Global solutions to the $n$-dimensional incompressible Oldroyd-B model without damping mechanism

Abstract

In this paper, we obtain the global solutions to the incompressible Oldroyd-B model without damping on the stress tensor in R n (n = 2, 3). This result allows to construct global solutions for a class of highly oscillating initial velocity. The proof uses the special structure of the system. Moreover, our theorem extends the previous result by Zhu [19] and covers the recent result by Chen and Hao [4].

Global wellposedness to the $n$-dimensional compressible Oldroyd-B model without damping mechanism

“…Remark 1.4. Let ρ be a constant, (1.1) reduces to the incompressible Oldroyd-B model without damping mechanism, the above theorem generalizes the Theorem 1.2 in [6] and coincides with the Theorem 1.2 in [40].…”

“…For the incompressible Oldroyd-B model without damping mechanism on the stress tensor, Zhu in [42] obtained the global small solutions in R 3 , by constructing two special time-weighted energies. This result was further generalized for a more general dimension by Chen and Hao [6] and Zhai [40], in the critical L 2 Besov spaces and critical L p Besov spaces, respectively. Recently, Zhu [43] and Pan et al [33] obtained the global small solutions to compressible viscoelastic flows without structure assumption, in R 3 with Sobolev initial data and in R n with Besov initial data, respectively.…”

“…From [6], [40], one can get the smoothing effect of (Pu, Λ −1 Pdiv τ) in the low frequencies, the smoothing effect of Pu and the damping effect of Λ −1 Pdiv τ in the high frequencies.…”

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.…”

Global well-posedness for the Phan-Thein-Tanner model in critical Besov spaces without damping

“…Recently, Q. Chen and X. Hao [5] and X. Zhai [28] study about the global well-posedness in the critical Besov spaces respectively. For the Oldroyd-B model, the global existence of strong solutions in two dimension without small conditions is still an open problem.…”