scholarly journals Simulation of the Navier-Stokes equations in three dimensions with a spectral collocation method

2013 ◽  
Vol 73 (2) ◽  
pp. 103-129 ◽  
Author(s):  
Christopher J. Subich ◽  
Kevin G. Lamb ◽  
Marek Stastna
Keyword(s):  
Collocation Method ◽  
Stokes Equations ◽  
Three Dimensions ◽  
Spectral Collocation Method ◽  
Navier Stokes ◽  
Spectral Collocation ◽  
Navier Stokes Equations

Singularities

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Stokes Equations ◽  
Negative Answer ◽  
Three Dimensions ◽  
Picard Iteration ◽  
Navier Stokes ◽  
Scale Invariant ◽  
Navier Stokes Equations ◽  
L2 Norm ◽  
Two Dimensional Flow ◽  
Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

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