论文标题
通过粗糙路径的单数SPDE的初始值问题
The Initial Value Problem for Singular SPDEs via Rough Paths
论文作者
论文摘要
在这一贡献中，我们为遇到初始条件的奇异准随机偏微分方程开发了解决方案理论。我们将解决方案理论作为对解决时空周期性问题开发的粗糙路径方法的扰动，这是Otto and Weber（2019）。与他们的工作一样，我们假设强迫是$ c^{α-2} $的$α\ in（\ frac {2} {3} {3}，1）$和时空周期性的，并且最初的条件是$ c^α$和周期性的。我们为理论的分析方面做出了贡献。实际上，我们表明我们可以通过使用初始边界层校正先前获得的时空周期解决方案来强制执行初始条件，该边界层可以完全确定性地处理。独特性是在通过初始边界层以这种方式校正的解决方案类别中获得的。此外，建立了解决方案在数据扰动方面的稳定性。
In this contribution we develop a solution theory for singular quasilinear stochastic partial differential equations subject to an initial condition. We obtain our solution theory as a perturbation of the rough path approach developed to handle the space-time periodic problem by Otto and Weber (2019). As in their work, we assume that the forcing is of class $C^{α-2}$ for $α\in (\frac{2}{3},1)$ and space-time periodic and, additionally, that the initial condition is of class $C^α$ and periodic. We contribute to the analytic aspects of the theory. Indeed, we show that we can enforce the initial condition via correcting the previously obtained space-time periodic solution with an initial boundary layer which may be handled in a completely deterministic manner. Uniqueness is obtained in the class of solutions which are corrected in this way by an initial boundary layer. Moreover, stability of the solutions with respect to perturbations of the data is established.
