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Description:
In this monograph, we study the notion of measure-valued solution for the compressible and incompressible Euler equations and investigate how such solutions can be generated by singular limits of weak solutions of related fluid models. Of particular interest will be limits of compressible weak solutions, vanishing viscosity, and low Mach number limits. After laying the abstract linear foundations used in this thesis we take a deeper look at the relationship between weak and measure-valued solutions of compressible fluid flows. First, we construct examples of energy admissible compressible measure-valued solutions arising from deterministic and continuous initial data, which cannot be generated by vanishing viscosity limits or limits of weak solutions. This is in sharp contrast to the incompressible situation, where every (classical) measure-valued solution is the limit of weak solutions. In general, any generable compressible measure-valued solution has to satisfy a necessary Jensen-type inequality. On the other hand, we obtain also sufficient conditions involving a slightly different Jensen inequality on the level of potential operators from an L^1-version of Fonseca-Müller’s characterization result for A-free Young measures together with a certain A-free truncation technique. Proving the aforementioned truncation method is a major milestone in our approach. Given a sequence of A-free functions in potential form converging in an L^1-sense to some limit set, the aim is to truncate this sequence in such a way that the new sequence converges uniformly to the limit set while preserving the A-freeness and remaining sufficiently close to the original sequence. We establish such a truncation result for potentials of first and second order, including the potential of the linearly relaxed Euler system. Moreover, we introduce a measure-valued framework in which the low Mach limit can be treated adequately. This will lead to the consideration of the novel concept of augmented measure-valued solutions. We give sufficient ...