In the recent years, significant attention has been directed again to the Euler system, which was derived more than 250 years ago. The system describes the motion of an inviscid fluid. The main attention has been directed to incompressible fluids. Nevertheless, also the system of compressible fluids is an emerging topic, however still very far from a complete understanding. The classical results of Scheffer and Schnirelman pointed out the problem of non-uniqueness of distributional solutions to incompressible the Euler system. However, the crucial step appeared to be an application of methods arising from differential geometry, namely the celebrated theorem by Nash and Kuiper. This brought Camillo De Lellis and Laszlo Szekelyhidi Jr. in 2010 to the proof of existence of bounded (even Hölder-continuous) nontrivial compactly supported in space and time solutions of the Euler equations (obviously not conserving physical energy!). Without a doubt, this result is a first step towards the conjecture of Lars Onsager, who in his 1949 paper on the theory of turbulence asserted the existence of such solutions for any Hölder exponent up to 1/3.
The talk is based on several recent joint results with Jose A. Carrillo, Eduard Feireisl, Piotr Gwiazda and Emil Wiedemann and concerns various notions of solutions to compressible Euler equations and some systems of a similar structure. Firstly, we shall concentrate on weak solutions and discuss the issue of non-uniqueness and the non-conservation of the energy. We show the existence of infinitely many global-in-time weak solutions for any bounded initial data by adapting the method of convex integration, used in the incompressible Euler system by De Lellis and Szekelyhidi. Then, we consider the class of dissipative solutions satisfying, in addition, the associated global energy balance (inequality). We give sufficient conditions on the regularity of solutions to the compressible isentropic Euler systems in order for the energy to be conserved.