Global wellposedness to the generalized Oldroyd type models in R3

Abstract

It remains unknown whether smooth solutions of the 3D Oldroyd models exist globally in viscoelastic fluids. Using the trick of transferring dissipation, we get the global strong solutions of the generalized Oldroyd models in three dimensions, under the assumption that initial data is sufficiently small. It improves the result in two dimensions obtained by Elgndi and Rousset (2015) [4].

Steady flows of an Oldroyd fluid with threshold slip

“…Suppose that (v, S, E, p) is a classical solution to problem (1)- (9). Following exactly the same steps as in [8, Remark 3], we obtain relations (15) and (16). On the other hand, it can be proved that if a weak solution ( v, S, E) of problem (1)-(9) is sufficiently smooth, then there exists a function p such that ( v, S, E, p) is a classical solution to (1)-(9).…”

“…Starting with the pioneering works of Renardy [35] and Guillopé & Saut [20], mathematical models of viscoelastic fluids of Oldroyd type have been studied by many authors. We mention here only the papers [2,3,5,9,10,12,13,14,15,16,17,18,21,22,26,28,30,39,40,41]. A detailed analysis of different problems and results related to the Oldroyd model and other similar non-Newtonian models can be found in the review article [37].…”

“…Putting v = v 0 , E = E 0 , ϕ = 2v 0 in inequality (15) and taking into account (54), we get the following inequality…”

Global existence of classical solutions for the 3D generalized compressible Oldroyd‐B model

“…When g ( τ ,∇ u )=0, they also got the similar result with general initial data. For the 3D case, Elgindi and Liu achieved some small initial data result.…”

Weak martingale solution of stochastic critical Oldroyd-B type models perturbed by pure jump noise

“…Authors in [4] have studied the two dimensional model when there is a lack of dissipation in the velocity equation only, i.e., when ¼ 0, j > 0, and when Q has a special structure (that is, in the corotational case). As an improvization of [4], the authors in [5] have exploited the method of transferring dissipation and have obtained the existence of a unique global solution in three dimensional space provided the initial data is small in the H s Sobolev-norm with s > 5 2 : It is worth noting that even when Q ¼ 0, the coupling is critical with respect to the smoothing effect provided by the partial parabolic regularization (see, for instance, [4] for details). When the diffusion is present in both the equations i.e., , j > 0, the problem is subcritical in dimension two, and global well-posedness of strong solutions can be obtained in [6] due to the smoothing effect provided by the fully parabolic system.…”