Abstract

This talk will survey some results for the two- or three-space dimensional compressible Euler equations, results both in theory and numerics. We shall present

• non-uniqueness results of weak entropy solutions for special initial data using convex integration
• introducing solution concepts beyond weak solutions that allows to show convergence to the incompressible limit of the compressible Euler equations with gravity
• the relationship between stationary preservation, maintaining vorticity, and asymptotic preserving numerical methods
• introduce a high order numerical method that holds promise to achieve this.

This is joint work among others with Simon Markfelder, Wasilij Barsukow, Eduard Feireisl and Phil Roe.

References

[1] W. Barsukow, J. Hohm, C. Klingenberg, and P. L. Roe. The active flux scheme on Cartesian grids and its low Mach number limit. Journal of Scientific Computing 81, pp. 594–622 (2019)

[2] E. Feireisl, C. Klingenberg, O. Kreml, and S. Markfelder. On oscillatory solutions to the complete Euler equations. Journal of Differential Equations 296 (2), pp. 1521-1543 (2020)