Applicable Analysis, 2015 Vol. 94, No. 4, 840–855, http://dx.doi.org/10.1080/00036811.2014.905677 Existence of solutions for a class of Navier–Stokes equations with infinite delay Sandro Marcos Guzzo a ∗ and Gabriela Planas b a Colegiado do Curso de Matemática, Universidade Estadual do Oeste do Paraná – UNIOESTE, Cx.P. 711, Cascavel PR 85819-110, Brazil; b Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda, 651, Campinas-SP 13083-859, Brazil Communicated by Roger Temam (Received 17 May 2013; accepted 11 March 2014) In this paper, a class of Navier–Stokes equations with infinite delay is considered. It includes delays in the convective and the forcing terms. We discuss the existence of mild and classical solutions for the problem. We establish the results for an abstract delay problem by using the fact that the Stokes operator is the infinitesimal generator of an analytic semigroup of bounded linear operators. Finally, we apply these abstract results to our particular situation. Keywords: Navier–Stokes equations; infinite delays; mild solution AMS Subject Classifications: 35R10; 35Q30; 47D05 1. Introduction In this paper, we study the existence of solutions for a class of Navier–Stokes equations with hereditary terms which can be described in the abstract form u ′ (t ) = A p u (t ) + F (u t , u (t )) + f (t , u t ), t > 0, (1) u 0 = ϕ ∈ B, (2) where A p is the Stokes operator with domain D( A p ) ⊂ X p being X p a suitable Banach space, the history u t , given by u t (θ) = u (t + θ), belongs to an abstract phase space B defined axiomatically, f is a continuous function defined on the phase space and F is an appropriate mapping which is related to the convective term of the Navier–Stokes equations involving delays. Navier–Stokes equations have been extensively studied over the last century due to their important role in fluid mechanics and turbulence. Recently, Navier–Stokes equations with a forcing term which contains some hereditary characteristics were considered by Caraballo and Real [1]. The asymptotic behaviour of solutions to Navier–Stokes equations with various types of time-delayed external forces was investigated by many authors, we ∗ Corresponding author. Email: smguzzo@gmail.com © 2014 Taylor & Francis