| We study the Navier-Stokes equations on half spaces with initial data in the nonhomogeneous Morrey spaces. We use wavelets satisfying some particular conditions to study so-called mild solutions of these equations when they satisfy the no-slip boundary condition. We extend the ideas of Cannone and Meyer in [Can] by using wavelet projections instead of Littlewood-Paley projections. This allows us to work on domains where Fourier based techniques could not be used. We give characterizations of Morrey spaces in terms of coefficients of wavelets that have compact support and zero average values on half spaces. Moreover, we define what it means for function norms to be adapted to bilinear operators when those operators are expressed in terms of their matrices in the wavelet basis and the norms are expressed in wavelet coordinates as well. We prove estimates for multiresolution projections of pointwise products of functions in the nonhomogeneous Morrey spaces analogous to those of Cannone and Meyer in [Can]. Also, we show that if a linear operator maps wavelets with specific properties into corresponding vaguelets, then it will be bounded on the homogeneous and nonhomogeneous Morrey spaces. Using these techniques, we can show the same results obtained by Federbush in [Fe]. Finally, with the exception of a term involving some commutator of two operators, we have obtained all the estimates needed for applying Picard's iteration method to get mild solutions for the Navier-Stokes equations in the nonhomogeneous Morrey spaces on half spaces. |
