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

By Philip T. Gressman, Robert M. Strain. 2009

231 citations recorded by the source corpus.

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 \cite{gsNonCutA}. We more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying $\gamma > -n$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann $H$-theorem. When $\gamma \ge -2s$, 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 $\gamma \ge -2s$, as conjectured in Mouhot-Strain \cite{MR2322149}. 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/92034f65f61c89b4bbf8ee2a41b9928ee5ffbb82

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Citations in the corpus

The Optimal Decay Estimates on the Framework of Besov Spaces for Generally Dissipative Systems (2014, methodology): “Based on the global existence of solutions in weighted Sobolev spaces of geometric fractional order (see [8]), Sohinger and Strain first introduced the mixed Besov space Ḃ−s 2,∞L 2 v(s > 0) to investigate the optimal time decay rates.”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, background): “This computation plays no role in this paper, but it is interesting to see how the metric introduced in [37] arises naturally from the geometry described above.”
Factorization for non-symmetric operators and exponential H-theorem (2010, methodology): “In the case of non-cutoff hard and moderately soft potentials, the linearized operator has a spectral gap [79, 48] and we expect our factorization method to be applicable in this case by using a different decomposition of the linearized collision operator, such as the one used in [74] in order to qu”
Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law (2013, background): “Such nonlinear energy framework has led to resolutions to several open problems in the Boltzmann theory [19, 20, 25].”