Skip to main content
SpringerLink
Log in

Vanishing viscosity limits for the free boundary problem of compressible viscoelastic fluids with surface tension

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate the regularity of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of the flow map so that the inherent structure of the elastic term improves the uniform regularity of normal derivatives in the limit of vanishing viscosity. This result indicates that the boundary layer does not appear in the free boundary problem of compressible viscoelastic fluids, which is different from the case studied by Mei et al. (2018) for the free boundary compressible Navier-Stokes system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cai Y, Lei Z, Lin F H, et al. Vanishing viscosity limit for incompressible viscoelasticity in two dimensions. Comm Pure Appl Math, 2019, 72: 2063–2120

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen Y M, Zhang P. The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions. Comm Partial Differential Equations, 2016, 31: 1793–1810

    Article  MathSciNet  MATH  Google Scholar 

  3. Ciampa G, Crippa G, Spirito S. Strong convergence of the vorticity for the 2D Euler equations in the inviscid limit. Arch Ration Mech Anal, 2021, 240: 295–326

    Article  MathSciNet  MATH  Google Scholar 

  4. Clopeau T, Mikelic A, Robert R. On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions. Nonlinearity, 1998, 11: 1625–1636

    Article  MathSciNet  MATH  Google Scholar 

  5. Constantin P, Drivas T D, Elgindi T. Inviscid limit of vorticity distributions in the Yudovich class. Comm Pure Appl Math, 2022, 75: 65–82

    Article  MathSciNet  MATH  Google Scholar 

  6. Constantin P, Wu J H. Inviscid limit for vortex patches. Nonlinearity, 1995, 8: 735–742

    Article  MathSciNet  MATH  Google Scholar 

  7. Di Iorio E, Marcati P, Spirito S. Splash singularities for a 2D Oldroyd-B model with nonlinear Piola-Kirchhoff stress. NoDEA Nonlinear Differential Equations Appl, 2017, 24: 60

    Article  MathSciNet  MATH  Google Scholar 

  8. Di Iorio E, Marcati P, Spirito S. Splash singularity for a free-boundary incompressible viscoelastic fluid model. Adv Math, 2020, 368: 107124

    Article  MathSciNet  MATH  Google Scholar 

  9. Di Iorio E, Marcati P, Spirito S. Splash singularities for a general Oldroyd model with finite Weissenberg number. Arch Ration Mech Anal, 2020, 235: 1589–1660

    Article  MathSciNet  MATH  Google Scholar 

  10. Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136: 521–573

    Article  MathSciNet  MATH  Google Scholar 

  11. Elgindi T, Lee D. Uniform regularity for free-boundary Navier-Stokes equations with surface tension. J Hyperbolic Differ Equ, 2018, 15: 37–118

    Article  MathSciNet  MATH  Google Scholar 

  12. Fei M W, Tao T, Zhang Z F. On the zero-viscosity limit of the Navier-Stokes equations in ℝ 3+ without analyticity. J Math Pures Appl (9), 2018, 112: 170–229

    Article  MathSciNet  MATH  Google Scholar 

  13. Filho M C L, Lopes H J N, Planas G. On the inviscid limit for two-dimensional incompressible flow with Navier friction condition. SIAM J Math Anal, 2005, 36: 1130–1141

    Article  MathSciNet  MATH  Google Scholar 

  14. Gallay T. Interaction of vortices in weakly viscous planar flows. Arch Ration Mech Anal, 2011, 200: 445–490

    Article  MathSciNet  MATH  Google Scholar 

  15. Gu X M, Lei Z. Local well-posedness of free-boundary incompressible elastodynamics with surface tension via vanishing viscosity limit. Arch Ration Mech Anal, 2022, 245: 1285–1338

    Article  MathSciNet  MATH  Google Scholar 

  16. Gu X M, Wang F. Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition. J Math Anal Appl, 2020, 482: 123529

    Article  MathSciNet  MATH  Google Scholar 

  17. Hao C C, Wang D H. A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamics. J Differential Equations, 2016, 261: 712–737

    Article  MathSciNet  MATH  Google Scholar 

  18. Hu X P, Huang Y T. Well-posedness of the free boundary problem for incompressible elastodynamics. J Differential Equations, 2019, 266: 7844–7889

    Article  MathSciNet  MATH  Google Scholar 

  19. Iftimie D, Planas G. Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity, 2006, 19: 899–918

    Article  MathSciNet  MATH  Google Scholar 

  20. Iftimie D, Sueur F. Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch Ration Mech Anal, 2011, 199: 145–175

    Article  MathSciNet  MATH  Google Scholar 

  21. Kato T. Nonstationary flows of viscous and ideal fluids in ℝ3. J Funct Anal, 1972, 9: 296–305

    Article  MathSciNet  MATH  Google Scholar 

  22. Le Meur H V J. Well-posedness of surface wave equations above a viscoelastic fluid. J Math Fluid Mech, 2011, 13: 481–514

    Article  MathSciNet  MATH  Google Scholar 

  23. Lei Z. Global well-posedness of incompressible elastodynamics in two dimensions. Comm Pure Appl Math, 2016, 69: 2072–2106

    Article  MathSciNet  MATH  Google Scholar 

  24. Lei Z, Liu C, Zhou Y. Global solutions for incompressible viscoelastic fluids. Arch Ration Mech Anal, 2008, 188: 371–398

    Article  MathSciNet  MATH  Google Scholar 

  25. Li H, Wang W, Zhang Z F. Well-posedness of the free boundary problem in incompressible elastodynamics. J Differential Equations, 2019, 267: 6604–6643

    Article  MathSciNet  MATH  Google Scholar 

  26. Lin F H, Liu C, Zhang P. On hydrodynamics of viscoelastic fluids. Comm Pure Appl Math, 2005, 58: 1437–1471

    Article  MathSciNet  MATH  Google Scholar 

  27. Lin F H, Zhang P. On the initial-boundary value problem of the incompressible viscoelastic fluid system. Comm Pure Appl Math, 2008, 61: 539–558

    Article  MathSciNet  MATH  Google Scholar 

  28. Maekawa Y. On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Comm Pure Appl Math, 2014, 67: 1045–1128

    Article  MathSciNet  MATH  Google Scholar 

  29. Masmoudi N, Rousset F. Uniform regularity for the Navier-Stokes equation with Navier boundary condition. Arch Ration Mech Anal, 2012, 203: 529–575

    Article  MathSciNet  MATH  Google Scholar 

  30. Masmoudi N, Rousset F. Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations. Arch Ration Mech Anal, 2017, 223: 301–417

    Article  MathSciNet  MATH  Google Scholar 

  31. McGrath F J. Nonstationary plane flow of viscous and ideal fluids. Arch Ration Mech Anal, 1968, 27: 329–348

    Article  MathSciNet  MATH  Google Scholar 

  32. Mei Y, Wang Y, Xin Z P. Uniform regularity for the free surface compressible Navier-Stokes equations with or without surface tension. Math Models Methods Appl Sci, 2018, 28: 259–336

    Article  MathSciNet  MATH  Google Scholar 

  33. Nguyen T T, Nguyen T T. The inviscid limit of Navier-Stokes equations for analytic data on the half-space. Arch Ration Mech Anal, 2018, 230: 1103–1129

    Article  MathSciNet  MATH  Google Scholar 

  34. Paddick M. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete Contin Dyn Syst, 2016, 36: 2673–2709

    Article  MathSciNet  MATH  Google Scholar 

  35. Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Comm Math Phys, 1998, 192: 433–461

    Article  MathSciNet  MATH  Google Scholar 

  36. Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution. Comm Math Phys, 1998, 192: 463–491

    Article  MathSciNet  MATH  Google Scholar 

  37. Sideris T C, Thomases B. Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit. Comm Pure Appl Math, 2005, 58: 750–788

    Article  MathSciNet  MATH  Google Scholar 

  38. Sideris T C, Thomases B. Global existence for three-dimensional incompressible isotropic elastodynamics. Comm Pure Appl Math, 2007, 60: 1707–1730

    Article  MathSciNet  MATH  Google Scholar 

  39. Simon J. Compact sets in the space Lp(0,T; B). Ann Mat Pura Appl (4), 1986, 146: 65–96

    Article  Google Scholar 

  40. Sueur F. On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain. J Math Fluid Mech, 2014, 16: 163–178

    Article  MathSciNet  MATH  Google Scholar 

  41. Swann H S G. The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R3. Trans Amer Math Soc, 1971, 157: 373–397

    MathSciNet  MATH  Google Scholar 

  42. Trakhinin Y. Well-posedness of the free boundary problem in compressible elastodynamics. J Differential Equations, 2018, 264: 1661–1715

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang C, Wang Y X, Zhang Z F. Zero-viscosity limit of the Navier-Stokes equations in the analytic setting. Arch Ration Mech Anal, 2017, 224: 555–595

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang D H, Xie F. Inviscid limit of compressible viscoelastic equations with the no-slip boundary condition. arXiv: 2106.08517, 2021

  45. Wang Y. Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in three dimensional bounded domain. Arch Ration Mech Anal, 2016, 221: 1345–1415

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang Y, Xin Z P, Yong Y. Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in three-dimensional domains. SIAM J Math Anal, 2015, 47: 4123–4191

    Article  MathSciNet  MATH  Google Scholar 

  47. Wang Y-G, Williams M. The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions. Ann Inst Fourier (Grenoble), 2012, 62: 2257–2314

    Article  MathSciNet  MATH  Google Scholar 

  48. Wang Y-G, Xin Z P. Zero-viscosity limit of the linearized compressible Navier-Stokes equations with highly oscillatory forces in the half-plane. SIAM J Math Anal, 2005, 37: 1256–1298

    Article  MathSciNet  MATH  Google Scholar 

  49. Wang Y J, Xin Z P. Vanishing viscosity and surface tension limits of incompressible viscous surface waves. SIAM J Math Anal, 2021, 53: 574–648

    Article  MathSciNet  MATH  Google Scholar 

  50. Xiao Y L, Xin Z P. On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Comm Pure Appl Math, 2007, 60: 1027–1055

    Article  MathSciNet  MATH  Google Scholar 

  51. Xin Z P, Yanagisawa T. Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane. Comm Pure Appl Math, 1999, 52: 479–541

    Article  MathSciNet  MATH  Google Scholar 

  52. Xu L, Zhang P, Zhang Z F. Global solvability of a free boundary three-dimensional incompressible viscoelastic fluid system with surface tension. Arch Ration Mech Anal, 2013, 208: 753–803

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhang J Y. Local well-posedness and incompressible limit of the free-boundary problem in compressible elastodynamics. Arch Ration Mech Anal, 2022, 244: 599–697

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Xumin Gu was supported by National Natural Science Foundation of China (Grant No. 12031006) and the Shanghai Frontier Research Center of Modern Analysis. Yu Mei was supported by National Natural Science Foundation of China (Grant No. 12101496) and the Fundamental Research Funds for the Central Universities (Grant No. G2021KY05101). The authors thank Professor Zhen Lei for many stimulating discussions.

Author information

Authors and Affiliations

  1. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Xumin Gu

  2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China

    Yu Mei

Authors
  1. Xumin Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu Mei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gu, X., Mei, Y. Vanishing viscosity limits for the free boundary problem of compressible viscoelastic fluids with surface tension. Sci. China Math. 66, 1263–1300 (2023). https://doi.org/10.1007/s11425-022-1998-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-022-1998-9

Keywords

MSC(2020)

Search

Navigation