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Abstract An important feature of the planetary oceanic dynamics is that the aspect ratio (the ratio of the depth to horizontal width) is very small. As a result, the hydrostatic approximation (balance), derived by performing the formal small aspect ratio limit to the Navier–Stokes equations, is considered as a fundamental component in the primitive equations of the large-scale ocean. In this paper, we justify rigorously the small aspect ratio limit of the Navier–Stokes equations to the primitive equations. Specifically, we prove that the Navier–Stokes equations, after being scaled appropriately by the small aspect ratio parameter of the physical domain, converge strongly to the primitive equations, globally and uniformly in time, and the convergence rate is of the same order as the aspect ratio parameter. This result validates the hydrostatic approximation for the large-scale oceanic dynamics. Notably, only the weak convergence of this small aspect ratio limit was rigorously justified before. [ABSTRACT FROM AUTHOR]

Abstract: We study global well-posedness and regularity of solutions for a family of incompressible three-dimensional Navier–Stokes-alpha-like models that employ fractional Laplacian operators. This family of equations depends on two parameters, and , which affect the strength of non-linearity (vorticity stretching) and the degree of viscous smoothing. Varying and interpolates between the incompressible Navier–Stokes equations and the incompressible (Lagrangian averaged) Navier–Stokes- model. Our main result, which contains previously established results of J.L. Lions and others, provides a relationship between and that is sufficient to guarantee global existence, uniqueness and regularity of solutions. [Copyright &y& Elsevier]

We show that the global attractor of a weakly damped and driven Korteweg–de Vries equation (KdV) is embedded in the long-time dynamics of an ordinary differential equation called a determining form. In particular, there is a one-to-one identification of the trajectories in the global attractor of the damped and driven KdV and the steady state solutions of the determining form. Moreover, we analyze a data assimilation algorithm (down-scaling) for the weakly damped and driven KdV. We show that given a certain number of low Fourier modes of a reference solution of the KdV equation, the algorithm recovers the full reference solution at an exponential rate in time. [ABSTRACT FROM AUTHOR]