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The existence of local unique mild solutions to the Navier-Stokes equationsin the whole space with an initial tempered distribution datum in criticalhomogeneous or inhomogeneous Sobolev spaces is shown. Especially, the case whenthe integral-exponent is less than 2 is investigated. The global existence isalso obtained for the initial datum in critical homogeneous Sobolev spaces witha norm small enough in suitable critical Besov spaces. The key lemma is toestablish the bilinear estimates in these spaces, due to the point-wise decayof the kernel of the heat semigroup.

On the initial value problem for the Navier-Stokes equations with the initial datum in critical Sobolev and Besov spaces