Stability and decay analysis of stress-diffusive viscoelastic rate fluids

Xi Wang; Xueli Ke; Xi Wang; Xueli Ke

doi:10.3934/math.20251153

AIMS Mathematics

2025, Volume 10, Issue 11: 26187-26236. doi: 10.3934/math.20251153

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Stability and decay analysis of stress-diffusive viscoelastic rate fluids

Xi Wang ¹,
Xueli Ke ^{2
,
,}

1.
School of Mathematics, Northwest University, Xi'an 710127, China
2.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

Received: 18 September 2025 Revised: 30 October 2025 Accepted: 06 November 2025 Published: 13 November 2025
MSC : 76A15, 35Q35, 35D30

This paper focused on a class of three-dimensional incompressible viscoelastic rate-type fluids with stress diffusion. To simplify the analysis, we considered a model where the elastic stress was spherical. It is worth noting that while the existence of solutions for such fluids has been studied, their stability properties remain largely unexplored. So inspired by ^[1], this paper employed the bootstrap argument, the Schonbek's method, and repeated use of Besov space properties to prove, for the first time, stability of the solution under some additional conditions on the initial data — but without further smallness restrictions. Our results showed that the velocity decays faster than the typical algebraic rate, while the spherical component of the elastic strain tensor exhibited global exponential decay. Finally, using the decay rates, we derived the stability result for any given globally smooth solution—namely, that a sufficiently small perturbation yielded a unique globally smooth solution which stayed close to the original reference solution. We thereby extended the analysis of ^[1] on inhomogeneous Navier-Stokes equations to a viscoelastic fluid with stress diffusion.
- viscoelastic fluid,
- Littlewood-Paley theory,
- decay rate,
- Besov space
Citation: Xi Wang, Xueli Ke. Stability and decay analysis of stress-diffusive viscoelastic rate fluids[J]. AIMS Mathematics, 2025, 10(11): 26187-26236. doi: 10.3934/math.20251153

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Abstract

This paper focused on a class of three-dimensional incompressible viscoelastic rate-type fluids with stress diffusion. To simplify the analysis, we considered a model where the elastic stress was spherical. It is worth noting that while the existence of solutions for such fluids has been studied, their stability properties remain largely unexplored. So inspired by ^[1], this paper employed the bootstrap argument, the Schonbek's method, and repeated use of Besov space properties to prove, for the first time, stability of the solution under some additional conditions on the initial data — but without further smallness restrictions. Our results showed that the velocity decays faster than the typical algebraic rate, while the spherical component of the elastic strain tensor exhibited global exponential decay. Finally, using the decay rates, we derived the stability result for any given globally smooth solution—namely, that a sufficiently small perturbation yielded a unique globally smooth solution which stayed close to the original reference solution. We thereby extended the analysis of ^[1] on inhomogeneous Navier-Stokes equations to a viscoelastic fluid with stress diffusion.

References

[1]	H. Abidi, G. Gui, P. Zhang, On the decay and stability of global solutions to the 3-D inhomogeneous Navier-Stokes equations, Commun. Pure Appl. Math., 64 (2011), 832–881. https://doi.org/10.1002/cpa.20351 doi: 10.1002/cpa.20351
[2]	J. K. G. Dhont, W. J. Briels, Gradient and vorticity banding, Rheol. Acta., 47 (2008), 257–281. https://doi.org/10.1007/s00397-007-0245-0 doi: 10.1007/s00397-007-0245-0
[3]	M. A. Fardin, T. J. Ober, C. Gay, G. Grégoire, G. H. McKinley, S. Lerouge, Potential "ways of thinking" about the shear-banding phenomenon, Soft Matter, 8 (2011), 910–922. https://doi.org/10.1039/C1SM06165H doi: 10.1039/C1SM06165H
[4]	T. Divoux, M. A. Fardin, S. Manneville, S. Lerouge, Shear banding of complex fluids, Annu. Rev. Fluid Mech., 48 (2016), 81–103. https://doi.org/10.1146/annurev-fluid-122414-034416 doi: 10.1146/annurev-fluid-122414-034416
[5]	L. He, J. Huang, C. Wang, Global stability of large solutions to the 3D compressible Navier-Stokes equations, Arch. Rational Mech. Anal., 234 (2019), 1167–1222. https://doi.org/10.1007/s00205-019-01410-8 doi: 10.1007/s00205-019-01410-8
[6]	G. Gui, P. Zhang, Stability to the global large solutions of 3-D Navier-Stokes equations, Adv. Math., 225 (2010), 1248–1284. https://doi.org/10.1016/j.aim.2010.03.022 doi: 10.1016/j.aim.2010.03.022
[7]	B. Dong, J. Wu, X. Xu, N. Zhu, Stability and exponential decay for the 2D anisotropic Navier-Stokes equations with horizontal dissipation, J. Math. Fluid Mech., 23 (2021), 100. https://doi.org/10.1007/s00021-021-00617-8 doi: 10.1007/s00021-021-00617-8
[8]	M. Bulíček, E. Feireisl, J. Málek, On a class of compressible viscoelastic rate-type fluids with stress-diffusion, Nonlinearity, 32 (2019), 4665. https://doi.org/10.1088/1361-6544/ab3614 doi: 10.1088/1361-6544/ab3614
[9]	C. Ai, Z. Tan, J. Zhou, Global existence and decay estimate of solution to rate type viscoelastic fluids, J. Differ. Equations, 377 (2023), 188–220. https://doi.org/10.1016/j.jde.2023.08.039 doi: 10.1016/j.jde.2023.08.039
[10]	W. Wang, H. Wen, The Cauchy problem for an Oldroyd-B model in three dimensions, Math. Mod. Meth. Appl. Sci., 30 (2020), 139–179. https://doi.org/10.1142/S0218202520500049 doi: 10.1142/S0218202520500049
[11]	P. Wang, J. Wu, X. Xu, Y. Zhong, Sharp decay estimates for Oldroyd-B model with only fractional stress tensor diffusion, J. Funct. Anal., 282 (2022), 109332. https://doi.org/10.1016/j.jfa.2021.109332 doi: 10.1016/j.jfa.2021.109332
[12]	G. M. Moatimid, Y. M. Mohamed, Nonlinear electro-rheological instability of two moving cylindrical fluids: an innovative approach, Phys. Fluids, 36 (2024), 024110. https://doi.org/10.1063/5.0188061 doi: 10.1063/5.0188061
[13]	G. M. Moatimid, Y. M. Mohamed, A novel methodology in analyzing nonlinear stability of two electrified viscoelastic liquids, Chin. J. Phys., 89 (2024), 679–706. https://doi.org/10.1016/j.cjph.2023.12.030 doi: 10.1016/j.cjph.2023.12.030
[14]	M. Bulíček, T. Los, Y. Lu, J. Málek, On planar flows of viscoelastic fluids of Giesekus type, Nonlinearity, 35 (2022), 6557. https://doi.org/10.1088/1361-6544/ac9a2c doi: 10.1088/1361-6544/ac9a2c
[15]	M. Bulíček, J. Málek, V. Pruša, E. Süli, PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion, Contemp. Math., 710 (2018), 25–51. https://doi.org/10.1090/conm/710/14362 doi: 10.1090/conm/710/14362
[16]	M. Bulíček, J. Málek, V. Pruša, E. Süli, On incompressible heat-conducting viscoelastic rate-type fluids with stress-diffusion and purely spherical elastic response, SIAM J. Math. Anal., 53 (2021), 3985–4030. https://doi.org/10.1137/20M1384452 doi: 10.1137/20M1384452
[17]	M. Bulíček, J. Woźnick, On various models describing behaviour of thermoviscoelastic rate-type fluids, arXiv, 2025. https://doi.org/10.48550/arXiv.2502.03735
[18]	H. Triebel, Theory of function spaces, Birkhäuser Basel, 1983. https://doi.org/10.1007/978-3-0346-0416-1
[19]	J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Ec. Norm. Super., 14 (1980), 209–246. https://doi.org/10.24033/asens.1404 doi: 10.24033/asens.1404
[20]	H. Bahouri, J. Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations, Springer, 2011. https://doi.org/10.1007/978-3-642-16830-7
[21]	J. Y. Chemin, Théorémes d'unicité pour le systéme de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27–50. https://doi.org/10.1007/BF02791256 doi: 10.1007/BF02791256
[22]	J. Y. Chemin, Perfect incompressibe fluids, Oxford University Press, 1998.
[23]	J. L. Lions, Methods of solution of nonlinear boundary-values problems, Mir, Moscow, 1972.
[24]	M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Commun. Part. Diff. Eq., 11 (1986), 733–763. https://doi.org/10.1080/03605308608820443 doi: 10.1080/03605308608820443
[25]	R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353–386. https://doi.org/10.57262/ade/1355867948 doi: 10.57262/ade/1355867948

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