10,653 research outputs found
Diffusion in multiscale spacetimes
We study diffusion processes in anomalous spacetimes regarded as models of
quantum geometry. Several types of diffusion equation and their solutions are
presented and the associated stochastic processes are identified. These results
are partly based on the literature in probability and percolation theory but
their physical interpretation here is different since they apply to quantum
spacetime itself. The case of multiscale (in particular, multifractal)
spacetimes is then considered through a number of examples and the most general
spectral-dimension profile of multifractional spaces is constructed.Comment: 23 pages, 5 figures. v2: discussion improved, typos corrected,
references adde
Critical Keller-Segel meets Burgers on : large-time smooth solutions
We show that solutions to the parabolic-elliptic Keller-Segel system on
with critical fractional diffusion
remain smooth for any initial data and any positive time. This disproves, at
least in the periodic setting, the large-data-blowup conjecture by Bournaveas
and Calvez. As a tool, we show smoothness of solutions to a modified critical
Burgers equation via a generalization of the method of moduli of continuity by
Kiselev, Nazarov and Shterenberg. over a setting where the considered equation
has no scaling. This auxiliary result may be interesting by itself. Finally, we
study the asymptotic behavior of global solutions, improving the existing
results.Comment: 17 page
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