Using periodic boundary conditions to approximate the Navier–Stokes equations on R3 and the transfer of regularity. (1st October 2021)

• Record Type: Journal Article
• Authors: Robinson, James C
• Is Part Of: Nonlinearity. Volume 34:Number 11(2021)
• Page Start: 7683
• Page End: 7704
• Publication Date: 2021-10-01
• Subjects: Navier–Stokes equations -- expanding domains -- strong convergence; 35Q30; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac2673
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: This paper considers solutions u α of the three-dimensional Navier–Stokes equations on the periodic domains Q α ≔ (− α, α ) 3 as the domain size α → ∞, and compares them to solutions of the same equations on the whole space. For compactly-supported initial data u α 0 ∈ H 1 ( Q α ), an appropriate extension of u α converges to a solution u of the equations on R 3, strongly in L r ( 0, T ; H 1 ( R 3 ) ), r ∈ [1, ∞). The same also holds when u α 0 is the velocity corresponding to a fixed, compactly-supported vorticity. A consequence is that if an initial compactly-supported velocity u 0 ∈ H 1 ( R 3 ) or an initial compactly-supported vorticity ω 0 ∈ H 1 ( R 3 ) gives rise to a smooth solution on [0, T *] for the equations posed on R 3, a smooth solution will also exist on [0, T *] for the same initial data for the periodic problem posed on Q α for α sufficiently large; this illustrates a 'transfer of regularity' from the whole space to the periodic case.