The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien

Abstract

In this paper, we consider the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid in two space dimensions. We focus upon the sharp time decay rates (upper and lower bounds) and global-in-time stability of large strong solutions for the PTT system with magnetic field. Firstly, the convergence of large solutions to the equilibrium have been investigated and these convergence rates are shown to be sharp. We then show that two large solutions converge globally in time as long as two initial data are close to each other. One of the main objectives of this paper is to develop a way to capture $L^{2}$ -convergence result via auxiliary logarithmic time decay estimates with the initial data in $L^{p} (R^{2}) \cap L^{2} (R^{2})$ . Improving time decay rates for the high-order derivatives of large solutions by using interpolation inequalities. In addition, time-weighted energy estimate, Fourier time-splitting method, semigroup method and iterative scheme have also been utilized.

Keywords

Viscoelastic fluids sharp time decay rates global-in-time stability large solutions

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References

Abidi ,

Gui and

Zhang , On the decay and stability of global solutions to the 3D inhomogeneous Navier–Stokes equations, Comm. Pure Appl. Math. 64 (2011), 832–881. doi:10.1002/cpa.20351.

Abidi and

Paicu , Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A 138(2008) (2008), 447–476. doi:10.1017/S0308210506001181.

Cao ,

Regmi and

Wu , The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differential Equations 254 (2013), 2661–2681. doi:10.1016/j.jde.2013.01.002.

Cao and

Wu , Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math. 226 (2011), 1803–1822. doi:10.1016/j.aim.2010.08.017.

J.-Y.

Chemin and

Masmoudi , About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal. 33 (2001), 84–112. doi:10.1137/S0036141099359317.

Chen and

Miao , Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces, Nonlinear Anal. 68 (2008), 1928–1939. doi:10.1016/j.na.2007.01.042.

Chen ,

Li ,

Yao and

Yao , Global well-posedness for the three-dimensional generalized Phan-Thien–Tanner model in critical Besov spaces, J. Math. Fluid Mech. 23 (2021), 55. doi:10.1007/s00021-021-00578-y.

Chen ,

Li ,

Yao and

Yao , Global-in-time stability and sharp rates of decay of large solutions to incompressible Phan-Thien–Tanner system in

R^{3}

, Preprint.

Chen ,

Luo and

Yao , Blow up and global existence for the periodic Phan-Thein–Tanner model, J. Differential Equations 267 (2019), 6758–6782. doi:10.1016/j.jde.2019.07.005.

10.

Chen ,

Luo and

Zhai , Global well-posedness for the Phan-Thein–Tanner model in critical Besov spaces without damping, J. Math. Physics 60 (2019), 061503. doi:10.1063/1.5094086.

11.

Chen and

Zhang , The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differential Equations 31 (2006), 1793–1810. doi:10.1080/03605300600858960.

12.

Ducomet and

Feireisl , The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. 266 (2006), 595–629. doi:10.1007/s00220-006-0052-y.

13.

Duvaut and

J.L.

Lions , Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal. 46 (1972), 241–279. doi:10.1007/BF00250512.

14.

Fang and

Zi , Global solutions to the Oldroyd-B model with a class of large initial data, SIAM J. Math. Anal. 48 (2016), 1054–1084. doi:10.1137/15M1037020.

15.

Feng ,

Zhu and

Zi , Blow-up criterion for the incompressible viscoelastic flows, J. Funct. Anal. 272 (2017), 3742–3762. doi:10.1016/j.jfa.2016.10.024.

16.

Fernández-Cara ,

Guillén and

R.R.

Ortega , Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 26 (1998), 1–29.

17.

Gui ,

Huang and

Zhang , Large global solutions to 3-D inhomogeneous Navier–Stokes equations slowly varying in one variable, J. Funct. Anal. 261 (2011), 3181–3210. doi:10.1016/j.jfa.2011.07.026.

18.

Gui and

Zhang , Stability to the global large solutions of 3-D Navier–Stokes equations, Adv. Math. 225 (2010), 1248–1284. doi:10.1016/j.aim.2010.03.022.

19.

Guillopé and

J.-C.

Saut , Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal. 15 (1990), 849–869. doi:10.1016/0362-546X(90)90097-Z.

20.

Guillopé and

J.-C.

Saut , Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Modél. Math. Anal. Numér. 24 (1990), 369–401. doi:10.1051/m2an/1990240303691.

21.

He ,

Huang and

Wang , Global stability of large solutions to the 3D compressible Navier–Stokes equations, Arch. Ration. Mech. Anal 234 (2019), 1167–1222. doi:10.1007/s00205-019-01410-8.

22.

Hieber ,

Wen and

Zi , Optimal decay rates for solutions to the incompressible Oldroyd-B model in

R^{3}

, Nonlinearity 32 (2019), 833–852. doi:10.1088/1361-6544/aaeec7.

23.

Hu and

Wang , Global solutions to the three-dimensional full compressible magnetohydroynamics flows, Comm. Math. Phys. 283 (2008), 253–284. doi:10.1007/s00220-008-0497-2.

24.

Hu and

Wang , Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal. 197 (2010), 203–238. doi:10.1007/s00205-010-0295-9.

25.

Hu and

Wang , Strong solutions to the three-dimensional compressible viscoelastic fluids, J. of Differential Equations 252 (2012), 4027–4067. doi:10.1016/j.jde.2011.11.021.

26.

Hu and

G.C.

Wu , Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal. 45 (2013), 2815–2833. doi:10.1137/120892350.

27.

Huang and

Li , Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier–Stokes and magnetohydrodynamic flows, Comm. Math. Phys. 324 (2013), 147–171. doi:10.1007/s00220-013-1791-1.

28.

Jiang and

Jiang , Strong solutions of the equations for viscoelastic fluids in some classes of large data, J. Differential Equations 282 (2021), 148–183. doi:10.1016/j.jde.2021.02.020.

29.

Kawashima , Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics, Japan J. Appl. Math. 1 (1984), 207–222. doi:10.1007/BF03167869.

30.

Lei ,

Liu and

Zhou , Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal. 188 (2008), 371–398. doi:10.1007/s00205-007-0089-x.

31.

Lei ,

Masmoudi and

Zhou , Remarks on the blowup criteria for Oldroyd models, J. Differential Equations 248 (2010), 328–341. doi:10.1016/j.jde.2009.07.011.

32.

Lin ,

Liu and

Zhang , On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math. 58 (2005), 1437–1471. doi:10.1002/cpa.20074.

33.

Lin and

Zhang , Global small solutions to MHD type system (I): 3-D case, Comm. Pure Appl. Math. 67 (2014), 531–580. doi:10.1002/cpa.21506.

34.

P.L.

Lions and

Masmoudi , Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B 21 (2000), 131–146. doi:10.1142/S0252959900000170.

35.

Lu and

Pokorný , Global existence of large data weak solutions for a simplified compressible Oldroyd-B model without stress diffusion, Anal. Theory Appl 36 (2020), 348–372. doi:10.4208/ata.OA-SU3.

36.

Lu and

Zhang , Relative entropy, weak-strong uniqueness, and conditional regularity for a compressible Oldroyd-B model, SIAM J. Math. Anal 50 (2018), 557–590. doi:10.1137/17M1128654.

37.

Paicu and

Zhang , Global solutions to the 3-D incompressible inhomogeneous Navier–Stokes system, J. Funct. Anal. 262 (2012), 3556–3584. doi:10.1016/j.jfa.2012.01.022.

38.

Pan ,

Zhou and

Zhu , Global classical solutions of three dimensional viscous MHD system without magnetic diffusion on periodic boxes, Arch. Ration. Mech. Anal. 227 (2018), 637–662. doi:10.1007/s00205-017-1170-8.

39.

Phan-Thien , A nonlinear network viscoelastic model, J. Rheol. 22 (1978), 259–283. doi:10.1122/1.549481.

40.

Phan-Thien and

R.I.

Tanner , A new constitutive equation derived from network theory, J. Non-Newton. Fluid Mech. 2 (1977), 353–365. doi:10.1016/0377-0257(77)80021-9.

41.

Qian and

Zhang , Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal. 198 (2010), 835–868. doi:10.1007/s00205-010-0351-5.

42.

Ren ,

Wu ,

Xiang and

Zhang , Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal. 267 (2014), 503–541. doi:10.1016/j.jfa.2014.04.020.

43.

Schonbek ,

L^{2}

decay for weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 88 (1985), 209–222. doi:10.1007/BF00752111.

44.

Sermange and

Temam , Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36 (1983), 635–664. doi:10.1002/cpa.3160360506.

45.

Wang and

Wen , The Cauchy problem for an Oldroyd-B model in three dimensions, Math. Models Methods Appl. Sci. 30 (2020), 139–179. doi:10.1142/S0218202520500049.

46.

Zhang and

Fang , Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical

L^{p}

framework, SIAM J. Math. Anal. 44 (2012), 2266–2288. doi:10.1137/110851742.

47.

Zhou ,

Zhu and

Zi , Global well-posedness and decay rates for the three dimensional compressible Oldroyd-B model, J. Differential Equations 265 (2018), 1259–1278. doi:10.1016/j.jde.2018.04.003.

48.

Zi ,

Fang and

Zhang , Global solution to the incompressible Oldroyd-B model in the critical

L^{p}

framework: The case of the non-small coupling parameters, Arch. Ration. Mech. Anal. 213 (2014), 651–687. doi:10.1007/s00205-014-0732-2.

The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien–Tanner system with magnetic field

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