Quasi-geostrophic equations with initial data in Banach spaces of local measures

Abstract

This paper studies the well posedness of the initial value problem for the quasi-geostrophic type equations ∂tθ + u∇θ + (-∆)γ θ = 0 on ℝd x]0, +∞[ θ(x, 0) = θ0(x), x ∈ ℝd where 0 < γ ≤ 1 is a fixed parameter and the velocity field u = (u1, u2,...,ud is divergence free; i.e., ∇u = 0). The initial data θ0 is taken in Banach spaces of local measures (see text for the definition), such as Multipliers, Lorentz and Morrey-Campanato spaces. We will focus on the subcritical case 1/2 < γ ≤ 1 and we analyse the well-posedness of the system in three basic spaces: Ld/r,∞, Ẋr and Mp,d/r, when the solution is global for sufficiently small initial data. Furthermore, we prove that the solution is actually smooth. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions.