论文标题
在粘性汉密尔顿 - 雅各布方程中多次形成和正则化
Singularity formation and regularization at multiple times in the viscous Hamilton-Jacobi equation
论文作者
论文摘要
在随机控制理论中具有重要应用的cauchy-dirichlet PBM，用于超质量粘性汉密尔顿 - 雅各布斯EQN（VHJ），它承认了一种独特的全球粘度解决方案。 sol。因此，在有限的时间内出现奇异性的弱意义上存在，这是通过边界上的梯度爆炸（GBU）发生的。而Visc理论。 sol。已经对许多PDE进行了广泛的研究并应用了，对SOL的精致行为的结果较少。特别是VISC的详细行为。 sol。 GBU之后的VHJ大部分是开放的。 在这里，对于每个$ m \ ge 1 $ we构建sol。至少在$ m $ $ times，然后恢复规律性以及SOL。该展示GBU在第一次爆炸时间没有LBC。在1D中，我们获得了VISC的完整分类。 sol。在每个时间，在较高d中延伸至径向病例。此外，对于每个$ m \ ge 2 $，并任意给出了GBU类型的组合，其中/不带LBC以$ m $ $ $ $ $ $ $ $ $的订单组合，我们表明存在。溶胶。与GBU的精确组合。一些溶胶。显示一种称为“弹跳”的新型行为。 全球弱溶胶。与富士式EQN相比，具有多个时间奇异性的VHJ表现出更大的行为。我们介绍了一种基于任意数量的关键参数的方法，其连续性需要一个微妙的论点。由于我们不依赖任何已知的特殊溶胶。与藤田eqn不同，我们的方法有望适用于其他eqn。 在VHJ和随机控制理论的背景下，多次的奇异行为是全新的。在此框架中，我们的结果表明，对于某些奖励的某些空间分布，如果对照的布朗粒子在边界附近开始，则净增益在不同的时间范围内获得有利可图的值，而在某些中间时间则不能。
The Cauchy-Dirichlet pbm for the superquadratic viscous Hamilton-Jacobi eqn (VHJ), which has important applications in stochastic control theory, admits a unique, global viscosity solution. Sol. thus exist in the weak sense after appearance of singularity in finite time, which occurs through gradient blow-up (GBU) on the boundary. Whereas theory of visc. sol. has been extensively studied and applied to many PDEs, there are less results on refined behavior of sol. In particular, detailed behavior of visc. sol. of VHJ after GBU has remained mostly open. Here, in general dim., for each $m\ge 1$ we construct sol. which undergo GBU and LBC at least at $m$ times and then recover regularity, as well as sol. that exhibit GBU without LBC at 1st blowup time. In 1d, we obtain the complete classification of visc. sol. at each time, which extends to radial cases in higher d. Furthermore for each $m\ge 2$ and arbitrarily given combination of GBU types with/without LBC at $m$ times in arbitrarily given order, we show exist. of a sol. with this exact combination of GBU. Some sol. display a new type of behavior called "bouncing". Global weak sol. of VHJ with multiple time singularity turn out to display larger variety of behaviors than for the Fujita eqn. We introduce a method based on an arbitrary number of critical parameters, whose continuity requires a delicate argument. Since we do not rely on any known special sol. unlike in Fujita eqn, our method is expected to apply to other eqns. Singular behaviors at multiple times are completely new in the context of VHJ but also of stochastic control theory. In this framework our results imply that for certain spatial distributions of rewards, if a controled Brownian particle starts near the boundary, then the net gain attains profitable values on different time horizons but not on some intermediate times.
