Quasi-Geostrophic Type Equations with Weak Initial Data
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We study the initial value problem for the quasi-geostrophic type equations ∂θ / ∂t + u • ∇θ + (-Δ)(λ)θ = 0, on ℝ(n) x (0, ∞), θ(x,0) = θ(0)(x), x ∈ ℝ(n), where λ(0 ≤ λ ≤ 1) is a fixed parameter and u = (u(j)) is divergence free and determined from θ through the Riesz transform u(j) = ±R(π)(j)θ, with π(j) a permutation of 1,2, •••, n. The initial data θ(0) is taken in the Sobolev space Ĺ(r),(p) with negative indices. We prove local well-posedness when 1/2 < λ ≤ 1, 1 < p < ∞, n/p ≤ 2λ - 1, r = n/p - (2λ - 1) ≤ 0. We also prove that the solution is global if θ(0) is sufficiently small.