154 research outputs found

    Explicit Barenblatt Profiles for Fractional Porous Medium Equations

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    Several one-parameter families of explicit self-similar solutions are constructed for the porous medium equations with fractional operators. The corresponding self-similar profiles, also called \emph{Barenblatt profiles}, have the same forms as those of the classic porous medium equations. These new exact solutions complement current theoretical analysis of the underlying equations and are expected to provide insights for further quantitative investigations

    A Variation Embedding Theorem and Applications

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    Fractional Sobolev spaces, also known as Besov or Slobodetzki spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces and discuss a few applications. First we show q-variation regularity of Cameron-Martin paths associated to fractional Brownian motion and other Volterra processes. This is useful, for instance, to establish large deviations for enhanced fractional Brownian motion. Second, the q-variation embedding, combined with results of rough path theory, provides a different route to a regularity result for stochastic differential equations by Kusuoka. Third, the embedding theorem works in a non-commutative setting and can be used to establish Hoelder/variation regularity of rough paths

    The fractional Keller-Segel model

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    The Keller-Segel model is a system of partial differential equations modelling chemotactic aggregation in cellular systems. This model has blowing up solutions for large enough initial conditions in dimensions d >= 2, but all the solutions are regular in one dimension; a mathematical fact that crucially affects the patterns that can form in the biological system. One of the strongest assumptions of the Keller-Segel model is the diffusive character of the cellular motion, known to be false in many situations. We extend this model to such situations in which the cellular dispersal is better modelled by a fractional operator. We analyze this fractional Keller-Segel model and find that all solutions are again globally bounded in time in one dimension. This fact shows the robustness of the main biological conclusions obtained from the Keller-Segel model
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