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The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. (English) Zbl 1055.35087
The authors give critical Sobolev inequalities in Besov spaces with logarithmic form similar to those of Brezis-Gallouet-Wainger and Beale-Kato-Majda. Applications to the regularity problem under the critical condition to the Navier-Stokes equations, the Euler equations and the gradient flow to the harmonic map to the sphere are discussed.

MSC:
35Q30 Navier-Stokes equations
35Q05 Euler-Poisson-Darboux equations
35L60 First-order nonlinear hyperbolic equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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