A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations

2010/11/23 by Zhen Lei, Qi S. Zhang, Lei, Zhen +1
Mathematics · Engineering · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1011.5066

Abstract

Let v(x, t)= v^r e_r + v^θe_θ+ v^z e_z be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by b = v^r e_r + v^z e_z the radial-axial vector field. Under a general scaling invariant condition on b, we prove that the quantity Γ= r v^θ is Hölder continuous at r = 0, t = 0. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \citeKNSS and Seregin-Sverak in \citeSS. As another application, we prove that if b ∈ L^∞([0, T], BMO^-1), then v is regular. This provides an answer to an open question raised by Koch and Tataru in \citeKochTataru about the uniqueness and regularity of Navier-Stokes equations in the axially-symmetric case.

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