154 research outputs found
Explicit Barenblatt Profiles for Fractional Porous Medium Equations
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
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
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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