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The first part of the book is devoted to the transport equation for a given vector field, exploiting the lagrangian structure of solutions. It also treats the regularity of solutions of some degenerate elliptic equations, which appear in the eulerian counterpart of some transport models with congestion. The second part of the book deals with the lagrangian structure of solutions of the Vlasov-Poisson system, which describes the evolution of a system of particles under the self-induced gravitational/electrostatic field, and the existence of solutions of the semigeostrophic system, used in meteorology to describe the motion of large-scale oceanic/atmospheric flows.

Mathematics. --- Partial differential equations. --- Calculus of variations. --- Geophysics. --- Partial Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Geophysics and Environmental Physics. --- Vector fields. --- Elliptic functions. --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Direction fields (Mathematics) --- Fields, Direction (Mathematics) --- Fields, Slope (Mathematics) --- Fields, Vector --- Slope fields (Mathematics) --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Vector analysis --- Differential equations, partial. --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Partial differential equations --- Isoperimetrical problems --- Variations, Calculus of --- Geological physics --- Terrestrial physics --- Earth sciences --- Physics