This paper establishes the existence and uniqueness, and also presents a blow-up criterion, for solutions of the quasi-geostrophic (QG) equation in a framework of Fourier type, specifically Fourier-Besov-Morey spaces. If it is assumed that the initial data 0 is small and belonging to the critical Fourier-Besov-Morrey spaces, we get the global well-posedness results of the QG equation (1). Moreover, we prove that there exists a time T > 0 such that the QG equation ( 1) admits a unique local solution for large initial data.

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In this paper, we study the analyticity of mild solutions to the Debye-Huckel system with small initial data in critical Fourier-Besov-Morrey spaces. Specifically, by using the Fourier localization argument, the Littlewood-Paley theory and bilinear-type fixed point theory, we prove that global-in-time mild solutions are Gevrey regular. As a consequence of analyticity, we get time decay of mild solutions in Fourier-BesovMorrey spaces. Finally, we show a blow-up criterion of the local-in-time mild solutions of the Debye-Huckel system.

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