Quasi-geostrophic equations with initial data in Banach spaces of local measures
Texas State University-San Marcos, Department of Mathematics
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.
Quasi-geostrophic equation, Local spaces, Mild solutions, Self-similar solutions
Gala, S. (2005). Quasi-geostrophic equations with initial data in Banach spaces of local measures. <i>Electronic Journal of Differential Equations, 2005</i>(63), pp. 1-22.