Home On the interior regularity criteria for the viscoelastic fluid system with damping
Article
Licensed
Unlicensed Requires Authentication

On the interior regularity criteria for the viscoelastic fluid system with damping

  • Gaocheng Yue EMAIL logo , Zixuan Pang and Yangyang Wu
Published/Copyright: April 24, 2024

Abstract

We consider a system of PDEs that model a viscoelastic fluid with damping mechanism. In 3 , we construct some new local energy bounds that enable us to improve several ϵ-regularity criteria for the Caffarelli–Kohn–Nirenberg theorem for weak solutions of this system.

MSC 2020: 35Q30; 35Q35

Communicated by Luis Silvestre


References

[1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. 10.1002/cpa.3160350604Search in Google Scholar

[2] H. J. Choe and J. L. Lewis, On the singular set in the Navier–Stokes equations, J. Funct. Anal. 175 (2000), no. 2, 348–369. 10.1006/jfan.2000.3582Search in Google Scholar

[3] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud. 105, Princeton University Press, Princeton, 1983. 10.1515/9781400881628Search in Google Scholar

[4] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, 2003. 10.1142/9789812795557Search in Google Scholar

[5] C. Guevara and N. C. Phuc, Local energy bounds and ϵ-regularity criteria for the 3D Navier–Stokes system, Calc. Var. Partial Differential Equations 56 (2017), no. 3, Paper No. 68. 10.1007/s00526-017-1151-7Search in Google Scholar

[6] R. Hynd, Partial regularity of weak solutions of the viscoelastic Navier–Stokes equations with damping, SIAM J. Math. Anal. 45 (2013), no. 2, 495–517. 10.1137/120875041Search in Google Scholar

[7] O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations, J. Math. Fluid Mech. 1 (1999), no. 4, 356–387. 10.1007/s000210050015Search in Google Scholar

[8] F. Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem, Comm. Pure Appl. Math. 51 (1998), no. 3, 241–257. 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-ASearch in Google Scholar

[9] F. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system, Comm. Pure Appl. Math. 61 (2008), no. 4, 539–558. 10.1002/cpa.20219Search in Google Scholar

[10] F.-H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math. 58 (2005), no. 11, 1437–1471. 10.1002/cpa.20074Search in Google Scholar

[11] W. S. Ożański, Partial regularity of Leray-Hopf weak solutions to the incompressible Navier–Stokes equations with hyperdissipation, Anal. PDE 16 (2023), no. 3, 747–783. 10.2140/apde.2023.16.747Search in Google Scholar

[12] N. C. Phuc and M. Torres, Characterizations of signed measures in the dual of BV and related isometric isomorphisms, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 1, 385–417. 10.2422/2036-2145.201508_005Search in Google Scholar

[13] V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations, Pacific J. Math. 66 (1976), no. 2, 535–552. 10.2140/pjm.1976.66.535Search in Google Scholar

[14] V. Scheffer, Hausdorff measure and the Navier–Stokes equations, Comm. Math. Phys. 55 (1977), no. 2, 97–112. 10.1007/BF01626512Search in Google Scholar

[15] V. Scheffer, The Navier–Stokes equations in space dimension four, Comm. Math. Phys. 61 (1978), no. 1, 41–68. 10.1007/BF01609467Search in Google Scholar

[16] V. Scheffer, The Navier–Stokes equations on a bounded domain, Comm. Math. Phys. 73 (1980), no. 1, 1–42. 10.1007/BF01942692Search in Google Scholar

[17] J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 9 (1962), 187–195. 10.1007/BF00253344Search in Google Scholar

[18] W. Wang and Z. Zhang, On the interior regularity criteria and the number of singular points to the Navier–Stokes equations, J. Anal. Math. 123 (2014), 139–170. 10.1007/s11854-014-0016-7Search in Google Scholar

Received: 2023-10-04
Accepted: 2024-03-10
Published Online: 2024-04-24
Published in Print: 2025-04-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 12.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/acv-2023-0107/html
Scroll to top button