The Fujita-Kato theorem for some Oldroyd-B model

In this paper, we investigate the Cauchy problem associated to a system of PDEs of Oldroyd type. The considered model describes the evolution of certain viscoelastic fluids within a corotational framework. The non-corotational setting is also addressed in dimension two. We show that some widespread results concerning the incompressible Navier-Stokes equations can be extended to the considered systems. In particular we show the existence and uniqueness of global-in-time classical solutions for large data in dimension two. This result is supported by suitable condition on the initial data to provide a global-in-time Lipschitz regularity for the flow, which allows to overcome specific challenging due to the lack of a decay in time of the main forcing terms. Secondly, we address the global-in-time well-posednessin dimension d >= 3. We prove the propagation of Lipschitz regularities for the flow. For this result, we just assume the initial data to be sufficiently small in a critical Lorentz space. (C) 2020 Elsevier Inc. All rights reserved.