Geodesic
Nonlinear diffusion and exact solutions to the Navier--Stokes equations
The Bulletin of Irkutsk State University. Series Mathematics, Tome 3 (2010) no. 1, pp. 61-69

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There are considered a number of invariant or partially invariant solutions to the Navier-Stokes equations (NSE) of rank two. These solutions are determined from one-dimensional linear or quasi-linear diffusion equations. Explicit solution, which describes smoothing of initial velocity discontinuity in a liquid with initial uniform vorticity, is constructed. This problem is reduced to a linear equation with coefficients depending on time. The global existence and non-existence theorems in the problem of a longitudinal strip deformation with free boundaries are formulated. In this case, the governing quasi-linear equation is turned out to be integro-differential one. Third example demonstrates process of axially symmetric spreading of a layer on a solid plane. The corresponding free boundary problem is reduced to the Cauchy problem for the second-order degenerate quasi-linear parabolic equation. It allows us to prove the global-in-time solvability of this problem.
Keywords: linear and nonlinear diffusion, Navier–Stokes equations, free boundary problems, invariant and partially invariant solutions. 
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     title = {Nonlinear diffusion and exact solutions to the {Navier--Stokes} equations},
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V. V. Pukhnachev. Nonlinear diffusion and exact solutions to the Navier--Stokes equations. The Bulletin of Irkutsk State University. Series Mathematics, Tome 3 (2010) no. 1, pp. 61-69. http://geodesic.mathdoc.fr/item/IIGUM_2010_3_1_a6/