We introduce a notion of super-potential for positive closed currents of bidegree (p, p) on projective spaces. This gives a calculus on positive closed currents of arbitrary bidegree. We define in particular the intersection of such currents and the pull-back operator by meromorphic maps. One of the main tools is the introduction of structural discs in the space of positive closed currents which gives a “geometry” on that space. We apply the theory of super-potentials to construct Green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms.

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in Rⁿ × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].

L’objet de cet article est de répondre à une question posée par Gelfond en 1968 en montrant que la somme des chiffres des carrés écrits en base q ⩾ 2 est équirépartie dans les progressions arithmétiques.

In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of squares written in a basis q ⩾ 2 is equidistributed in arithmetic progressions.