We study the well-posedness and regularity of the generalized Navier-Stokes equations with initial data in a new critical space Q β,−1 α;∞ (R n ) = ∇ · (Q β α (R n )) n , β ∈ ( 1 2 , 1), which is larger than some known critical homogeneous Besov spaces. Here Q β α (R n ) is a space defined as the set of all measurable functions with sup l(I )where the supremum is taken over all cubes I with edge length l(I ) and edges parallel to the coordinate axes in R n . In order to study the well-posedness and regularity, we give a Carleson measure characterization of Q β α (R n ) by investigating a new type of tent spaces and an atomic decomposition of the predual for Q β α (R n ). In addition, our regularity results apply to the incompressible Navier-Stokes equations with initial data in Q 1,−1 α;∞ (R n ).
Keywords:Fractional Laplacian Gagliardo-Nirenberg inequality Sobolev inequality Logarithmic Sobolev inequality Hardy inequality a b s t r a c tIn this paper, we establish the Gagliardo-Nirenberg inequality under Lorentz norms for fractional Laplacian. Based on special cases of this inequality under Lebesgue norms, we prove the L p -logarithmic Gagliardo-Nirenberg and Sobolev inequalities. Motivated by the L 2 -logarithmic Sobolev inequality, we obtain a fractional logarithmic Sobolev trace inequality in terms of the restriction τ k u of u from R n to R n−k . Finally, we prove the fractional Hardy inequality under Lorentz norms.
In this brief note we study the n-dimensional magnetohydrodynamic equations with hyper-viscosity and zero resistivity. We prove global regularity of solutions when the hyper-viscosity is sufficiently strong.
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