Unique continuation results for abstract quasi-linear evolution equations in Banach spaces

TL;DR

This paper investigates unique continuation properties for a broad class of evolution equations in Banach spaces, utilizing conserved quantities and well-posedness, with applications to physical models like wave and fluid dynamics.

Contribution

It introduces novel unique continuation results for quasi-linear evolution equations in Banach spaces, connecting conservation laws and well-posedness to physical models.

Findings

Unique continuation results established for various evolution equations.

Applications to physical systems such as wave propagation and hydrodynamics.

Conservation of norms plays a key role in the analysis.

Abstract

Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into the conservation of some norm of the solutions of the system in a suitable Banach space. The second one is regarded to well-posed problems. Our results are then applied to some equations, most of them describing physical processes like wave propagation, hydrodynamics, and integrable systems, such as the $b-$; Fornberg-Whitham; potential and $\pi-$Camassa-Holm; generalised Boussinesq equations; and the modified Euler-Poisson system.