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In this chapter, we survey recent results concerning the existence and properties of so-called wild solutions to the compressible Euler system in two spatial dimensions. These solutions are constructed using the convex integration method developed in this context by De Lellis and Székelyhidi (Ann Math 170:1417–1436;2009, Arch Rational Mech Anal 195:225–260;2010). We introduce the Riemann problem for the compressible Euler system, classify its one-dimensional self-similar solutions, and summarize results related to the uniqueness and nonuniqueness of these solutions within the class of admissible weak solutions. We also discuss the existence of wild solutions for regular initial data and the failure of criteria based on maximal dissipation of energy to select physically relevant solutions.

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On Wild Solutions to the Compressible Euler System

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  • 发表日期：

    2025年

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  • Ondřej Kreml

摘要

In this chapter, we survey recent results concerning the existence and properties of so-called wild solutions to the compressible Euler system in two spatial dimensions. These solutions are constructed using the convex integration method developed in this context by De Lellis and Székelyhidi (Ann Math 170:1417–1436;2009, Arch Rational Mech Anal 195:225–260;2010). We introduce the Riemann problem for the compressible Euler system, classify its one-dimensional self-similar solutions, and summarize results related to the uniqueness and nonuniqueness of these solutions within the class of admissible weak solutions. We also discuss the existence of wild solutions for regular initial data and the failure of criteria based on maximal dissipation of energy to select physically relevant solutions.