Viscous Burgers equation physical meaning The viscous Burgers' equation:
$$ q_{t}+q\:q_{x}~=~\nu\:q_{xx}, \mbox{ where } \:\:\nu >0, $$
combines the nonlinear propagation of $q(x,t)$ and the diffusion.


*

*What is this equation for? (in modelling context)

*What is so special about it? (other than if $\nu=0$ then the solution can form a shockwave)
 A: 1D Burger's equation is not meant to model a physical phenomenon. Rather, it is a simplification of homogeneous incompressible Navier-Stokes equations that preserves (some of) its mathematical structure: the non-linear convection term and the second order derivative of viscous forces. 
It was initially intended as a useful simplification to try to understand the mathematical problems present in N-S equations, e.g. the limit of small viscosities (high Reynolds number). It is not anymore a tool for researchers studying existence and uniqueness of solutions of N-S equations, but is still useful to test numerical schemes designed for N-S.
Its multidimensional version also arises in some other physical contexts, see http://arxiv.org/abs/nlin/0012033
A: The convection-advection equation resembles the mathematical structure of Burgers equation. Despite differences (convection-advection has an "extra" variable $c(x,t)$) you can think that Burgers describes compressible material fluid transportation by means of advection (the $ \vec{v} \cdot \nabla $ term) and diffusion (the $D \nabla^2$ term) without mechanical pressure gradients applied. 
