We show how the philosophy of the theory of differential inclusions for Lipschitz mappings can be used in order to construct weak solutions for the Euler equations, based on the articles of C. De Lellis and L. Szekelyhidi.

In the second Chapter we describe the Baire category theory, and the approach via the Banach-Mazur game.

In the third Chapter we treat the theory of rank-one convex and quasiconvex measures (i.e. gradient Young measures), and their duality with classes of semiconvex functions. We describe the theory of extreme points and we give results about the stability of hulls.

In the fourth Chapter we describe the theory of differential inclusions for Lipschitz mappings and we also treat the cases when some restrictions are imposed to the gradients of our mappings. We then describe the convex integration results and the theory of stability near extreme points due to Kirchheim. We also state general theorems about nonhomogeneous inclusions.

In Chapter 5 we describe the theory of Compensated Compactness, and we show how its use in the case of the Euler equation is analogous to the theory of differential inclusions. We describe the results of De Lellis and Szekelyhidi about nonuniqueness of weak solutions, and the construction of solutions with a prescribed energetical behavior.

In Appendix A we show a counterexample for regularity in the case of nonlinear systems of equations, which uses convex integration.

In Appendix B we describe the previous construction of Shnirelman of Euler solutions like those of Chapter 5.