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)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.
CitationWu, J. (1998). Quasi-geostrophic type equations with weak initial data. Electronic Journal of Differential Equations, 1998(16), pp. 1-10.
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