论文标题
在多维无压力欧拉方程的粘性粒子解决方案上
On the sticky particle solutions to the multi-dimensional pressureless Euler equations
论文作者
论文摘要
在本文中，我们考虑了多维无压力欧拉系统，我们解决了一般测量型初始数据的存在问题和粘性粒子溶液的独特性。尽管\ cite {bressan-nguyen}是已知的，但对存在和独特性的明确反例，但人们是否仍然可以找到大量数据的粘性粒子解决方案以及如何选择它们的问题是否完全开放。 在本文中，我们证明，对于弱拓扑中的一组合并的初始数据集，压力无EULER系统承认由自由流提供的独特粘性粒子解决方案，其中轨迹是截然不同的直线。 的确，这种存在和独特性的结果可用于更广泛的解决方案，从而减少了它们的动能，我们称之为耗散溶液，事实证明这是经典粘性粒子溶液的紧凑弱闭合。因此，能源为L.S.C.的任何方案并消散将收敛于一组数据，即我们的解决方案，即自由流。
In this paper we consider the multi-dimensional pressureless Euler system and we tackle the problem of existence and uniqueness of sticky particle solutions for general measure-type initial data. Although explicit counterexamples to both existence and uniqueness are known since \cite{Bressan-Nguyen}, the problem of whether one can still find sticky particle solutions for a large set of data and of how one can select them was up to our knowledge still completely open. In this paper we prove that for a comeager set of initial data in the weak topology the pressureless Euler system admits a unique sticky particle solution given by a free flow where trajectories are disjoint straight lines. Indeed, such an existence and uniqueness result holds for a broader class of solutions decreasing their kinetic energy, which we call dissipative solutions, and which turns out to be the compact weak closure of the classical sticky particle solutions. Therefore any scheme for which the energy is l.s.c. and is dissipated will converge, for a comeager set of data, to our solution, i.e. the free flow.
