Assuming some initial data u 0 ∈Ḣ 1/2 (R 3 ) lead to a singularity for the 3d Navier-Stokes equations, we show that there are also initial data with the minimalḢ 1/2 -norm which will produce a singularity.

ABSTRACT. We address the well-posedness theory for the magento-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (−∆) γ with 0 < γ < 1, we discover that for γ > 1/2 the equations are locally well-posed, while for γ < 1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when γ goes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar is given by an unbounded Fourier multiplier which is both even and anisotropic. Lastly, we note that the anisotropy of the constitutive law for the velocity may be explored in order to obtain an improvement in the regularity of the solutions when the initial data and the force have thin Fourier support, i.e. they are supported on a plane in frequency space. In particular, for such well-prepared data one may prove the local existence and uniqueness of solutions for all values of γ ∈ (0, 1).

We address the well-posedness of the primitive equations of the ocean with continuous initial data. We show that the splitting of the initial data into a regular finite energy part and a small bounded part is preserved by the equations, thus leading to existence and uniqueness of solutions.

Abstract. We address the global solvability of the Kuramoto-Sivashinsky equation in a rect-We give sufficient conditions on the width L 2 of the domain, depending on the length L 1 , so that the obtained solutions are global. Our proofs are based on anisotropic estimates.

In this paper, we consider the Cauchy problem for a nonlinear parabolic system u t − Δu + u · ∇uWe analyze the convergence of its solutions to a solution of the incompressible Navier-Stokes system as → 0.Mathematics Subject Classification (2010). Primary 35Q30; Secondary 76D05.