We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator for our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral operators in which this correspondence seems to take place. Also, motivated by Mulase's work on the KP hierarchy, we prove a group factorization theorem for our group of Fourier integral operators.

An exact solution of the nonlinear governing equations in the βplane approximation is established and analysed in this paper.Such solution describes a purely azimuthal equatorial two-layer flows with free surface and discontinuous stratification.We present the explicit solution for the velocity field and the pressure.Moreover, We derive the implicit formulas for the shape of the free surface and the interface.Finally, the monotonicity properties between the free surface and the pressure on the free surface, as well as the regularity result for the interface are proved, respectively.

In this paper we study the inhomogeneous incompressible Euler equations in the whole space R n with n ≥ 3. We obtain well-posedness and blow-up results in a new framework for inhomogeneous fluids, more precisely Besov-Herz spaces that are Besov spaces based on Herz ones, covering particularly critical cases of the regularity.Comparing with previous works on Besov spaces, our results provide a larger initial data class for a well-defined flow.For that, we need to obtain suitable linear estimates for some conservation-law models in our setting such as transport equations and the linearized inhomogeneous Euler system.