List of issues > Series «Mathematics». 2010. Vol. 1
Nonlinear diffusion and exact solutions to the Navier-Stokes equations
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 onedimensional 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 quasilinear 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.
1. Ovsiannikov L. V. Group Analysis of Differential Equations / L. V. Ovsiannikov.– Academic Press. – 1982.
2. Andreev V. K. Application of Group-Theoretic Methods in Hydrodynamics /V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, A. A. Rodionov. – KluwerAcademic. – 1998.
3. Pukhnachov V. V. On a problem of viscous strip deformation with a free boundary/ V. V. Pukhnachov // C. R. Acad. Sci. Paris. – 1999. – T. 328, Serie 1. – P.357-362.
4. Galaktionov V. A. Blow-up for a class of solutions with free boundary for theNavier-Stokes equations / V. A. Galaktionov, J. L. Vazquez // Adv. Diff. Eq. –1999. – T. 4 – P. 297-321.
5. Ladyzhenskaya O. A. Linear and Quasilinear Equations of Parabolic Type / O.A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva. – AMS. – 1968.
6. Galaktionov V. A. Regularity of interfaces in diffusion processes under theinfluence of strong absorption / V. A. Galaktionov, S. I. Shmarev, J. L. Vazquez// Arch. Ration. Mech. Anal. – 1999. – T. 149. – P. 183-212.