Functional and measure-valued solutions of the euler equations for flows of incompressible fluids

Abstract

We consider the notion of a functional solution of the Euler equations for incompressible fluid flows. We show that a functional solution can be constructed under “very weak” a priori estimates on approximate solution sequences of the equation; an estimate uniform in L ¹_loc together with weak consistency with the equation is sufficient to construct a solution. We prove that if we have an estimate uniform in L ²_loc available for the approximate solution sequence, then the structured functional solution just described becomes a measure-valued solution in the sense of DiPerna & Majda. We also show that a functional solution can be obtained from a measure-valued solution. We give an example showing that a much higher concentration of energy than in the case of measure-valued solutions is allowed by the approximation procedure of a functional solution.