The Initial Value Problem for the Euler Equations of Incompressible Fluids Viewed as a Concave Maximization Problem
The Initial Value Problem for the Euler Equations of Incompressible Fluids Viewed as a Concave Maximization Problem
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不可压缩流体欧拉方程的初值问题被视为凹最大化问题
DOI：
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发表时间：
2017
影响因子：
2.4
通讯作者：
Y. Brenier
中科院分区：
文献类型：
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作者：
Y. Brenier
我们考虑不可压缩流体的欧拉方程（Arnold 和 Khesin，《流体动力学中的拓扑方法》，Springer，柏林，1998 年；《流体力学数学主题中的 Lions》，第 1 卷。不可压缩模型，牛津大学出版社，牛津，1996 年），并尝试借助凹最大化问题来解决初始值问题。我们在“Benamou-Brenier”公式中证明了该问题与具有二次成本的最优传输问题具有相似的结构（Ambrosio 等人，在度量空间和概率测度空间中的梯度流中，Birkhäuser，2008 年；Benamou 和 Brenier 在 Numer Math 84:375–393, 2000 中；Otto 和 Westdickenberg 在 SIAM J Math Anal 中） 37:1227–1255, 2005；Villani in Topics in optimization communications, AMS, Providence, 2003），总是承认可以用凸积分理论意义上的欧拉方程的子解来解释的松弛解（De Lellis et al. in Ann Math 170(2):1417–1436, 2009）。这个想法被扩展到达弗莫斯书中考虑的“凸熵守恒定律”类别（连续介质物理中的双曲守恒定律中的达弗莫斯，施普林格，柏林，2000）。在所有情况下，都表明初始值问题的任何平滑解决方案都可以从这种最大化问题中恢复，至少在短时间内是这样。最后，在所谓的无粘伯格斯方程的非常简单的情况下，表明克鲁日科夫意义上的每个熵解都可以在不受时间间隔限制的情况下恢复。
We consider the Euler equations of incompressible fluids (Arnold and Khesin in Topological methods in hydrodynamics, Springer, Berlin, 1998; Lions in Mathematical topics in fluid mechanics, vol 1. Incompressible models, Oxford University Press, Oxford, 1996) and attempt to solve the initial value problem with the help of a concave maximization problem. We show that this problem, which shares a similar structure with the optimal transport problem with quadratic cost, in its “Benamou–Brenier” formulation (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures, Birkhäuser, 2008; Benamou and Brenier in Numer Math 84:375–393, 2000; Otto and Westdickenberg in SIAM J Math Anal 37:1227–1255, 2005; Villani in Topics in optimal transportation, AMS, Providence, 2003), always admits a relaxed solution that can be interpreted in terms of sub–solution of the Euler equations in the sense of convex integration theory (De Lellis et al. in Ann Math 170(2):1417–1436, 2009). This idea is extended to the class of “conservation laws with convex entropy” considered in Dafermos’ book (Dafermos in Hyperbolic conservation laws in continuum physics, Springer, Berlin, 2000). In all cases, it is shown that any smooth solution to the initial value problem can be recovered from such a maximization problem, at least for short times. Finally, in the very simple case of the so-called inviscid Burgers equation, it is shown that every entropy solution, in the sense of Kruzhkov, can be recovered without any restriction on the time interval.