Global Well-Posedness in the Critical Besov Spaces for the Incompressible Oldroyd-B Model Without Damping Mechanism

Abstract

We prove the global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism on the stress tensor in R d for the small initial data. Our proof is based on the observation that the behaviors of Green's matrix to the system of u, (−∆) − 1 2 P∇ · τ as well as the effects of τ change from the low frequencies to the high frequencies and the construction of the appropriate energies in different frequencies.

Global well-posedness, stability and instability for the non-viscous Oldroyd-B model

“…Zhu [30] got small global smooth solutions of the 3D Oldroyd-B model with η = 0, µ = 0 by observing the linearization of the system satisfies the damped wave equation. Inspired by the work of Zhu [30] and Danchin in [10], Chen and Hao [6] extended this small data global solution in Sobolev spaces to the critical Besov spaces. Moreover, Zhai [27] constructs global solutions for a class of highly oscillating initial velocities by observing the special structure of the system.…”

Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model

“…Miao and Chen [3] obtain the existence and uniqueness of a small global solution in the Besov spaces B s 2,∞ (s > d 2 ) by a regularization method in conjunction with Bony's paraproduct decomposition. Chen and Hao [4] prove the global well-posedness in the critical Besov spaces without damping mechanism for the small initial data. Recently, Zhai [28] obtain the global solutions for a class of highly oscillating initial velocity by using the special structure of the system, and also established the optimal timedecay of solutions by a new energy approach involved in the high frequencies and low frequencies decomposition in the Besov spaces(see [29]).…”

High Reynolds number and high Weissenberg number Oldroyd-B model with dissipation

“…A rich array of results have been established on the well-posedness and closely related problems. Interested readers can consult some of the references listed here, see, e.g., [1,3,4,5,6,8,9,11,12,13,14,15,16,17,22,18,19,23,24,25,27,28,29,30,31,32,33,34]. This list is by no means exhaustive.…”