Existence and stability results for 3-D regular flows of viscoelastic fluids of White–Metzner type Original Research Article

This paper is concerned with regular flows of incompressible viscoelastic fluids which obey a differential constitutive law of White–Metzner type. We study the existence and uniqueness of local solutions in 3-D domain as well as the global existence for small data. By a classical stability argument, we deduce the existence and stability of small periodic and stationary solutions. We also generalize the 2-D results. More precisely, we prove that the results obtained by Hakim (J. Math. Anal. Appl. 185 (1994) 675) remain true without any restriction on the smallness of the retardation parameter which is the linking coefficient between the equation of velocity (Navier–Stokes equation) and the transport equation verified by the extra-stress tensor. The hypothesis of boundedness on the relaxation function is also dropped.