A factorization theorem for operators occurring in the Stokes, Brinkman and Oseen equations

In many physical problems one is faced with solving partial differential equations of the form L1(L1+L2)u=0, where L1 and L2 are linear operators. It is found in many cases that the solution u is of the form u1+u2 where L1u1=0 and (L1+L2)u2=0. In this paper we present sufficient conditions under which such a splitting is possible. Moreover, we give explicit formulae for u1 and u2 for a given u. We also show in some examples where the operators satisfy the sufficient conditions and such a splitting is used extensively. In particular, we find a class of solutions for the unsteady Brinkman and unsteady Oseen equations using the splitting that we propose.