On the L_p-L_q decay estimate for the Stokes equations with free boundary conditions in an exterior domain

This paper deals with the L_p-L_q decay estimate of the C⁰ 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 ||δ^jT(t)f||_Lp ≤ C_p,q^t - j/2 - N/2(1/q - 1/p) ||f||_Lq (j = 0,1) provided that 1 < q ≤ p ≤ ∞ and q ≠ ∞. 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.