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Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. (English) Zbl 1106.35061

Summary: We study the two-dimensional dissipative quasi-geostrophic equations

θ t +u·θ=0,θ t +u·θ+κ(-Δ) α θ=0,
u=(u 1 ,u 2 )=- ψ x 2 , ψ x 1 ,(-Δ) 1/2 ψ=-θ,

in the Sobolev space H s ( 2 ). Existence and uniqueness of the solution local in time is proved in H s when s>2(1-α). Existence and uniqueness of the solution global in time is also proved in H s when s2(1-α) and the initial data Λ s θ 0 L 2 is small. For the case, s>2(1-α), we also obtain the unique large global solution in H s provided that θ 0 L 2 is small enough.

MSC:
35Q35PDEs in connection with fluid mechanics
86A05Hydrology, hydrography, oceanography
37L30Attractors and their dimensions, Lyapunov exponents
76D03Existence, uniqueness, and regularity theory
76D05Navier-Stokes equations (fluid dynamics)
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