Abstract Framework for the Theory of Statistical Solutions

Anne C. Bronzi, Cecilia F. Mondaini, Ricardo M. S. Rosa

Published 2015-03-23Version 1

An abstract framework for the theory of statistical solutions is developed for general evolution equations. The motivation for this concept is to model the evolution of uncertainties on the initial conditions for systems which have global solutions that are not known to be unique. The typical example is the three-dimensional incompressible Navier-Stokes equations, for which the theory was initially developed. The aim here is to extend this theory to general evolution equations. Both concepts of statistical solution in trajectory space and in phase space are given, and the corresponding results of existence of statistical solution for the associated initial value problems are proved. The conditions of the theorems are natural and not difficult to verify, relying essentially on properties of the set of individual solutions of the system. The proof also significantly simplifies the classical known proofs for the Navier-Stokes equations and other systems. The wide applicability of the abstract theory is illustrated in three examples: the very three-dimensional incompressible Navier-Stokes equations, a reaction-diffusion equation, and a nonlinear wave equation, all displaying the property of global existence of weak solutions without a known result of global uniqueness.