Abstract Framework for the Theory of Statistical Solutions

TL;DR

This paper develops a general abstract framework for statistical solutions applicable to various evolution equations, extending prior work on Navier-Stokes equations to broader systems with uncertain initial conditions.

Contribution

It introduces a unified theory of statistical solutions in trajectory and phase space for general evolution equations with non-unique global solutions.

Findings

Existence of statistical solutions for initial value problems is established.

Framework applies to Navier-Stokes, reaction-diffusion, and nonlinear wave equations.

Demonstrates the broad applicability of the theory to systems with uncertain initial data.

Abstract

An abstract framework for the theory of statistical solutions is developed for general evolution equations, extending the theory initially developed for the three-dimensional incompressible Navier-Stokes 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. 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 wide applicability of the theory is illustrated with the very 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.