Strong solutions of the Navier-Stokes equation in Morrey spaces

Abstract

It is shown that the nonstationary Navier-Stokes equation (NS) in ℝ⁺×ℝ^m is well posed in certain Morrey spacesM _p,λ (ℝ⁺×ℝ^m) (see the text for the definition: in particularM _p,0=L^p ifp>1 andM _1,0 is the space of finite measures), in the following sense. Given a vectora∈M _p,m-p with diva=0 and with certain supplementary conditions, there is a unique local (in time) solution (velocity field)u(t, ·)∈M _{p, m-p}, which is smooth fort>0 and takes the initial valuea at least in a weak sense.u is a global solution ifa is sufficiently small. Of particular interest is the spaceM _1,m−1, which admits certain measures; thusa may be a surface measure on a smooth (m−1)- dimensional surface in ℝ⁺×ℝ^m. The regularity of solutions and the decay of global solutions are also considered. The associated vorticity equation (for the vorticity ζ=∂∧u) can similarly be solved in (tensor-valued)M _1,m−2, which is also a space of measures of another kind.