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$\frac{d}{dt} u(t) = A(t) u(t) + f(t)$
on an infinite dimensional Banach space with no nonlinearity existence and uniqueness of solutions was proven probably in many ways but in 1960 Tanabe tanabe-evolution-equations-banach-space. and cwiszewski-evolution-equations-infinite-dimensions We’re not quite interested in these but the nonlinear equations in infinite dimensions that serve as the limit of Navier-Stokes type equations on graphs of finite number of nodes where the Laplacian is a numerical matrix as we let $n$ tend to infinity.  We would like some control of the Laplace spectrum while taking the graph limit in the sense of Lovasz.  Wolfe et. al. and Chatterjee, Diaconis, et. al. have considered the statistical modeling and nonparametric estimation on the limit graph where the Bernoulli parameters for edge existence limit to an object called a graphon I think the term is due to Wolfe.  We are interested in nonlinear PDE on the graph and its limit.  We’re interested in ergodicity properties of the dynamical systems in the limit.