Nonlocal Burgers equations and generalized Hopf-Cole transforms

Tuesday, February 23, 2010 - 9:00am - 9:45am
EE/CS 3-180
Koji Okitani (University of Sheffield)
Keywords: maximum principle, blow-up, nonlocality

Abstract: We consider nonlocal versions of Burgers equations in a preliminary attempt to
1) generalize Hopf-Cole transforms for incompressible flows, and 2) to assess
the effect of nonlocality on the breakdown of maximum principle leading
to blow-up.

It is well-known that by the Forsyth-Florin-Hopf-Cole transform 1D Burgers
equation is integrable through a Hamilton-Jacobi-like equation for
the velocity potential. In higher spatial dimensions, on top of a potential
component, there appears a solenoidal component. For the former, a similar
method of solution works for multi-dimensional Burgers equation. To treat the
latter, we recast e.g. 2D incompressible Navier-Stokes equations as a nonlocal
Hamilton-Jacobi-like equation using the stream function. This form apparently
exhibits nontrivial cancellations of nonlinear terms, known as nonlinearity

On this basis, we propose a nonlocal model equation in 1D and study its
behavior numerically. It is shown that this model is equivalent to another
model equation which is known to blow up. We derive a Hopf-Cole-like transform
to recast the model as close as a heat diffusion equation, but with an
additional dangerous term. Attempts are made to explore possible transforms
for the 2D Navier-Stokes equations. Time permitting, we may describe yet
another model (joint work with M. Dowker) where a nonlocal term, mimicking the
pressure, leads apparently to blow-up in finite time.