yet another look at the Sugitani 1974 paper for arbitrary nonlinear forcing term F

9:57 PM (1 hour ago)



We have a few different approaches to the problem of phase transition of nonlinear fractional diffusion. It’s nice to know that KPP forcing term F(u)=u^2u can be solved stochastically (Cipriano et. al.) and analytically has a turbulent phase transition ( http://arxiv.org/pdf/1111.0408.pdf) but for more general F I don’t know proven results on phase transiton. So Sugitani 1974 is a good point to return:
So the fundamental solution for arbitrary F solves the integral equation using that of F==0 say p(t,x) as
q(t,x) = p(t,x) + \int ds \int dy p(ts,xy) F(q(s,y))
This is useful probably because p(t,x) is known. So what to do from here to show there is a phase transition? Open question.
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10:56 PM (1 minute ago)



SIMPLISTIC REASONING
Ok so the solution can probably be produced by a totally stupid iteration scheme:
q_0(t,x) = p(t,x)
q_1 = p(t,x) + \int p(…) F(q_0)
…
q_n = p(t,x) + \int p(…) F(q_{n1})
Let’s be bad mathematicians and just assume this works for now without rigorous justification, and let’s just do some simplistic reasoning. When F=0 not much is going on. When F is not zero, sometimes this scheme will converge nicely and sometimes it will not converge nicely. In the case F=u^2u nice and not nice convergence (phase transition to turbulence) will depend on the parameter alpha attached to p(t,x) so this is a subtle issue and not dependent on F per se. So the question of phase transition boils down to this subtle issue. When F(u) is generic nonlinear, assuming the iteration scheme above converges, it looks to me like a funky version of polynomial iteration, and intuition from polynomial iterations applied sloppily would tell us that if F(z) is a polynomial, and the scheme were simply
z_k = F(z_{k1})
then there is sensitive dependence on initial z_0 known from the rational iteration literature. Supposing this carried over to the modification of the scheme by also { \int p(…) } operator, we expect turbulence quite often. This extremely intuitive and sloppy idea is the first that comes to mind.
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