Measure-Valued singular limits for compressible fluids
| Autor: | Gallenmüller, Dennis |
|---|---|
| Přispěvatelé: | Wiedemann, Emil, Swierczewska-Gwiazda, Agnieszka, Klingenberg, Christian |
| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: |
Young-Maß
selection criterion Eulersche Bewegungsgleichungen augmented measure-valued solution compressible Euler equations incompressible Euler equations Young measure Hydrodynamik Fluid dynamics low Mach limit isentropic Euler equations DDC 510 / Mathematics measure-valued solution ddc:510 vanishing viscosity limit |
| Popis: | 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 conditions on low Mach sequences for generating such an augmented solution. More importantly, we also obtain necessary Jensen-type conditions on augmented solutions to arise from low Mach sequences. Our discussion of the low Mach limit will shed some new light on the role of the pressure in the compressible and incompressible situation. As a consequence of our results, we propose two selection criteria in order to discard some compressible measure-valued solutions and augmented incompressible solutions out of the in general infinitely many as unphysical. To a certain extent, this tackles the question of uniqueness for measure-valued solutions. |
| Databáze: | OpenAIRE |
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