In most cases, the global analysis for linearized equations is used as the foundation for the nonlinear evolution partial differential equations analysis. In order to achieve this goal, one makes an effort to create and investigate global solutions for relevant linear equations. In general, solutions to different types of partial differential equations lead to representations of the solutions in a variety of distinct forms. Partial elliptic differential equations, for instance, result in parametrices that take the form of pseudo-differential operators. On the other hand, it is possible to create propagators for hyperbolic equations in the form of Fourier integral operators. Other equations, such as the Schrödinger equation or the linearized Korteweg-de Vries equation, lead to oscillatory integrals that are more generic in nature. The solutions of Schrödinger equations may be shown in the form of Legendrian oscillatory integrals, but the oscillatory integrals that are generated by more generic evolution partial differential equations are of a more broad nature. Methods of representing solutions, calculus of solution operators and propagators, global weighted L2 and other estimates, spectral qualities, functional analytic properties, and so on are often included as components of the needed analysis.