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 depletion. 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.