Row-finite systems of ordinary differential equations in a scale of Banach spaces

2018/03/19 by Daletskii, Alexei, Finkelshtein, Dmitri
#34A12 #34A30 #82C22 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1803.06909

Abstract

Motivated by the study of dynamics of interacting spins for infinite particle systems, we consider an infinite family of first order differential equations in a Euclidean space, parameterized by elements x of a fixed countable set. We suppose that the system is row-finite, that is, the right-hand side of the x-equation depends on a finite but in general unbounded number n_x of variables. Under certain dissipativity-type conditions on the right-hand side and a bound on the growth of n_x, we show the existence of the solutions with infinite life-time, and prove that they live in an increasing scale of Banach spaces. For this, we obtain uniform estimates for solutions to approximating finite systems using a version of Ovsyannikov's method for linear systems in a scale of Banach spaces. As a by-product, we develop an infinite-time generalization of the Ovsyannikov method.

Related

• Row finite systems of stochastic differential equations with dissipative drift
• Stochastic differential equations in a scale of Hilbert spaces
• Stochastic differential equations in a scale of Hilbert spaces 2. Global solutions
• Stochastic dynamics of particle systems on unbounded degree graphs
• Around Ovsyannikov's method