Wu, Jiahong2019-03-252019-03-251998-06-12Wu, J. (1998). Quasi-geostrophic type equations with weak initial data. <i>Electronic Journal of Differential Equations, 1998</i>(16), pp. 1-10.1072-6691https://hdl.handle.net/10877/7949We study the initial value problem for the quasi-geostrophic type equations ∂θ/∂t + u · ∇θ + (-Δ)λθ = 0, on ℝn x (0, ∞), θ(x, 0) = θ0(x), x ∈ ℝn, where λ(0 ≤ λ ≤ 1) is a fixed parameter and u = (uj) is divergence free and determined from θ through the Riesz transform uj = ±Rπ(j)θ, with π(j) a permutation of 1,2, ···, n. The initial data θ0 is taken in the Sobolev space Ĺr,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 θ₀ is sufficiently small.Text10 pages1 file (.pdf)enAttribution 4.0 InternationalQuasi-geostrophic equationsWeak dataWell-posednessArticle