On the Regularity of the Entropy Solution of the Fractional-Semigeostrophic Equation Using Ultrafilters

This paper explores the regularity properties of the entropy solution to the fractional semigeostrophic (FSG) equation, leveraging the advanced mathematical framework of ultrafilters. The FSG equation, which extends the classical semigeostrophic model by incorporating fractional derivatives, is critical in modeling geophysical fluid dynamics with more accuracy, particularly in scenarios involving anomalous diffusion. Traditional methods to study entropy solutions often face limitations due to the complex nature of fractional derivatives and the nonlinearity inherent in the FSG equation. By employing ultrafilters, a powerful tool in non-standard analysis, we overcome these challenges and provide new insights into the regularity of entropy solutions. Our results demonstrate that the use of ultrafilters not only simplifies the analysis but also enhances the understanding of the smoothness and stability of the entropy solutions. This work lays the groundwork for future studies on fractional partial differential equations in geophysical contexts, potentially leading to more accurate predictive models in meteorology and oceanography.