The Cauchy problem for the complete Euler system is in general ill-posed in the class of admissible (entropy producing) weak solutions. This suggests that there might be sequences of approximate solutions that develop fine-scale oscillations. Accordingly, the concept of measure-valued solution that captures possible oscillations is more suitable for analysis. We study the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. Here dissipative means that a suitable form of the second law of thermodynamics is incorporated in the definition of the measure-valued solutions. In particular, using the recently established weak-strong uniqueness principle, we show that the numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter.

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耗散量值解的Euler方程有限体积格式的收敛性

完整的Euler系统的柯西问题通常在可容许的（产生熵的）弱解中是不适当的。这表明可能存在产生精细振荡的近似解序列。因此，量值解决方案的概念捕获可能的振荡更适合分析。我们研究了多维情况下正压和完整可压缩Euler方程的一类熵稳定有限体积格式的收敛性。我们建立了合适的稳定性和一致性估计，并表明由数值解生成的Young度量表示欧拉系统的耗散度量值解。在此，耗散的意思是将热力学第二定律的合适形式并入到度量值解的定义中。特别是，使用最近建立的弱强唯一性原理，我们证明了数值解至少在极限系统的寿命上逐点收敛于极限系统的正则解。