Anomalous Diffusion by Fractal Homogenization

被引:0
作者
Armstrong, Scott [1 ]
Vicol, Vlad [1 ]
机构
[1] NYU, Courant Inst Math Sci, New York, NY 10012 USA
基金
美国国家科学基金会;
关键词
Anomalous diffusion; Passive scalar; Iterative quantitative homogenization; EXACT RENORMALIZATION; MATHEMATICAL-MODELS; PASSIVE SCALARS; DISSIPATION; TRANSPORT; FLOWS; UNIQUENESS; EQUATION; FIELDS; TRACER;
D O I
10.1007/s40818-024-00189-6
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
For every alpha < 1/3, we construct an explicit divergence-free vector field b (t, x) which is periodic in space and time and belongs to (CtCx alpha)-C-0 boolean AND C(t)(alpha)C(x )(0)such that the corresponding scalar advection-diffusion equation partial derivative(t)theta(kappa )+ b<middle dot>del theta(kappa)-kappa Delta theta(kappa )= 0 exhibits anomalous dissipation of scalar variance for arbitrary H-1 initial data: lim sup(kappa -> 0 )integral(1)(0)integral(Td)kappa divided by del theta(kappa)(t,x)divided by(2)dx dt > 0. The vector field is deterministic and has a fractal structure, with periodic shear flowsalternating in time between different directions serving as the base fractal. Theseshear flows are repeatedly inserted at infinitely many scales in suitable Lagrangiancoordinates. Using an argument based on ideas from quantitative homogenization, thecorresponding advection-diffusion equation with small kappa is progressively renormal-ized, one scale at a time, starting from the (very small) length scale determined by themolecular diffusivity up to the macroscopic (unit) scale. At each renormalization step,the effective diffusivity is enhanced by the influence of advection on that scale. By iter-ating this procedure across many scales, the effective diffusivity on the macroscopicscale is shown to be of order one.
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页数:145
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