For the ideal compressible Euler systems, which are fundamental in fluid-dynamics and pro-type examples of nonlinear hyperbolic systems, one of the main features is that the characteristic speeds of a wave propagation depend on the wave itself which leads to the finite-time formation of shocks in general. Thus one has to work with weak solutions globally. Yet the uniqueness of the "physical" solutions becomes a challenging issue. In the one-dimensinal case, various admissible criterion have been introduced to rule out the non-physical solutions. In particular, the physical entropy can guarantee the uniqueness of weak solutions at least in the case of weak solution with small variations.

However, in higher space dimensions, for some given initial data, there are infinitely many highly oscillatory solutions (wild solutions) which are bounded, measurable and satisfying the physical entropy. In this talk, I will review some progress on the constructions of such "wild solutions" by a method of convex integration; present some results on the structure of such "wild solutions"; and investigate the effects of lower order dissipations. Some open problems will be discussed.