Fractional heat semigroup kernels on metric measure spaces satisfy Euclidean-style decay, yielding Strichartz regularity and capacity-based trace criteria for fractional dissipative Cauchy problems.
Fractional and fractal derivatives modeling of turbulence
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abstract
This study makes the first attempt to use the 2/3-order fractional Laplacian modeling of enhanced diffusing movements of random turbulent particle resulting from nonlinear inertial interactions. A combined effect of the inertial interactions and the molecule Brownian diffusivities is found to be the bi-fractal mechanism behind multifractal scaling in the inertial range of scales of moderate Reynolds number turbulence. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 2/3-order fractional Laplacian representation is also used to construct a fractional Reynolds equation for nonlinear interactions of fluctuating velocity components, underlying turbulence spacetime fractal structures of Levy 2/3 stable distribution. The new perspective of this study is that the fractional calculus is an effective approach modeling of chaotic fractal phenomena induced by nonlinear interactions.
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Fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations
Fractional heat semigroup kernels on metric measure spaces satisfy Euclidean-style decay, yielding Strichartz regularity and capacity-based trace criteria for fractional dissipative Cauchy problems.