The velocity field of a dense swarm can be modeled in the continuum limit by a variant of the Navier-Stokes equations for fluid flow, in which there is no pressure term and the fluid is allowed to be compressible. Under the additional hypothesis that the velocity field is conservative, these nonlinear equations reduce to a single nonlinear scalar equation that has exact analytic solutions related to the heat equation.

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1998-1999 Mathematics in Biology