Functional and measure-valued solutions of the Euler equations for flows of incompressible fluids
The notion of a functional solution of the Euler equations for incompressible fluids flows and other relevant aspects are considered. The two-dimensional, nonstationary Euler equation, taken as a functional, the continuity equation, and the boundary and initial conditions are established as basic equations. The existence of a functional solution of the problem is proved, and its relation to the so-called measure-valued solutions is investigated. It is shown that a functional solution can be constructed using very weak a priori estimates of approximate solution sequences for the equation. For brevity, only some names and topics occuring in the paper may be mentioned here, such as DiPerna and Majda, the Banach space, a generalized Young measure, and a concentration of energy of Dirac-measure type. Several definitions and theorems are stated, and a lemma occasionally is established and proved. Using these means, it is shown that the described method to gain a functional solution is of an approximating and stable type, and that a functional solution allows an arbitrarily strong concentration of energy during the approximation procedure. As an example, for an exact stationary solution of a special flow affected by a vorticity, a stationary functional solution of two- dimensional Euler equations is explicitly constructed. Finally, as a second example of the construction of a functional solution, the family of solutions of the Navier-Stokes equations with vanishing viscosity is obtained by the method of parametrization. It is shown that the method of vanishing viscosity is approximating and stable.
scientific article; zbMATH DE number 775511