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

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  • 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.
    Mathematics Subject Classification: Primary: 35Q30, 76D03, 76D05.


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