Abstract

We consider the Showalter–Sidorov problem for the stochastic version of the linear Ginzburg–Landau equation in Hilbert spaces of smooth differential forms defined on a compact oriented Riemannian manifold without boundary with stochastic processes serving as coefficients. This equation is reduced to an abstract Sobolev type stochastic equation with a relatively radial operator on the right-hand side, for which the solvability of the Showalter–Sidorov problem is established and the stability of solutions is investigated using dichotomies. Differentiation of stochastic processes that are the coefficients of differential forms is understood in the sense of the Nelson–Gliklikh derivative.

中文翻译：

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微分形式空间中具有相对$$ p $$-径向算子的Sobolev型随机方程

摘要

我们考虑在光滑定向微分形式的希尔伯特空间的希尔伯特空间中的线性Ginzburg-Landau方程的随机版本的Showalter-Sidorov问题，该线性微分形式定义在无定向随机边界的紧凑定向黎曼流形上，并作为系数。该方程简化为抽象的Sobolev型随机方程，在右侧具有一个相对径向的算符，为此建立了Showalter-Sidorov问题的可解性，并使用二分法研究了解的稳定性。随机过程的微分是微分形式的系数，这在Nelson–Gliklikh导数的意义上可以理解。