IJPAM: Volume 24, No. 2 (2005)

EXPLICIT GENERAL SERIES SOLUTION
FOR EULER AND NAVIER-STOKES EQUATIONS

G.F.D. Duff$^1$, R.B. Leipnik$^2$
$^1$(Deceased) University of Toronto
Toronto, CANADA
$^2$Department of Mathematics
University of California
Santa Barbara, CA 93106 USA
e-mail: [email protected]


Abstract.Explicit formulas for the coefficients of general series solutions of the Navier-Stokes and Euler equations in a general domain of $R^{3}$ (or $R^{n}$) are developed and analysed. The series are eigenvector expansions in spatial variables and orthogonal or Taylor series in time. While elaborate, the coefficient formulas offer a new avenue of analysis of the vector solutions of time-dependent motions of an incompressible fluid, viscous or not. Evaluation is made for a rectilinear filled domain, and a filled spherical domain. Approximations are developed for cubical and $\overline{C^{\infty }}$ Riemannian domains, operators, and solutions.

Received: August 22, 2005

AMS Subject Classification: 34C20, 35Q35, 37N10

Key Words and Phrases: exact explicit Euler dynamic solutions, Navier-Stokes solutions, Helmholtz basis, algebraic series coefficients, approximations in 3-D, differential geometry

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 24
Issue: 2