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Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes–Korteweg System on \({\mathbb{R}^3}\)

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Abstract

The compressible Navier–Stokes–Korteweg system is considered on \({\mathbb{R}^3}\) when the external force is periodic in the time variable. The existence of a time periodic solution is proved for a sufficiently small external force by using the time-T-map related to the linearized problem around the motionless state with constant density and absolute temperature. The spectral properties of the time-T-map is investigated by a potential theoretic method and an energy method in some weighted spaces. The stability of the time periodic solution is proved for sufficiently small initial perturbations. It is also shown that the \({L^\infty}\) norm of the perturbation decays as time goes to infinity.

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References

  1. Cai H., Tan Z., Xu Q.: Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type. Kinet. Relat. Models 8, 29–51 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, Z., Xiao, Q., Zhao, H.: Time periodic solutions of compressible fluid models of Korteweg type, preprint, 2012, Analysis of PDEs.

  3. Chen Z., Zhao H.: Existence and nonlinear stability of stationary solutions to the full compressible Navier–Stokes–Korteweg system. J. Math. Pures Appl. 101, 330–371 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chiu P.H., Lin Y.T.: A conservative phase field method for solving incompressible two-phase flows. J. Comput. Phys. 230, 185–204 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Dunn J.E., Serrin J.: On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88, 95–133 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hattori H., Li D.N.: Solutions for two-dimensional system for materials of Korteweg type. SIAM J. Math. Anal. 25, 85–98 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hattori H., Li D.N.: Global solutions of a high dimensional system for Korteweg materials. J. Math. Anal. Appl. 198, 84–97 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hattori H., Li D.N.: The existence of global solutions to a fluid dynamic model for materials for Korteweg type. J. Partial Differ. Equ. 9, 323–342 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Heida M., Málek J.: On compressible Korteweg fluid-like materials. Int. J. Eng. Sci. 48, 1313–1324 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kagei Y., Kawashima S.: Stability of planar stationary solutions to the compressible Navier–Stokes equation on the half space. Commun. Math. Phys. 266, 401–430 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Kagei Y., Kobayashi T.: Asymptotic behavior of solutions of the compressible Navier–Stokes equation on the half space. Arch. Ration. Mech. Anal. 177, 231–330 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kagei Y., Tsuda K.: Existence and stability of time periodic solution to the compressible Navier-Stokes equation for time periodic external force with symmetry. J. Differ. Equ. 258, 399–444 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Kato, T.: Perturbation Theory for Liner Operators, Classics math., Springer, Berlin (reprint of the 1980 edition) (1995)

  14. Korteweg, D.J.: Sur la forme que prennent les équations du mouvement des fluides si lfon tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarite dans lfhypothèse dfune variation continue de la densité, Archives Néerlandaises des sciences exactes et naturelles, Ser 2 (6), pp. 1–24 (1901)

  15. Kotschote M.: Existence and time-asymptotics of global strong solutions to dynamic Korteweg models. Indiana Univ. Math. J. 63, 21–51 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Matsumura A., Nishida T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Jpn. Acad. Ser. A 55, 337–342 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Okita M.: On the convergence rates for the compressible Navier- Stokes equations with potential force. Kyushu J. Math. 68, 261–286 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tsuda, K.: On the existence and stability of time periodic solution to the compressible Navier–Stokes equation on the whole space, to appear in Arch. Rational Mech. Anal

  19. Shibata Y., Shimizu S.: A decay property of the fourier transform and its application to the Stokes problem. J. Math. Fluid Mech. 3, 213–230 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Shibata Y., Tanaka K.: On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance. J. Math. Soc. Japan 55, 797–826 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Van der Waals, J.D.: Théorie thermodynamique de la capillarité, dans lfhypothèse dfune variation continue de la densité. Archives Néerlandaises des sciences exactes et naturelles XXVIII, pp. 121–209 (1893)

  22. Wang Y., Tan Z.: Optimal decay rates for the compressible fluid models of Korteweg type. J. Math. Anal. Appl. 379, 256–271 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan

    Kazuyuki Tsuda

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  1. Kazuyuki Tsuda
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Correspondence to Kazuyuki Tsuda.

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Communicated by Y. Shibata

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Tsuda, K. Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes–Korteweg System on \({\mathbb{R}^3}\) . J. Math. Fluid Mech. 18, 157–185 (2016). https://doi.org/10.1007/s00021-015-0244-3

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  • DOI: https://doi.org/10.1007/s00021-015-0244-3

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