Global unique solutions for the 2-D inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion

Xi Wang; Xueli Ke; Xi Wang; Xueli Ke

doi:10.3934/math.20241443

AIMS Mathematics

2024, Volume 9, Issue 11: 29806-29819. doi: 10.3934/math.20241443

Previous Article Next Article

Research article Special Issues

Global unique solutions for the 2-D inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion

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: 07 September 2024 Revised: 13 October 2024 Accepted: 15 October 2024 Published: 21 October 2024
MSC : 76A15, 35Q35, 35D30

We establish the global unique solution for the 2-D inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion by employing the standard energy method and the standard compactness arguments.
- viscoelastic rate-type fluids,
- energy method,
- global unique solutions,
- Friedrich's method,
- compactness argument
Citation: Xi Wang, Xueli Ke. Global unique solutions for the 2-D inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion[J]. AIMS Mathematics, 2024, 9(11): 29806-29819. doi: 10.3934/math.20241443

Related Papers:

Abstract

We establish the global unique solution for the 2-D inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion by employing the standard energy method and the standard compactness arguments.

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]	H. Bahouri, J. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations, Springer, Berlin Heidelberg, 2011.
[3]	M. Bulí$\breve{\rm{c}}$ek, E. Feireisl, J. Málek, On a class of compressible viscoelastic rate-type fluids with stress-diffusion, Nonlinearity, 32 (2019), 4665–4681. https://doi.org/10.1088/1361-6544/ab3614 doi: 10.1088/1361-6544/ab3614
[4]	M. Bulí$\breve{\rm{c}}$ek, J. Málek, V. Pru$\breve{\rm{s}}$a, E. Süli, PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion, Amer. Math. Soc., 710 (2018), 25–51. https://doi.org/10.1090/conm/710/14362 doi: 10.1090/conm/710/14362
[5]	M. Bulí$\breve{\rm{c}}$ek, J. Málek, C. Rodriguez, Global well-posedness for two-dimensional flows of viscoelastic rate-type fluids with stress diffusion, J. Math. Fluid Mech., 24 (2022), 24–61. https://doi.org/10.1007/s00021-022-00696-1 doi: 10.1007/s00021-022-00696-1
[6]	J. Chemin, Perfect incompressible fluids, Oxford University Press, New York, 1998.
[7]	R. Danchin, Local and global well-posedness reuslts for flows of inhomogeneous viscous fluids, Adv. Differ. Equ., 9 (2004), 353–386. https://doi.org/10.57262/ade/1355867948 doi: 10.57262/ade/1355867948
[8]	R. Danchin, The inviscid limit for density-dependent incompressible fluids, Annal. Fac. Sci. Toulouse Math., 15 (2006), 637–688. https://doi.org/10.5802/afst.1133 doi: 10.5802/afst.1133
[9]	R. Danchin, P. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Commun. Pur. Appl. Math., 65 (2012), 1458–1480. https://doi.org/10.1002/cpa.21409 doi: 10.1002/cpa.21409
[10]	R. Danchin, P. Mucha, Incompressible flows with piecewise constant density, Arch. Ration. Mech. Anal., 207 (2013), 991–1023. https://doi.org/10.1007/s00205-012-0586-4 doi: 10.1007/s00205-012-0586-4
[11]	G. Gui, Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity, J. Funct. Anal., 267 (2014), 1488–1539. https://doi.org/10.1016/j.jfa.2014.06.002 doi: 10.1016/j.jfa.2014.06.002
[12]	H. Kozono, T. Ogawa, Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some sem-linear evolution equations, Math. Z., 242 (2002), 251–278. https://doi.org/10.1007/s002090100332 doi: 10.1007/s002090100332
[13]	A. Majda, A. Bertozzi, Vorticity and incompressible flow, Cambridge University Press, Cambridge, 2002.
[14]	P. Marius, P. Zhang, Striated regularity of 2-D inhomogeneous incompressible Navier-Stokes system with variable viscosity, Commun. Math. Phys., 376 (2020), 385–439. https://doi.org/10.1007/s00220-019-03446-z doi: 10.1007/s00220-019-03446-z
[15]	J. Málek, V. Pru$\breve{\rm{s}}$a, T. Sk$\breve{\rm{r}}$ivan, E. Süli, Thermodynamics of viscoelastic rate-type fluids with stress diffusion, Phys. Fluids, 30 (2018), 023101. https://doi.org/10.1063/1.5018172 doi: 10.1063/1.5018172
[16]	G. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co. Inc., 1996. https://doi.org/10.1142/3302
[17]	H. Triebel, Theory of function spaces, Monogr. Math., Birkhäuser Verlag, Basel, Boston, 1983. https://doi.org/10.1007/978-3-0346-0416-1
[18]	C. Wang, Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43–62. https://doi.org/10.1016/j.aim.2011.05.008 doi: 10.1016/j.aim.2011.05.008

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)