The local exact controllability of the steady-state solutions for the magnetohydrodynamic equations was established by the authors in a previous paper. However, it was subsequently found that an argument given there was relying on an uncertain estimate. The purpose of this paper is to correct that argument.

Various nonhomogeneous variants of the div-curl lemma are proved, in which the local Hardy space of Goldberg is the target space. This is done both in the whole Euclidean space and in domains.

This note deals with the solvability of a class of quasivariational evolution problems in Hilbert spaces. In particular, we address the so-called quasivariational sweeping process and prove suitable well-posedness results by means of an order method.

In this paper we are concerned with a system of second-order differential equations of the form $x''+A(t,x)x=0$, $t\in [0,\pi]$, $x\in {{\bf R}}^N$, where $A(t,x)$ is a symmetric $N\times N$ matrix. We concentrate on an asymptotically linear situation and we prove the existence of multiple solutions to the Dirichlet problem associated to the system. Multiplicity is obtained by a comparison between the number of moments of verticality of the matrices $A_0(t)$ and $A_\infty(t)$, which are the uniform limits of $A(t,x)$ for $|x|\to 0$ and $|x|\to +\infty$, respectively. For the proof, which is based on a generalized shooting approach, we provide a theorem on the existence of zeros of a class of $N$-dimensional vector fields.

In this article we consider the Primitive Equations without horizontal viscosity but with a mild vertical viscosity added in the hydrostatic equation, as in [13] and [16], which are the so-called $\delta-$Primitive Equations. We prove that the problem is well posed in the sense of Hadamard in certain types of spaces. This means that we prove the finite-in-time existence, uniqueness and continuous dependence on data for appropriate solutions. The results given in the 3D periodic space easily extend to dimension 2. We also consider a Boussinesq type of equation, meaning that the mild vertical viscosity present in the hydrostatic equation is replaced by the time derivative of the vertical velocity. We prove the same type of results as for the $\delta-$Primitive Equations; periodic boundary conditions are similarly considered.