Geodesic
Reduction of basic initial-boundary value problems for the Navier--Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudodifferential equations
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 20, Tome 171 (1989), pp. 36-52

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We consider initial-boundary value problems for the Navier—Stokes equations prescribing velocities, stresses, or normal component of the velocity and tangential stresses on the boundary. We show that they can be reduced to initial boundary value problems for systems of the form $v_t+Av+Kv=f$ where $A$ is a linear elliptic operator containing a non-local term and $K$ is a nonlinear operator. For these problems we prove a local existence theorem in Sobolev–Slobodetski spaces $W_2^{l,l/2}$. 
@article{ZNSL_1989_171_a2,
     author = {Gerd Grubb and V. A. Solonnikov},
     title = {Reduction of basic initial-boundary value problems for the {Navier--Stokes} equations to initial-boundary value problems for nonlinear parabolic systems of pseudodifferential equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {36--52},
     publisher = {mathdoc},
     volume = {171},
     year = {1989},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1989_171_a2/}
}
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Gerd Grubb; V. A. Solonnikov. Reduction of basic initial-boundary value problems for the Navier--Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudodifferential equations. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 20, Tome 171 (1989), pp. 36-52. http://geodesic.mathdoc.fr/item/ZNSL_1989_171_a2/