JAMP  Vol.5 No.4 , April 2017
Classical Fundamental Unique Solution for the Incompressible Navier-Stokes Equation in RN
Abstract: We present a class of non-convective classical solutions for the multidimensional incompressible Navier-Stokes equation. We validate such class as a representative for solutions of the equation in bounded and unbounded domains by investigating the compatibility condition on the boundary, the smoothness of the solution inside the domain and the boundedness of the energy. Eventually, we show that this solution is indeed the unique classical solution for the problem given some appropriate and convenient assumptions on the data.
Cite this paper: S. Khedr, W. (2017) Classical Fundamental Unique Solution for the Incompressible Navier-Stokes Equation in RN. Journal of Applied Mathematics and Physics, 5, 939-952. doi: 10.4236/jamp.2017.54083.

[1]   Gurtin, M. (1982) An Introduction to Continuum Mechanics, Mathematics in Science and Engineering. Academic Press, Cambridge.

[2]   Kambe, T. (2007) Elementary Fluid Mechanics. World Scientific Publishing, Singapore.

[3]   Kambe, T. (2010) A New Formulation of Equations of Compressible Fluids by Analogy with Maxwell’s Equations. Fluid Dynamics Research, 42.

[4]   Kambe, T. (1983) Axisymmetric Vortex Solution of Navier-Stokes Equation. Journal of Physical Society of Japan, 53, 13-15.

[5]   Ladyzhenskaya, O. (1969) The Mathematical Theory of Viscous Incompressible Flow, Translated from the Russian by Richard A. Silverman, Mathematics and its Applications. 2nd Edition, Gordon and Breach.

[6]   Galdi, G.P. (2011) An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Springer Monographs in Mathematics, Springer New York.

[7]   Majda, A.J. and Bertozzi, L. (2001) Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge.

[8]   Tao, T. (2016) Finite Time Blowup for an Averaged Three-Dimensional Navier-Stokes Equation. Journal of American Mathematical Society, 29, 601-674.

[9]   Constantin, P. and Fefferman, C. (1993) Direction of Vorticity and the Problem of Global Regularity for the Na-vier-Stokes Equations. Indiana University Mathematics Journal, 42, 775-789.

[10]   Constantin, P. (1990) Navier-Stokes Equations and Area of Interfaces. Communications in Mathematical Physics, 129, 241-266.

[11]   Constantin, P. (2001) Some Open Problems and Research Directions in the Mathematical Study of Fluid Dynamics, Mathematics Unlimited-2001 and Beyond. Springer Verlag, Berlin, 353-360.

[12]   Caffarelli, L., Kohn, R. and Nirenberg, L. (1982) Partial Regularity of Suitable Weak Solutions of the Navier-Stokes Equations. Communication on Pure & Applied Mathematics, 35, 771-831.

[13]   Evans, L. (2010) Partial Differential Equations. 2nd Edition, Graduate Studies in Mathematics. American Mathematical Society, 19.

[14]   John, F. (1978) Partial Differential Equations. Vol. 1, 4th Edition, Applied Mathematical Sciences, Springer US, New York.