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Heat equation governs Brownian spreading of a drop of ink in water.  Still water.  The heat operator is $H_t = \partial_t - \Delta$.  It is an extraordinary operator whose sample paths are continuous and the modern mathematical infrastructure of finance is based on measures on the path space adapted to these called the risk neutral measure.  In no-arbitrage theories of valuation of financial derivatives, one calculates — assuming that heat equation governs sample paths of various fluctuating quantities — the expected value using this measure of discounted future transactions.
Heat equation is nice.  Too nice, in the author’s opinion, since the heat semigroup $P_t$ is not just smoothing but extremely smoothing.  It maps $L^2(X)$ to bounded functions.  The heat equation is associated with essentially the gas phase.
If we change the phase to liquid, things become nastier.  The Navier-Stokes equation is a highly nonlinear perturbation of the heat equation $H_t f = A(f)$ where the right hand side is like $f\nabla f$.  Now the heat semigroup being so smoothing implies that this equation is not going to have solutions anywhere as nice as the heat equation.