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Abstract

This paper deals with the $L_{p}$ - $L_{q}$ decay estimate of the $C^{0}$ analytic semigroup ${T (t)}_{t ⩾ 0}$ associated with the perturbed Stokes equations with free boundary conditions in an exterior domain. The problem arises in the study of free boundary problem for the Navier–Stokes equations in an exterior domain. We proved that $\begin{matrix} {‖ \nabla^{j} T (t) f ‖}_{L_{p}} ⩽ C_{p, q} t^{- \frac{j}{2} - \frac{N}{2} (\frac{1}{q} - \frac{1}{p})} ‖ f ‖_{L_{q}} (j = 0, 1) \end{matrix}$ provided that $1 < q ⩽ p ⩽ \infty$ and $q \neq \infty$ . Compared with the non-slip boundary condition case, the gradient estimate is better, which is important for the application to proving global well-posedness of free boundary problem for the Navier–Stokes equations. In our proof, it is crucial to prove the uniform estimate of the resolvent operator, the resolvent parameter ranging near zero.

Keywords

Exterior domains Stokes equations free boundary problem without surface tension

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On the L p - L q decay estimate for the Stokes equations with free boundary conditions in an exterior domain

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