Abstract
We study solutions of the initial value problem for the 2D regularized surface quasi-geostrophic (RSQG) equation. For Ḣ 1 ( Ω ) ∩ L q ( Ω ) ( q < 2 < ∞ ) Open image in new window initial data, we prove the global existence and uniqueness of weak solution for RSQG equation with subcritical powers. For RSQG equation, we establish some regularization results and prove the inviscid limit of the RSQG equation to the classical quasi-geostrophic equation.
Highlights
The quasi-geostrophic equation (QG) with periodic boundary conditions on a basic period box Ω = [0, 2π]2 ⊂ R2 is ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ θt + (− div 1 )2 = φ = θ, ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩∇⊥φ = θ dx θ (x, 0) u, = 0, = θ0 (x) φdx . =
The quasigeostrophic equation with dissipative term (-Δ)aθ has received an extensive amount of attentions and has many results in theory and numerical analysis, but there are few results on the surface quasi-geostrophic equation with regularized term (-Δ)aθt
In [1], Wu reformulates the problem as an integral equation and applies the Banach contraction mapping principle to prove local existence with initial value θ0 ÎHs(s > 1)
Summary
The quasi-geostrophic equation (QG) with periodic boundary conditions on a basic period box Ω = [0, 2π]2 ⊂ R2 is. Where θ(x, t) is a real-valued function of x and t, which represents the potential temperature, and u represents the incompressible horizontal velocity at the surface. The advective velocity u in these equations is determined from θ by a stream function via the auxiliary relations. Where 0 ≤ a ≤ 1 and > 0 are real numbers. The quasigeostrophic equation with dissipative term (-Δ)aθ has received an extensive amount of attentions and has many results in theory and numerical analysis (see e.g., [2,3,4,5,6,7] for further references), but there are few results on the surface quasi-geostrophic equation with regularized term (-Δ)aθt. The key issue is still whether weak solutions are regular for all the time
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