Well-Posedness and Long-Time Dynamics

This book focuses on the initial boundary value problem to the 3D quasigeostrophic and rotating Boussinesq equations. The equations are derived from conservation laws in continuum physics followed by the formulation of the problem as initial value problem on Hilbert spaces. By using Faedo-Galerkin approximation and semigroup technique, existence and uniqueness of solutions are proved. Additionally, the book outlines how Lyapunov functions can be used to assess energy stability criteria. The author also addresses singular problems for which the equation has parabolic structure (rotating Boussinesq equations for the atmosphere and ocean) and the singular limit is hyperbolic (quasigeostrophic equations for the atmosphere and ocean) in the asymptotic limit of small Rossby number. In particular, this approach gives as a corollary a constructive proof of the well-posedness of the problem of quasigeostrophic potential vorticity equations governing modons.