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An abstract framework for the theory of statistical solutions is developedfor general evolution equations, extending the theory initially developed forthe three-dimensional incompressible Navier-Stokes equations. The motivationfor this concept is to model the evolution of uncertainties on the initialconditions for systems which have global solutions that are not known to beunique. Both concepts of statistical solution in trajectory space and in phasespace are given, and the corresponding results of existence of statisticalsolution for the associated initial value problems are proved. The wideapplicability of the theory is illustrated with the very incompressibleNavier-Stokes equations, a reaction-diffusion equation, and a nonlinear waveequation, all displaying the property of global existence of weak solutionswithout a known result of global uniqueness.

Abstract Framework for the Theory of Statistical Solutions