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## More reasons why nonlinear forcing term F for nonlinear fractional diffusions has turbulence

### Zulfikar Ahmed<zulfikar.moinuddin.ahmed@gmail.com>

11:44 AM (0 minutes ago)

 to donoho, bcc: aimee
Recall that for KPP nonlinearity F(u)=u^2-u the instability is known and recent, and for more general F I was intuitively seeking analogue to Julia sets with an intuitive scheme for solving the integral equation.  Let’s take a look quickly at when the integral equation is known to be solvable:

http://www.m-hikari.com/imf-password2009/17-20-2009/awawdehIMF17-20-2009.pdf

So notice that the condition where standard method using a Banach fixed point theorem solves the integral equation is when F is Lipschitz continuous meaning

| F(x) – F(y)| <= C |x-y|

This sort of thing is FALSE for nonlinearities like KPP and possibly for others.  In other words, the standard technique for proving that a solution to the nonlinear fractional diffusion equation with general nonlinear F fails.  Therefore although this does not prove that instability must occur in the solution, it does gives us some further intuition that beyond the KPP instability for non-Lipschitz F, especially when iterations of F produces a complicated Julia set would produce instability/turbulence, and this phenomenon may be provably wider than the case of KPP F=u^2-u.  I’m led to the conjecture that whenever F has a complicated and large Julia set, instability would be expected in the solution (even without any probabilistic interpretation such as branching process).

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Screenshot 2015-08-06 11.33.46.png
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