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Existence of Incompressible and Immiscible Flows in Critical Function Spaces on Bounded Domains

  • Myong-Hwan Ri
  • Ping ZhangEmail author
Article

Abstract

We study global existence and uniqueness of solutions to inhomogeneous incompressible Navier–Stokes equations on bounded domains of \(\mathbb {R}^n, n\ge 2\), with initial velocity in the Besov space \(B^0_{q,\infty }(\Omega )\), \(q\ge n\), and piecewise constant initial density. Existence of solutions is proved when \(B^0_{n,\infty }\)-norm of initial velocity and initial density difference are small, and for uniqueness we require that \(q>n\). The proof of existence of solutions is done via an iterative scheme based on maximal \(L^\infty _\gamma \)-regularity of the Stokes operator in little Nicolskii spaces and on solvability for transport equations in the spaces of pointwise multipliers for little Nicolskii spaces, while the proof of uniqueness is done via a Lagrangian approach using the result of an time-evolutionary Stokes system with nonzero divergence obtained in this paper.

Keywords

Existence Uniqueness Inhomogeneous Navier–Stokes equations Immiscible flow Divergence problem 

Mathematics Subject Classification

35Q30 35B35 76D03 76D07 76E99 

Notes

Acknowledgements

P. Zhang is partially supported by NSF of China under Grants 11371347 and 11688101, Morningside Center of Mathematics of The Chinese Academy of Sciences and innovation grant from National Center for Mathematics and Interdisciplinary Sciences.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Abidi, H.: Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique. Rev. Mat. Iberoam. 23, 537–586 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abidi, H., Gui, G., Zhang, P.: On the wellposedness of three-dimensional inhomogeneous Navier–Stokes equations in the critical spaces. Arch. Ration. Mech. Anal. 204, 189–230 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abidi, H., Gui, G., Zhang, P.: Well-posedness of 3-D inhomogeneous Navier–Stokes equations with highly oscillatory initial velocity field. J. Math. Pures Appl. 100, 166–203 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Amann, H.: On the strong solvability of the Navier–Stokes equations. J. Math. Fluid Mech. 2, 16–98 (2000)MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Amann, H.: Linear and Quasilinear Parabolic Problems, I. Abstract Linear Theory. Birkhäuser, Basel (1995)CrossRefGoogle Scholar
  6. 6.
    Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Berlin (1977)zbMATHGoogle Scholar
  7. 7.
    Bogovskii, M.E.: Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR 248, 1037–1040 (1979)MathSciNetGoogle Scholar
  8. 8.
    Borchers, W., Sohr, H.: On the equations \(\text{ rot } v = g\) and \(\div u = f\) with zero boundary conditions. Hokkaido Math. J. 19, 67–87 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Caetano, A.M.: Approximation by functions of compact support in Besov–Triebel–Lizorkin spaces on irregular domains. Stud. Math. 142, 47–63 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Danchin, R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. R. Soc. Edinb. Sect. A 133, 1311–1334 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Danchin, R.: Density-dependent incompressible fluids in bounded domains. J. Math. Fluid Mech. 8, 333–381 (2006)MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Danchin, R., Mucha, P.B.: A critical functional framework for the inhomogeneous Navier–Stokes equations in the half-space. J. Funct. Anal. 256, 881–927 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Danchin, R., Mucha, P.B.: A Lagrangian approach for the incompressible Navier–stokes equations with variable density. Commun. Pure. Appl. Math. 65, 1458–1480 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Danchin, R., Mucha, P.B.: Incompressible flows with piecewise constant density. Arch. Ration. Mech. Anal. 207, 991–1023 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Danchin, R., Zhang, P.: Inhomogeneous Navier-Stokes equations in the half-space, with only bounded density. J. Funct. Anal. 267, 2371–2436 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press Inc, London (1992)zbMATHGoogle Scholar
  18. 18.
    Farwig, R., Sohr, H.: Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Jpn. 46, 607–643 (1994)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady-State Problems, 2nd edn. Springer, Berlin (2011)zbMATHGoogle Scholar
  20. 20.
    Geissert, M., Heck, H., Hieber, M.: On the equation \(\div u = g\) and Bogovskii’s operator in Sobolev spaces of negative order, partial differential equations and functional analysis. Oper. Theory Adv. Appl. 168, 113–121 (2006)zbMATHGoogle Scholar
  21. 21.
    Giga, Y., Miyakawa, T.: Solutions in Lr of the Navier–Stokes initial value problem. Arch. Ration. Mech. Anal. 89, 267–281 (1985)CrossRefGoogle Scholar
  22. 22.
    Huang, J., Paicu, M., Zhang, P.: Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity. Arch. Ration. Mech. Anal. 209, 631–682 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kazhikhov, A.V.: Solvability of the initial-boundary value problem for the equations of an inhomogeneous fluid. Dokl Akad. Nauk. SSSR 216, 1008–1010 (1974). (Russian) MathSciNetADSGoogle Scholar
  24. 24.
    Kato, T.: Strong \(L^p\)-solutions of the Navier-Stokes equations in \(\mathbb{R}^m\), with applications to weak solutions. Math. Z. 187, 471–480 (1984)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ladyzhenskaya, O.A., Solonnikov, V.A.: V.A.: The unique solvability of an initial-boundary value problem for viscous incompresible fluids. (Russian) Boundary value problems of mathematical physics and related questions of the theory of functions, 8. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 52, 52–109 (1975)MathSciNetGoogle Scholar
  26. 26.
    Ri, M.-H., Zhang, P., Zhang, Z.: Global well-posedness for Navier-Stokes equations with small initial data in \(B^0_{n,\infty }(\Omega )\). J. Math. Fluid Mech. 18, 103–131 (2016)MathSciNetCrossRefADSGoogle Scholar
  27. 27.
    Temam, R.: Navier–Stokes Equations. Noth-Holland, Amsterdam (1977)zbMATHGoogle Scholar
  28. 28.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam (1983)zbMATHGoogle Scholar
  29. 29.
    Triebel, H.: Function spaces in Lipschitz domains and on Lipschitz manifolds, characteristic functions as a pointwise multiplers. Rev. Math. Comput. 15, 475–524 (2002)zbMATHGoogle Scholar
  30. 30.
    Yagi, A.: Abstract Parabolic Evolution Equations and their Applications. Springer, Berlin (2010)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of MathematicsState Academy of SciencesPyongyangDemocratic People’s Republic of Korea
  2. 2.Academy of Mathematics and Systems Science and Hua Loo-Keng Center for Mathematical SciencesChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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