Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics

We give lower bounds for the lifespan of a solution to the inviscid Boussinesq system. In dimension two, we point out that it tends to infinity when the initial (relative) temperature tends to zero. This is, to the best of our knowledge, the first result of this kind for the inviscid Boussinesq system. In passing, we provide continuation criteria (of independent interest) in the N -dimensional case. In the second part of the paper, our method is adapted to handle the axisymmetric incompressible Euler equations with swirl. Introduction The evolution of the velocity u = u(t, x) and pressure P = P (t, x) fields of a perfect homogeneous incompressible fluid is governed by the following Euler equations: (0.1) { ∂tu+ u · ∇u+∇P = 0, div u = 0. There is a huge literature concerning the well-posedness issue for Euler equations. Roughly, they may be solved locally in time in any reasonable Banach space embedded in the set C0,1 of bounded Lipschitz functions (see e.g. [1, 4, 6, 12, 13, 17, 19, 22]). In the two-dimensional case, it is well known that Euler equations are globally well-posed for sufficiently smooth initial data. This noticeable fact relies on the conservation of the vorticity ω := ∂1u 2 − ∂2u 1 along the flow of the velocity field, and has been first proved rigorously in the pioneering works by W. Wolibner [20] and V. Yudovich [21]. This conservation property is no longer true, however, in more physically relevant contexts such as (1) the three-dimensional setting for (0.1), (2) nonhomogeneous incompressible perfect fluids, (3) inviscid fluids subjected to a buoyancy force which is advected by the velocity fluid (the so-called inviscid Boussinesq system below). As a consequence, the problem of global existence for general (even smooth or small) data is still open for the above three cases. In a recent work [9], it has been shown that for slightly nonhomogeneous two-dimensional incompressible fluids, the lifespan tends to infinity when the nonhomogeneity tends to zero. The present paper is mainly dedicated to the study of the lifespan for the first and third item. More precisely, in the first section of the paper, we shall consider the inviscid Boussinesq system: (0.2)    ∂tθ + u · ∇θ = 0, ∂tu+ u · ∇u+∇P = θeN , div u = 0. Here the relative temperature θ = θ(t, x) is a real valued function and eN stands for the unit vertical vector. 2010 Mathematics Subject Classification. 35Q35,76B03. 1It need not be nonnegative as it designates the discrepancy to some reference temperature. 1