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We show the existence of strong solutions for the nonhomogeneous Navier-Stokes equations in three-dimensional domains with boundary uniformly of class C 3 . Under suitable assumptions, uniqueness is also proved.

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We derive new results about existence and uniqueness of local and global solutions for the nonlinear Schrödinger equation, including self-similar solutions. Our analysis is performed in the framework of weak-L p spaces.

We consider spectral semi-Galerkin approximations for the strong solutions of the nonhomogeneous Navier-Stokes equations. We derive an optimal uniform in time error bound in the H 1 norm for approximations of the velocity. We also derive an error estimate for approximations of the density in some spaces L r . (2000). 35Q30, 76M22, 65M15, 65M60. Mathematics Subject Classification