On the regularity of a class of generalized quasi-geostrophic equations

By Changxing Miao, Liutang Xue. 2010

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Abstract In this article we consider the following generalized quasi-geostrophic equation ∂ t θ + u ⋅ ∇ θ + ν Λ β θ = 0 , u = Λ α R ⊥ θ , x ∈ R 2 , where ν > 0 , Λ : = − Δ , α ∈ ] 0 , 1 [ and β ∈ ] 0 , 2 [ . We first show a general conditional criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case α ∈ ] 0 , 1 [ and β ∈ ] α + 1 , 2 ] we obtain the global well-posedness of the system with smooth initial data; and for the case α ∈ ] 0 , 1 [ and β ∈ ] 2 α , α + 1 ] we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5c5b8ad94ec62c4826b203c868a19fdf9533d056