Quasi-geostrophic Type Equations with Weak Initial Data
نویسنده
چکیده
We study the initial value problem for the quasi-geostrophic type equations ∂θ ∂t + u · ∇θ + (−∆)θ = 0, on R × (0,∞), θ(x, 0) = θ0(x), x ∈ R n , 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 L̇r,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.
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