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A note on the Navier-Stokes IBVP with small data in $L^n$

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  • We study existence and uniqueness of regular solutions to the Navier-Stokes initial boundary value problem in bounded or exterior domains $\Omega$ ($\partial\Omega$ sufficiently smooth) under the assumption $v_\circ$ in $L^n(\Omega)$, sufficiently small, and we prove global in time existence. The results are known in literature (see Remark 3), however the proof proposed here seems shorter, and we give a result concerning the behavior in time of the $L^q$-norm ($q\in[n,\infty]$) of the solutions and of the $L^n$-norm of the time derivative, with a sort of continuous dependence on the data, which, as far as we know, are new, and are close to the ones of the solution to the Stokes problem. Moreover, the constant for the $L^q$-estimate is independent of $q$.
    Mathematics Subject Classification: 35Q30, 76D05, 76N10.


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