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We address the regularity for the 2D Boussinesq equations with zero diffusivity and positive viscosity on an open bounded smooth domain Ω. In particular, we prove that for the initial velocity in H2(Ω) and the initial density in H1(Ω) there exists a global in time solution for which the velocity and the density are locally in time bounded functions in H2 and H1, respectively.

We address a fluid-structure interaction model describing the motion of an elastic body immersed in an incompressible fluid. We establish a priori estimates for the local existence of solutions for a class of initial data which also guarantees uniqueness.

Lions, Temam, and Wang in [“Probleme a frontiere libre pour les modeles couples de l'ocean et de l'atmosphere,” Acad. Sci., Paris, C. R. 318(12), 1165–1171 (1994)] introduced a free surface model for the primitive equations of the ocean. In this paper, we establish the local well-posedness of the model with analytic initial data.

We consider sufficient conditions for the regularity of Leray-Hopf solutions of the Navier-Stokes equations. We prove that if the third derivative of the velocity ∂u∕∂x3 belongs to the space Lts0Lxr0, where 2∕s0+3∕r0⩽2 and 9∕4⩽r0⩽3, then the solution is regular. This extends a result of Beirao da Veiga [Chin. Ann. Math., Ser. B 16, 407–412 (1995); C. R. Acad. Sci, Ser. I: Math. 321, 405–408 (1995)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative ∂u∕∂x3 can be substituted with any directional derivative of u.