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October  2010, 26(4): 1471-1490. doi: 10.3934/dcds.2010.26.1471

On the Lipschitzness of the solution map for the 2 D Navier-Stokes system

Atanas Stefanov 1,

1. 

Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7523

Received  November 2008 Revised  May 2009 Published  December 2009

We consider the Navier-Stokes system on R2. It is well-known that solutions with $L^2$ data become instantly smooth and persist globally. In this note, we show that the solution map is Lipschitz, when acting in $L^\infty $Hσ (R2) and $L^2_t$Hσ+1 (R2), when $0\leq $ σ<1. This generalizes an earlier result of Gallagher and Planchon [7], who showed the Lipschitzness in $L^2$(R2). The question for the Lipschitzness of the map in Hσ (R2), σ$\geq 1$ remains an interesting open problem, which hinges upon the validity of an endpoint estimate for the trilinear form $(\phi, v, w)\to \int$R2(∂Φ/∂x ∂v/∂y - ∂Φ/∂y ∂v/∂x)wdx.
Keywords: $L^2$ solution maps, 2 D Navier-Stokes equation, stability of solutions..
Mathematics Subject Classification: Primary: 35Q30, 76D03, 76D0.
Citation: Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471
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