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The Exact Limits and Improved Decay Estimates for All Order Derivatives of the Global Weak Solutions to a Two-Dimensional Incompressible Dissipative Quasi-Geostrophic Equation
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KeyWord:Incompressible dissipative quasi-geostrophic equation  All order derivatives of global weak solution  Primary decay estimates  exact limits  Improved decay estimates with sharp rates
Author NameAffiliation
Linghai Zhang Department of Mathematics, Lehigh University, 17 Memorial Drive East, Bethlehem, PA USA 18015 
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      We will accomplish the exact limits for all order derivatives of the global weak solutions to a two-dimensional incompressible dissipative quasi-geostrophic equation. We will also establish the improved decay estimates with sharp rates for all order derivatives. We will consider two cases for the initial function and the external force and prove the optimal results for both cases. We will couple together existing ideas (including the Fourier transformation and its properties, Parseval's identity, iteration technique, Lebesgue's dominated convergence theorem, Gagliardo-Nirenberg-Sobolev interpolation inequality, squeeze theorem, Cauchy-Schwartz's inequality, etc) existing results (the existence of global weak solutions, the existence of local smooth solution on $(T,\infty)$ and the elementary decay estimate with a sharp rate) and a few novel ideas to obtain the main results.