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On the initial value problem for the Navier-Stokes equations with the initial datum in critical Sobolev and Besov spaces. (English) Zbl 1342.35220
Summary: The existence of local unique mild solutions to the Navier-Stokes equations in the whole space with an initial tempered distribution datum in critical homogeneous or inhomogeneous Sobolev spaces is shown. Especially, the case when the integral-exponent is less than 2 is investigated. The global existence is also obtained for the initial datum in critical homogeneous Sobolev spaces with a norm small enough in suitable critical Besov spaces. The key lemma is to establish the bilinear estimates in these spaces, due to the point-wise decay of the kernel of the heat semigroup.

MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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