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. 
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/
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