Anomalous dimensions and the renormalization group in a nonlinear diffusion process

Abstract

We present a renormalization-group (RG) approach to the nonlinear diffusion process BtU^DdxU, with Z) =* J for diu > 0 and Z) = (1 +6)/2 for dxU < 0, which describes the pressure during the filtration of an elastic fluid in an elastoplastic porous medium. Our approach recovers Barenblatt's long-time result that, for a localized initial pressure distribution, w(x,r)~/ ~^°^^'^^f{x/^,€), where / is a scaling function and a'=6/(2;r^)'^^ + 0(6^) is an anomalous dimension, which we compute perturbatively using the RG. This is the first application of the RG to a nonlinear partial differential equation in the absence of noise. PACS numbers: 47.55.Mh, 47.25.Cg, 64.60.Ak, 64.60.HtBuckingham's FI theorem' states that the dependence of a physical quantity on a set of dimensionful parameters may be expressed as the dependence of a dimensionless quantity fl on dimensionless combinations nclli, .. . ,n" of the governing parameters:

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

“…These are of the same form as treated in § §4.1, hence readily solved to be 21) where r = r(x + h − 2) and h = h(t 0 ), by choosing the initial value 22) and v 0,1 (x, t) with r → ℓ. Thus, we have…”

“…Recent development of the theories of pattern formation with dissipative structures gives a good example how to reduce complicated ordinary and partial differential equations to simple equations with slow variables, such as Landau-Stuart equation, the time-dependent Ginzburg-Landau equation and so on. [18] Some years ago, it was shown by an Illinois group [21,22,23] and Bricmont and Kupiainen [24]that the RG equations can be used for a global and asymptotic analysis of ordinary and partial differential equations, hence giving a reduction of evolution equations of some types. A unique feature of the Illinois group's method is to start with the naive perturbative expansion and allow secular terms to appear; the secular terms correspond to the logarithmically divergent terms in QFT.…”

Asymptotic Scaling in a Model Class of Anomalous Reaction-Diffusion Equations

“…Our method is based on the abstract approach developed in [10], based itself on ideas and previous work by several authors [2,3,8,12].…”

Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs

“…We assume them to be small. The nonlinearity exponent is α > 1 (α is not necessary integer) [10] - [15]. The relevance of nonlinear interaction with respect to the long time large scale asymptotic behavior can be justified by means of dimensional analysis [11,15].…”

“…In the previous studies of nonlinear diffusion [10] - [15], the authors had introduced the various nonlinear terms into the diffusion equation modelling the possible fluctuations of transport coefficient, the diffusion-reaction processes, and queuing due to a bounded transport capacity of edges. We also introduce it for accounting the effect of varying dimension of space in the complex network (see the discussion below).…”