World Scientific
  • Search
  •   
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×

On the construction of suitable weak solutions to the 3D Navier–Stokes equations in a bounded domain by an artificial compressibility method

    https://doi.org/10.1142/S0219199716500644Cited by:7 (Source: Crossref)

    We prove that suitable weak solutions of 3D Navier–Stokes equations in bounded domains can be constructed by a particular type of artificial compressibility approximation.

    AMSC: 35Q30, 35A35, 76M20

    References

    • 1. C. Amrouche and A. Rejaiba, Lp-theory for Stokes and Navier–Stokes equations with Navier boundary condition, J. Differential Equations 256(4) (2014) 1515–1547. Crossref, Web of ScienceGoogle Scholar
    • 2. H. Beirão da Veiga, On the suitable weak solutions to the Navier–Stokes equations in the whole space, J. Math. Pures Appl. (9) 64(1) (1985) 77–86. Google Scholar
    • 3. H. Beirão da Veiga, On the construction of suitable weak solutions to the Navier–Stokes equations via a general approximation theorem, J. Math. Pures Appl. (9) 64(3) (1985) 321–334. Google Scholar
    • 4. H. Beirão da Veiga, Local energy inequality and singular set of weak solutions of the boundary non-homogeneous Navier–Stokes problem, in Current Topics in Partial Differential Equations, eds. Y. Ohya, K. Kasahara, and N. Shimakura (Kinokuniya Company, Tokyo, 1986), pp. 91–105; Papers dedicated to Professor Sigeru Mizohata on the occasion of his sixtieth birthday. Google Scholar
    • 5. H. Beirão da Veiga, Regularity for Stokes and generalized Stokes systems under non-homogeneous slip-type boundary conditions, Adv. Differential Equations 9(9–10) (2004) 1079–1114. CrossrefGoogle Scholar
    • 6. H. Beirão da Veiga, Vorticity and regularity for flows under the Navier boundary condition, Commun. Pure Appl. Anal. 5(4) (2006) 907–918. Crossref, Web of ScienceGoogle Scholar
    • 7. L. C. Berselli, Some results on the Navier–Stokes equations with Navier boundary conditions, Riv. Math. Univ. Parma (N.S.) 1(1) (2010) 1–75. Google Scholar
    • 8. L. C. Berselli and S. Spirito, Weak solutions to the Navier–Stokes equations constructed by semi-discretization are suitable, in Recent Advances in Partial Differential and Applications, Contemporary Mathematics, Vol. 66 (American Mathematical Society, Providence, RI, 2016), pp. 85–97. CrossrefGoogle Scholar
    • 9. L. C. Berselli and S. Spirito, An elementary approach to inviscid limits for the 3D Navier–Stokes equations with slip boundary conditions and applications to the 3D Boussinesq equations, NoDEA Nonlinear Differential Equations Appl. 21 (2014) 149–166. Crossref, Web of ScienceGoogle Scholar
    • 10. L. C. Berselli and S. Spirito, On the vanishing viscosity limit for the Navier–Stokes equations under slip boudnary conditions in general domains, Comm. Math. Phys. 316 (2012) 171–198. Crossref, Web of ScienceGoogle Scholar
    • 11. L. C. Berselli and S. Spirito, Suitable weak solutions to the 3D Navier–Stokes equations are constructed with the Voigt approximation (2016); ArXiv 1605.04688; to appear in J. Differential Equations. Google Scholar
    • 12. A. Biryuk, W. Craig and S. Ibrahim, Construction of suitable weak solutions of the Navier–Stokes equations, in Stochastic Analysis and Partial Differential Equations, Contemporary Mathematics, Vol. 429 (American Mathematical Society, Providence, RI, 2007), pp. 1–18. CrossrefGoogle Scholar
    • 13. M. Bulíček, J. Málek and K. R. Rajagopal, Navier’s slip and evolutionary Navier–Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J. 56(1) (2007) 51–85. Crossref, Web of ScienceGoogle Scholar
    • 14. L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35(6) (1982) 771–831. Crossref, Web of ScienceGoogle Scholar
    • 15. T. Chacón Rebollo and R. Lewandowski, Mathematical and Numerical Foundations of Turbulence Models and Applications, Modeling and Simulation in Science, Engineering and Technology (Birkhäuser/Springer, New York, 2014). CrossrefGoogle Scholar
    • 16. A. J. Chorin, Numerical solution of the Navier–Stokes equations, Math. Comp. 22 (1968) 745–762. Crossref, Web of ScienceGoogle Scholar
    • 17. A. J. Chorin, On the convergence of discrete approximations to the Navier–Stokes equations, Math. Comp. 23 (1969) 341–353. Crossref, Web of ScienceGoogle Scholar
    • 18. D. Donatelli and P. Marcati, A dispersive approach to the artificial compressibility approximations of the Navier–Stokes equations in 3D, J. Hyperbolic Differ. Equ. 3(3) (2006) 575–588. Link, Web of ScienceGoogle Scholar
    • 19. D. Donatelli and P. Marcati, Leray weak solutions of the incompressible Navier–Stokes system on exterior domains via the artificial compressibility method, Indiana Univ. Math. J. 59 (2010) 1831–1852. Crossref, Web of ScienceGoogle Scholar
    • 20. D. Donatelli and S. Spirito, Weak solutions of Navier–Stokes equations constructed by artificial compressibility method are suitable, J. Hyperbolic Differ. Equ. 8(1) (2011) 101–113. Link, Web of ScienceGoogle Scholar
    • 21. E. Feireisl and H. Petzeltov á, On integrability up to the boundary of the weak solutions of the Navier–Stokes equations of compressible flow, Comm. Partial Differential Equations 25(3–4) (2000) 755–767. Crossref, Web of ScienceGoogle Scholar
    • 22. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-state Problems, Springer Monographs in Mathematics (Springer-Verlag, New York, 2011). CrossrefGoogle Scholar
    • 23. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics (Springer-Verlag, Berlin, 2001); Reprint of the 1998 edition. CrossrefGoogle Scholar
    • 24. J.-L. Guermond, Finite-element-based Faedo–Galerkin weak solutions to the Navier–Stokes equations in the three-dimensional torus are suitable, J. Math. Pures Appl. (9) 85(3) (2006) 451–464. Crossref, Web of ScienceGoogle Scholar
    • 25. J.-L. Guermond, Faedo–Galerkin weak solutions of the Navier–Stokes equations with Dirichlet boundary conditions are suitable, J. Math. Pures Appl. (9) 88(1) (2007) 87–106. Crossref, Web of ScienceGoogle Scholar
    • 26. J.-L. Guermond, On the use of the notion of suitable weak solutions in CFD, Internat. J. Numer. Methods Fluids 57(9) (2008) 1153–1170. Crossref, Web of ScienceGoogle Scholar
    • 27. E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951) 213–231. CrossrefGoogle Scholar
    • 28. D. Iftimie and G. Planas, Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions, Nonlinearity 19(4) (2006) 899–918. Crossref, Web of ScienceGoogle Scholar
    • 29. O. A. Ladyženskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations, J. Math. Fluid Mech. 1(4) (1999) 356–387. Crossref, Web of ScienceGoogle Scholar
    • 30. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63(1) (1934) 193–248. CrossrefGoogle Scholar
    • 31. P.-L. Lions, Mathematical Topics in Fluid Mechanics: Compressible Models, Vol. 2, Oxford Lecture Series in Mathematics and its Applications, Vol. 10 (The Clarendon Press, New York, 1998). Google Scholar
    • 32. W. Rusin, Incompressible 3D Navier–Stokes equations as a limit of a nonlinear parabolic system, J. Math. Fluid Mech. 14(2) (2012) 383–405. Crossref, Web of ScienceGoogle Scholar
    • 33. H. Sohr and W. von Wahl, On the regularity of the pressure of weak solutions of Navier–Stokes equations, Arch. Math. (Basel) 46(5) (1986) 428–439. Crossref, Web of ScienceGoogle Scholar
    • 34. V. A. Solonnikov and V. E. Ščadilov, A certain boundary value problem for the stationary system of Navier–Stokes equations, Trudy Mat. Inst. Steklov. 125 (1973) 196–210, 235. Google Scholar
    • 35. R. Temam, Une méthode d’approximation de la solution des équations de Navier–Stokes, Bull. Soc. Math. France 96 (1968) 115–152. Crossref, Web of ScienceGoogle Scholar
    • 36. R. Temam, Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires. I, Arch. Ration. Mech. Anal. 32 (1969) 135–153. Crossref, Web of ScienceGoogle Scholar
    • 37. Y. Xiao and Z. Xin, On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition, Comm. Pure Appl. Math. 60(7) (2007) 1027–1055. Crossref, Web of ScienceGoogle Scholar