On well-posedness of non-linear theories in Physics

RUBIO, M. E; REULA, OSCAR A.

Santiago de Chile

Conferencia; 2nd JNMP: The 2nd Conference on non-linear mathematical Physics; 2019

One of the most challenging purposes of Theoretical Physics is to develop theories that attempt to explain and understand, as accurately as possible, the evolution of physical systems given a certain initial configuration. This aspect is crucial and inevitable in order to guarantee the predictability power of the theory. The theory of partial differential equations provides results and powerful tools to study this problem, which is known as the Cauchy Problem or Initial Value Problem. Although Functional Analysis helps to formalize several aspects associated with the functional spaces in which the existence, uniqueness and continuity of the evolution with respect to the initial data is guaranteed, there is a series of purely algebraic tools that univocally characterize well-posed systems.In this talk we will review some motivational and relevant results when approaching and studying the initial value problem in Physics, and after commenting on open problems and difficulties that appear, we will illustrate their use for the study of non-linear relativistic extensions of classical theories such as Electromagnetism and Hydrodynamics.