10,653 research outputs found

    Diffusion in multiscale spacetimes

    Get PDF
    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 S1{\mathbb S}^1: large-time smooth solutions

    Get PDF
    We show that solutions to the parabolic-elliptic Keller-Segel system on S1{\mathbb S}^1 with critical fractional diffusion (−Δ)12(-\Delta)^\frac{1}{2} 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
    • …
    corecore