We more generally cover collision kernels with parameters \( \sin (0,1)\) and \( \gamma\) satisfying \( \gamma > -n\) in arbitrary dimensions \( \mathbb{T}^n \times \mathbb{R}^n\) with \( n\geq 2\). Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann \( H\)-theorem. When \( \gamma \geq -2s\), we have exponential time decay to the Maxwellian equilibrium states. When \( \gamma <-2s\), our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when \( \gamma \geq -2s\), as conjectured by C. Mouhot and the second author [J. Math. Pures Appl. (9) 87, No. 5, 515–535 (2007; Zbl 1388.76338)]. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory.

MSC:

35Q20 Boltzmann equations
35R11 Fractional partial differential equations
35Q82 PDEs in connection with statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35H20 Subelliptic equations
35B65 Smoothness and regularity of solutions to PDEs
26A33 Fractional derivatives and integrals

Citations:

Zbl 1388.76338
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References:

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