The 3-D Spectrally-Hyperviscous Navier-Stokes Equations on Bounded Domains with Zero Boundary Conditions

TL;DR

This paper extends the spectrally-hyperviscous Navier-Stokes equations (SHNSE) to bounded domains with zero boundary conditions, establishing existence, convergence, and computationally advantageous reformulations, thus broadening their theoretical and practical applicability.

Contribution

It provides a rigorous formulation of SHNSE on general bounded domains with zero boundary conditions and proves foundational results including existence and convergence to NSE solutions.

Findings

Established existence of globally regular solutions.

Proved uniform convergence of SHNSE to NSE as parameters vary.

Reformulated SHNSE for better computational and boundary condition handling.

Abstract

We develop a mathematically and physically sound definition of the spectrally-hyperviscous Navier-Stokes equations (SHNSE) on general bounded domains \Omega with zero (no-slip) boundary conditions prescribed on \varGamma=\partial\varOmega. Previous successful studies of the SHNSE have been limited to periodic-box domains. There are significant theoretical obstacles to overcome in extending the SHNSE beyond this case. We solve them with the help of the Helmholtz decomposition, and with our new formulation of the SHNSE on general bounded domains in hand we then establish foundational results, beginning with the existence of globally regular solutions. Given that the SHNSE is meant to approximate the NSE for small \mu or large m, we establish this rigorously in the general case by adapting the weak subsequence convergence results of [5] to hold here. On intervals [0,T] with a common H^{1}-bound we deepen this sense of approximation by obtaining strong convergence. First, by using estimates depending only on the common H^{1}-bound to maximize computational applicability we show that SHNSE solutions converge uniformly in H^{1} to the NSE solution as either \mu\rightarrow0 or m\rightarrow\infty. Then in cases in which bootstrapped higher-order bounds can be readily used we show that higher-order convergence results hold. Our final results use the Stokes-pressure methodology developed in [29], [30] to recast the SHNSE in a form which like the NSE reformulation in [29], [30] is more adaptable to computation and the specification of boundary values for the pressure.