A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm of measure-valued solutions for these problems.We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

中文翻译：

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关于测度值解的计算

流体动力学中存在解的标准范式是基于近似解或近似最小值序列的构造。这种方法面临着严重的障碍，尤其是在多维问题中，在更精细的尺度上持续存在振荡会阻碍紧凑性。实际上，与最近的理论结果一致，这些振荡表明在可积函数的标准框架内可能缺乏解的存在/唯一性。正是在这种情况下，Young 测量——可以描述这种振荡序列的极限的参数化概率测量——提供了更一般的范式测量值解决方案对于这些问题。我们提出了可行的数值算法来计算近似测量值的解决方案，基于将近似测量作为蒙特卡洛采样随机场定律的实现。我们证明了这些算法对可压缩和不可压缩无粘性流体动力学方程的测量值解的收敛性，并提供了大量数值实验，为新范式的可行性提供了令人信服的证据。我们还讨论了这些算法及其扩展在流体动力学以外的不确定性量化和上下文中的使用，例如材料科学中的非凸变分问题。