Global classical solutions of the Boltzmann equation without angular cut-off

Journal of the American Mathematical Society - Tập 24 Số 3 - Trang 771-847
Philip T. Gressman1, Robert M. Strain1
1Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395

Tóm tắt

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)r^{-(p-1)}withp>2p>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 parameterss∈(0,1)s\in (0,1)andγ\gammasatisfyingγ>−n\gamma > -nin arbitrary dimensionsTn×Rn\mathbb {T}^n \times \mathbb {R}^nwithn≥2n\ge 2. Moreover, we prove rapid convergence as predicted by the celebrated BoltzmannHH-theorem. Whenγ≥−2s\gamma \ge -2s, we have exponential time decay to the Maxwellian equilibrium states. Whenγ>−2s\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 whenγ≥−2s\gamma \ge -2s, as conjectured by Mouhot and Strain. 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.

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Journal of the American Mathematical Society

Tập 24 Số 3

771-847

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