Global Existence for the Critical Dissipative Surface Quasi-Geostrophic Equation |
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Authors: | Omar Lazar |
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Institution: | 1. Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-77454, Marne-la-Vallée, France
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Abstract: | In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in ${\mathbb{R}^2}$ R 2 . Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data θ 0 liying in the space ${\Lambda^{s} (\dot{H}^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}$ Λ s ( H ˙ u l o c s ( R 2 ) ) ∩ L ∞ ( R 2 ) the critical (SQG) has a global weak solution in time for 1/2 < s < 1. Our proof is based on an energy inequality verified by the equation ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? which is nothing but the (SQG) equation with truncated and regularized initial data. By classical compactness arguments, we show that we are able to pass to the limit ( ${R \rightarrow \infty}$ R → ∞ , ${\epsilon \rightarrow 0}$ ? → 0 ) in ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? and that the limit solution has the desired regularity. |
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