Global solutions and large time behavior for some Oldroyd-B type models in $R^2$

Abstract

In this paper, we are concerned with global solutions to the co-rotation Oldroyd-B type model and large time behavior for the general Oldroyd-B type model. We first establish the energy estimate and B-K-M criterion for the 2-D co-rotation Oldroyd-B type model. Then, we obtain global solutions by proving the boundedness of vorticity. In general case, we apply Fourier spiltting method to prove the H 1 decay rate for global solutions constructed in [7].

Global Regularity and instability for the incompressible non-viscous Oldroyd-B model

“…This implies the global existence for some large initial data. Combining Remark 1.1, we obtain the global large solutions for (1.2) with a = 0 in critical Besov spaces, which implies some additions and improvements to [16,14].…”

“…Recently, Deng, Luo and Yin [14] obtain the global small solutions for (1.2) with Q(τ, ∇u) = τ Ω(u) − Ω(u)τ + b(D(u)τ + τ D(u)), b = 0, and give an estimation of the H 1 decay for the global solutions with any b ∈ [−1, 1] . Combining their results, and after some calculation, one can get the following theorem: Theorem 1.1.…”

“…Combining their results, and after some calculation, one can get the following theorem: Theorem 1.1. [16,14]…”

“…Then, combing the critical estimation of the transport equation in Besov space and the bootstrap argument, we obtain the global existence of (1.2). Compared with the results in [16,14], we needn't the small condition of u 0 , τ 0 Ḣ1 . Finally, we obtain the large time behavior such that τ (t)…”