Mathematical challenges for the theory of hyperbolic balance laws in fluid mechanics: complexity, scales, randomness (Q6646760)

This paper is a survey on recent progress in the field of hyperbolic conservation laws described as Euler-type equations. Multidimensional Euler equations are used as an example to demonstrate the concepts. Several concepts are considered from both analytical and numerical points of view, including the weak-strong uniqueness, asymptotic regimes, entropy dissipation, non-uniqueness, probabilistic and chaotic approaches. The focus is on weak and entropy solutions because the blow-up possibility is taken into account and thus strong classical solutions might be impossible. The survey covers the state-of-the-art for non-experts in the field and so is a convenient introduction to the topic.