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 (Journal of Nonlinear Modeling and Analysis, 2024)
We will accomplish the exact limits for all order derivatives of theglobal weak solutions to a two-dimensional incompressible dissipative quasi-geostrophic equation. We will also establish the improved decay estimateswith sharp rates for all order derivatives. We will consider two cases for theinitial function and the external force and prove the optimal results for bothcases. We will couple together existing ideas (including the Fourier transformation and its properties, Parseval’s identity, iteration technique, Lebesgue’sdominated convergence theorem, Gagliardo-Nirenberg-Sobolev interpolationinequality, squeeze theorem, Cauchy-Schwartz’s inequality, etc) existing results (the existence of global weak solutions, the existence of local smoothsolution on $(T, ∞)$ and the elementary decay estimate with a sharp rate) anda few novel ideas to obtain the main results.