论文标题
在存在奇异性的情况下传输方程的解决方案
On solutions of the transport equation in the presence of singularities
论文作者
论文摘要
我们在$ [0，t] \ times \ mathbb {r}^n $上的传输方程式在矢量字段为$ bv $ off a set $ s \ subset [0，t] \ times \ times \ mathbb {r}^n $的情况下。我们证明了解决方案的存在，并且是独一无二的，前提是奇异性的集合具有足够小的各向异性分形维度，并且矢量场的正常成分在奇异性附近足够整合。这一结果改善了Ambrosio的最新结果，Ambrosio要求矢量场到处都是有限变化的。此外，我们证明，在这些条件下，几乎相关的常规拉格朗日流的每个轨迹都不会与奇点的集合$ s $相交。最后，我们考虑了随时间发展的初始奇点集的特殊情况，因此奇点由相位空间中的曲线组成，这在诸如涡流动力学之类的应用中是典型的。我们证明了传输方程的解决方案存在，并且只要奇异点的盒子计数维度是曲线的Hölder指数界定的。
We consider the transport equation on $[0,T]\times \mathbb{R}^n$ in the situation where the vector field is $BV$ off a set $S\subset [0,T]\times \mathbb{R}^n$. We demonstrate that solutions exist and are unique provided that the set of singularities has a sufficiently small anisotropic fractal dimension and the normal component of the vector field is sufficiently integrable near the singularities. This result improves upon recent results of Ambrosio who requires the vector field to be of bounded variation everywhere. In addition, we demonstrate that under these conditions almost every trajectory of the associated regular Lagrangian flow does not intersect the set $S$ of singularities. Finally, we consider the particular case of an initial set of singularities that evolve in time so the singularities consists of curves in the phase space, which is typical in applications such as vortex dynamics. We demonstrate that solutions of the transport equation exist and are unique provided that the box-counting dimension of the singularities is bounded in terms of the Hölder exponent of the curves.
