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We consider a Keller-Segel model coupled to the incompressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criteria for both cases that equations of oxygen concentration is of parabolic or hyperbolic type. We also prove that solutions exist globally in time and upper bounds of temporal decays are obtained under the some smallness conditions of initial data.

We prove sharp interpolatory estimates between Riesz Transforms and directional Haar projections. We obtain applications to the theory of compensated compactness and prove a conjecture of L. Tartar on semi-continuity of separately convex integrands.

We obtain the interior regularity criteria for the vorticity of “suitable” weak solutions to the Navier–Stokes equations. We prove that if two components of a vorticiy belongs to in a neighborhood of an interior point with 3/p + 2/q ≤ 2 and 3/2

We study the singular limit of viscous polytropic fluids without thermal conductivity as the Mach number tends to zero. A uniform existence result for the Cauchy problem in R 3 is proved under the assumption that the initial data belongs uniformly to H k (R 3) with k = 2, 3 and is well-prepared in H 1 (R 3).