Invariant solutions of the Navier-Stokes equations

We examine certain invariant solutions of the Navier-Stokes equations. We prove theorems concerning the existence of solutions of boundary-value problems of the corresponding S/H systems.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 56–64, November–December, 1972.In conclusion the author thanks V. V. Pukhnachev for a discussion of the results and for his advice.     

On some new exact solutions of incompressible steady state Navier-Stokes equations

In the present work, Lie symmetries are constructed for the steady state incompressible Navier-Stokes equations in two dimensional space. The Navier-Stokes equations are transformed into a fourth order quasilinear partial differential equation through well known approach of stream function. Seven similarity solutions are obtained using one-parameter Lie group of transformations with commuting infinitesimal operators. Among the seven solutions so obtained, three are new and analysed physically. Also, the remaining four solutions are reported in Polyanin and Zaitsev (Handbook of nonlinear partial differential equations, Chapman and Hall/CRC Press, Boca Raton, pp. 607–608, 2004) and interpreted hydrostatically.     

Discontinuous solutions of the Navier-Stokes equations for compressible flow

We prove local existence and study properties of discontinuous solutions of the Navier-Stokes equations for one-dimensional, compressible, nonisentropic flow. We assume that, modulo a step function, the initial data is in L² and the initial velocity and density are in the space BV. We show that the velocity and the temperature become smoothed out in positive time, and that discontinuities in the density, pressure, and gradients of the velocity and temperature persist for all time. We also show that for stable gases these discontinuities decay exponentially in time, more rapidly for smaller viscosities.     

Spatial local solutions of the Navier-Stokes equations

This paper considers solutions of the Navier-Stokes equations polynomial in the coordinates, which. are called local solutions. For an incompressible fluid, all higher-order terms (sums of higher-order. monomials) of degree 2 are found and it is proved that nontrivial axisymmetric higher-order terms. of degree higher than 2 do not exist. Nonsolenoidal axisymmetric solutions are listed, which can be. treated as steady-state barotropic gas flows in a potential external-force field. All elliptic vortices. generalizing the well-known Kirchhoff solution are calculated. All solutions of degree 3 with the. higher-order term of partial form are found. Some of these solutions break down in a finite time. regardless of the value and sign of viscosity. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 109–119, March–April, 2009.