Electron. J. Diff. Equ., Vol. 2013 (2013), No. 83, pp. 1-17.

Existence, uniqueness and smoothness of a solution for 3D Navier-Stokes equations with any smooth initial velocity

Arkadiy Tsionskiy, Mikhail Tsionskiy

Abstract:
Solutions of the Navier-Stokes and Euler equations with initial conditions for 2D and 3D cases were obtained in the form of converging series, by an analytical iterative method using Fourier and Laplace transforms in [28,29]. There the solutions are infinitely differentiable functions, and for several combinations of parameters numerical results are presented. This article provides a detailed proof of the existence, uniqueness and smoothness of the solution of the Cauchy problem for the 3D Navier-Stokes equations with any smooth initial velocity. When the viscosity tends to zero, this proof applies also to the Euler equations.

Submitted December 10, 2012. Published April 5, 2013.
Math Subject Classifications: 35Q30, 76D05.
Key Words: 3D Navier-Stokes equations; Fourier transform; Laplace transform; fixed point principle.

An addendum as posted on December 31, 2013. It indicates that some results are incorrect. See the last page of this article.

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Arkadiy Tsionskiy
Department of Civil
Environmental and Ocean Engineering
Stevens Institute of Technology
Hoboken, NJ 07030, USA
email: amtsionsk@gmail.com
Mikhail Tsionskiy
Department of Civil
Environmental and Ocean Engineering
Stevens Institute of Technology
Hoboken, NJ 07030, USA
email: amtsionsk@gmail.com

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