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(loc)(1) together with weak consistency with the equation is sufficient to construct a solution. We prove that if we have an estimate uniform in L(loc)(2) 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.