Existence and amplitude bounds for large-amplitude non-breaking Euler solutions

Establish the existence of large-amplitude non-breaking solutions—smooth solutions without shock formation that persist for long times—for the one-dimensional compressible Euler equations, and ascertain upper bounds on the amplitude of such solutions beyond which shock formation necessarily occurs.

Background

Numerical experiments in the paper suggest that for moderate amplitudes relative to a spatially periodic entropy/density background, solutions to the 1D compressible Euler equations avoid shock formation and develop persistent solitary waves, while larger amplitudes lead to shocks. The authors validate a shock-formation criterion in related settings and present strong computational evidence of non-decaying, non-breaking behavior.

A rigorous existence theory for large-amplitude, non-breaking solutions and a precise characterization of the amplitude threshold separating non-breaking and shock-forming regimes remain open. Such results would complement the homogenized model analysis and provide mathematical guarantees for the observed long-time behavior.

References

Many interesting open questions are raised by this work. For instance, is it possible to prove that there exist large-amplitude non-breaking solutions of the 1D Euler equations, and is there a limit to how large they can be?

— Solitary wave formation in the compressible Euler equations (2412.11086 - Ketcheson et al., 15 Dec 2024) in Section “Conclusion”