Global Well-posedness for the Three-Dimensional Generalized Phan–Thien–Tanner Model in Critical Besov Spaces
Tóm tắt
A new Generalized Phan–Thien–Tanner (GPTT) model is derived from a Lodge–Yamamoto type of network theory for the polymeric fluids. The GPTT model is developed to describe the rheological behavior of the viscoelastic fluids. In this paper, we investigate the initial-value problem for the GPTT model. Under the norm of the initial data is a small perturbation around some particular solution, we prove that the strong solution exists globally in critical Besov spaces.
Tài liệu tham khảo
Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften, vol. 343. Springer, Heidelberg (2011)
Bautista, O., Sánchez, S., Arcos, J.C., Méndez, F.: Lubrication theory for electro-osmotic flow in a slit microchannel with the Phan-Thien and Tanner model. J. Fluid Mech. 722, 496–532 (2013)
Bird, R.B., Armstrong, R.C., Hassager, O.: Dynamics of Polymeric Liquids, vol. 1. Wiley, New York (1977)
Chemin, J.-Y., Masmoudi, N.: About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33(1), 84–112 (2001)
Chen, Q., Hao, X.: Global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism. J. Math. Fluid Mech., 21(3), (2019)
Chen, Q., Miao, C., Zhang, Z.: Global well-posedness for compressible Navier–Stokes equations with highly oscillating initial velocity. Commun. Pure Appl. Math. 63(9), 1173–1224 (2010)
Chen, Q., Miao, C., Zhang, Z.: Well-posedness in critical spaces for the compressible Navier–Stokes equations with density dependent viscosities. Rev. Mat. Iberoam. 26(3), 915–946 (2010)
Chen, Q., Miao, C., Zhang, Z.: On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces. Rev. Mat. Iberoam. 31(4), 1375–1402 (2015)
Chen, Y., Luo, W., Yao, Z.-A.: Blow up and global existence for the periodic Phan–Thein–Tanner model. J. Differ. Equ. 267(11), 6758–6782 (2019)
Chen, Y., Luo, W., Zhai, X.: Global well-posedness for the Phan–Thein–Tanner model in critical Besov spaces without damping. J. Math. Phys. 60(6), 061503 (2019)
Danchin, R.: Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141(3), 579–614 (2000)
Danchin, R.: Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch. Ration. Mech. Anal. 160(1), 1–39 (2001)
Danchin, R., He, L.: The incompressible limit in \(L^p\) type critical spaces. Math. Ann. 366(3–4), 1365–1402 (2016)
Fang, D., Hieber, M., Zi, R.: Global existence results for Oldroyd-B fluids in exterior domains: the case of non-small coupling parameters. Math. Ann. 357(2), 687–709 (2013)
Fang, D., Zi, R.: Global solutions to the Oldroyd-B model with a class of large initial data. SIAM J. Math. Anal. 48(2), 1054–1084 (2016)
Fernández-Cara, E., Guillén, F., Ortega, R.. R.: Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(1), 1–29 (1998)
Garduño, I.E., Tamaddon-Jahromi, H.R., Walters, K., Webster, M.F.: The interpretation of a long-standing rheological flow problem using computational rheology and a PTT constitutive model. J. Non-Newton. Fluid Mech. 233, 27–36 (2016)
Guillopé, C., Saut, J.-C.: Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15(9), 849–869 (1990)
Guillopé, C., Saut, J.-C.: Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type. RAIRO Modél. Math. Anal. Numér. 24(3), 369–401 (1990)
Lei, Z., Liu, C., Zhou, Y.: Global solutions for incompressible viscoelastic fluids. Arch. Ration. Mech. Anal. 188(3), 371–398 (2008)
Lei, Z., Zhou, Y.: Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit. SIAM J. Math. Anal. 37(3), 797–814 (2005)
Lin, F.-H., Liu, C., Zhang, P.: On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58(11), 1437–1471 (2005)
Lions, P.L., Masmoudi, N.: Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math. Ser. B 21(2), 131–146 (2000)
Mu, Y., Zhao, G., Chen, A., Wu, X.: Modeling and simulation of three-dimensional extrusion swelling of viscoelastic fluids with PTT, Giesekus and FENE-P constitutive models. Int. J. Numer. Methods Fluids 72(8), 846–863 (2013)
Mu, Y., Zhao, G., Wu, X., Zhai, J.: Modeling and simulation of three-dimensional planar contraction flow of viscoelastic fluids with PTT, Giesekus and FENE-P constitutive models. Appl. Math. Comput. 218(17), 8429–8443 (2012)
Oldroyd, J.G.: Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. R. Soc. Lond. Ser. A 245, 278–297 (1958)
Oliveira, P.J., Pinho, F.T.: Analytical solution for fully developed channel and pipe flow of Phan–Thien–Tanner fluids. J. Fluid Mech. 387, 271–280 (1999)
Phan-Thien, N.: A nonlinear network viscoelastic model. J. Rheol. 22(3), 259–283 (1978)
Phan-Thien, N., Tanner, R.I.: A new constitutive equation derived from network theory. J. Nonnewton. Fluid Mech. 2(4), 353–365 (1977)
Zhai, X.: Global solutions to the \(n\)-dimensional incompressible Oldroyd-B model without damping mechanism. arXiv:1810.08048, (2018)
Zhang, T., Fang, D.: Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical \(L^p\) framework. SIAM J. Math. Anal. 44(4), 2266–2288 (2012)
Zhu, Y.: Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism. J. Funct. Anal. 274(7), 2039–2060 (2018)