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Finite-time condensation in 1D Fokker-Planck model for bosons. (arXiv:1901.11098v1 [math.AP])
来源于:arXiv
We consider a one-dimensional analogue of the three-dimensional Fokker-Planck
equation for bosons. The latter is still only partially understood, and, in
particular, the physically relevant question of whether this equation has
solutions which form a Bose-Einstein condensate has remained unanswered. After
a change of variables, we establish global-in-time existence and uniqueness for
our 1D model (and generalisations thereof) using the concept of viscosity
solutions. We show that such solutions enjoy good regularity properties, which
guarantee that in the original variables blow-up can only occur at the origin
and with a fixed spatial profile, up to leading order, following a power law
linked to the steady states of the equation. This enables us to extend entropy
methods beyond the first blow-up time. As a consequence, in the
mass-supercritical case, solutions will blow up in $L^\infty$ in finite time
and - understood in an extended, measure-valued sense - they will eventually
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