Exponential dichotomies of a non-classical equation in spaces of differential forms on a two-dimensional torus with “noises”. (English) Zbl 1499.58026

Summary: We study the stability of solutions to the linear stochastic Barenblatt-Zheltov-Cochina equation in spaces of smooth differential forms defined on a two-dimensional torus. We show the existence of stable and unstable invariant spaces of solutions in spaces of “noises” for various parameters characterizing the medium and properties of the fluid. Also, we prove the existence of exponential dichotomies of solutions, which consist in the splitting of the phase space into a direct sum of two invariant spaces. Moreover, solutions starting in one of these spaces increase exponentially, remaining in this space, and solutions starting in another space decrease exponentially, also remaining in this space. We construct an algorithm to find stable and unstable solutions to the stochastic Barenblatt-Zheltov-Cochina equation on one of the maps of a two-dimensional torus. The algorithm takes into account that the initial data belong to the phase space. The algorithm is implemented in the Maple environment. For various values of the parameters included in the Barenblatt-Zheltov-Cochina equation, we present graphs of exponentially stable and exponentially unstable solutions that belong to stable and unstable invariant spaces. Also, we present the graphs of solutions having exponential dichotomy.


58J65 Diffusion processes and stochastic analysis on manifolds


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