1,420 research outputs found
Strichartz type estimates for fractional heat equations
We obtain Strichartz estimates for the fractional heat equations by using
both the abstract Strichartz estimates of Keel-Tao and the
Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous
Strichartz estimate via replacing by
and a parabolic homogeneous Strichartz estimate.
Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue
spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the
Strichartz estimates for the fractional heat equations with a time dependent
potential of an appropriate integrability. As an application, we prove the
global existence and uniqueness of regular solutions in spatial variables for
the generalized Navier-Stokes system with data.Comment: 20 page
Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type
We study the uniqueness, existence, and properties of bounded distributional
solutions of the initial value problem problem for the anomalous diffusion
equation . Here
can be any nonlocal symmetric degenerate elliptic operator including the
fractional Laplacian and numerical discretizations of this operator. The
function is only assumed to be continuous
and nondecreasing. The class of equations include nonlocal (generalized) porous
medium equations, fast diffusion equations, and Stefan problems. In addition to
very general uniqueness and existence results, we obtain -contraction and
a priori estimates. We also study local limits, continuous dependence, and
properties and convergence of a numerical approximation of our equations.Comment: To appear in "Advances in Mathematics
Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB)
equations, i.e. scalar conservation laws with diffusive-dispersive
regularization. We review the existence of traveling wave solutions for these
two classes of evolution equations. For classical equations the traveling wave
problem (TWP) for a local KdVB equation can be identified with the TWP for a
reaction-diffusion equation. In this article we study this relationship for
these two classes of evolution equations with nonlocal diffusion/dispersion.
This connection is especially useful, if the TW equation is not studied
directly, but the existence of a TWS is proven using one of the evolution
equations instead. Finally, we present three models from fluid dynamics and
discuss the TWP via its link to associated reaction-diffusion equations
On Rosenau-Type Approximations to Fractional Diffusion Equations
Owing to the Rosenau argument in Physical Review A, 46 (1992), pag. 12-15,
originally proposed to obtain a regularized version of the Chapman-Enskog
expansion of hydrodynamics, we introduce a non-local linear kinetic equation
which approximates a fractional diffusion equation. We then show that the
solution to this approximation, apart of a rapidly vanishing in time
perturbation, approaches the fundamental solution of the fractional diffusion
(a L\'evy stable law) at large times
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