Stability of certain geophysical structures in fluid dynamics via dispersive analysisdissertationsKo, HaramcreatorPausader, BenoitAdvisorGomez-Serrano, JavierReaderGuo, YanReaderBrown University. Department of Mathematicssponsor2026187, 177 p.born digitalThesis (Ph. D.)--Brown University, 2026thesesThis thesis establishes the stability of rotation in the 3D compressible Euler equation and incompressible Navier-Stokes equation, and the stability of linear stratification in the 2D Boussinesq equation. Using the common factor that these steady states induce dispersion, I establish a systematic method for linear analysis. This is explained through two different systems to highlight that the method is versatile enough to work for different dispersion relations in different systems. Combined with the energy estimates, this produces long-time stabilities of the respective steady states. Finally, I prove the global stability for one of these systems using a mix of paradifferential calculus, the spacetime resonance method, and the aforementioned linear theory.Differential equations, PartialEnglishRPB20260516