2016/12/27 by Chen, Gui-Qiang G., Slemrod, Marshall, Wang, Dehua
#35D30 #35L45 #35L65 #35M10 (Primary) #35Q31 #35Q35 #53C21 #53C42 #53C45 #57R40 #57R42 #58J32 #76F02 #76H05 #76N10 (Secondary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1612.08757
A theory for the evolution of a metric g driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash-Kuiper theorem is then applied to this Riemannian manifold to produce a wild evolving C1 manifold. The theory is applied to the incompressible Euler and Navier-Stokes equations. One practical outcome of the theory is a computation of critical profile initial data for what may be interpreted as the onset of turbulence for the classical incompressible Navier-Stokes equations.