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arXiv · 2112.12715

Measure-valued low Mach number limits of ideal fluids

Dennis Gallenmüller
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Abstract

As a framework for handling low Mach number limits we consider a notion of measure-valued solution of the incompressible Euler system which explicitly formulates the usually suppressed dependence on the pressure variable. We clarify in which sense such a measure-valued solution is generated as a low Mach limit and state sufficient conditions for this convergence. For the special case of pressure-free solutions we are able to give such sufficient conditions only on the incompressible solution itself. As a necessary condition for low Mach limits we obtain a Jensen-type inequality. The same necessary condition actually holds true for limits of weak solutions or vanishing viscosity limits. We illustrate that this Jensen inequality is not trivially fulfilled. Since measure-valued solutions in the classical sense are always generated by weak solutions, this shows that our solution concept contains more information and low Mach limits lead to a partial selection criterion for incompressible solutions.

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Dennis Gallenmüller. 2021-12-23. Measure-valued low Mach number limits of ideal fluids. https://arxiv.org/abs/2112.12715

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