Global Regularity for Some Oldroyd‐B Type Models

Abstract

We investigate some critical models for visco-elastic flows of Oldroyd-B type in dimension 2. We use a transformation that exploits the Oldroyd-B structure to prove an L 1 bound on the vorticity which allows us to prove global regularity for our systems.

Global wellposedness to the generalized Oldroyd type models in R3

“…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]. …”

“…Using the similar method, the authors got the global existence of smooth solution to system (1.3) without the term Q(∇u, τ ) under the arbitrary initial data, and system (1.3) with small initial data in [4].…”

“…An improvement of the Chemin-Masmoudi blow-up criterion was presented recently by Lei et al [12]. Much work has been limited to global existence of weak solutions and global well-posedness for smooth solutions to Oldroyd model and related models ( [2], [5], [7], [9], [10], [13], [14], [15], [16], [17], [19], [21], [26], [27], [28]). …”

Global regularity for the 2D Oldroyd‐B model in the corotational case

“…The system with μ = η = 0 is still a challenging open problem as discussed in . In this paper, we would like to show that for any small power α > 0, the system always admits a unique global smooth solution.…”

“…We mention that global weak for system with η ( τ Ω−Ω τ ) replaced by general Q , namely, , is still open up to now. As pointed out in , in the case where η = 0 and ν > 0, the global existence of smooth solutions to system is also open and quite challenging. Because there is no global existence result for general initial data, we would like to add the fractional dissipation (−Δ) α τ to the stress tensor τ equation to guarantee the global well‐posedness result.…”