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Non-existence of Classical Solutions with Finite Energy to the Cauchy Problem of the Compressible Navier–Stokes Equations

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Abstract

The well-posedness of classical solutions with finite energy to the compressible Navier–Stokes equations (CNS) subject to arbitrarily large and smooth initial data is a challenging problem. In the case when the fluid density is away from vacuum (strictly positive), this problem was first solved for the CNS in either one-dimension for general smooth initial data or multi-dimension for smooth initial data near some equilibrium state (that is, small perturbation) (Antontsev et al. in Boundary value problems in mechanics of nonhomogeneous fluids, North-Holland Publishing Co., Amsterdam, 1990; Kazhikhov in Sibirsk Mat Zh 23:60–64, 1982; Kazhikhov et al. in Prikl Mat Meh 41:282–291, 1977; Matsumura and Nishida in Proc Jpn Acad Ser A Math Sci 55:337–342, 1979, J Math Kyoto Univ 20:67–104, 1980, Commun Math Phys 89:445–464, 1983). In the case that the flow density may contain a vacuum (the density can be zero at some space-time point), it seems to be a rather subtle problem to deal with the well-posedness problem for CNS. The local well-posedness of classical solutions containing a vacuum was shown in homogeneous Sobolev space (without the information of velocity in L2-norm) for general regular initial data with some compatibility conditions being satisfied initially (Cho et al. in J Math Pures Appl (9) 83:243–275, 2004; Cho and Kim in J Differ Equ 228:377–411, 2006, Manuscr Math 120:91–129, 2006; Choe and Kim in J Differ Equ 190:504–523 2003), and the global existence of a classical solution in the same space is established under the additional assumption of small total initial energy but possible large oscillations (Huang et al. in Commun Pure Appl Math 65:549–585, 2012). However, it was shown that any classical solutions to the compressible Navier–Stokes equations in finite energy (inhomogeneous Sobolev) space cannot exist globally in time since it may blow up in finite time provided that the density is compactly supported (Xin in Commun Pure Appl Math 51:229–240, 1998). In this paper, we investigate the well-posedess of classical solutions to the Cauchy problem of Navier–Stokes equations, and prove that the classical solution with finite energy does not exist in the inhomogeneous Sobolev space for any short time under some natural assumptions on initial data near the vacuum. This implies, in particular, that the homogeneous Sobolev space is as crucial as studying the well-posedness for the Cauchy problem of compressible Navier–Stokes equations in the presence of a vacuum at far fields even locally in time.

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References

  1. Antontsev, S.N., Kazhikhov, A.V., Monakhov, V.N.: Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, vol. 22 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1990 (Translated from the Russian).

  2. Cho, Y., Choe, H.J., Kim, H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl. 9(83), 243–275 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cho, Y., Jin, B.J.: Blow-up of viscous heat-conducting compressible flows. J. Math. Anal. Appl. 320, 819–826 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cho, Y., Kim, H.: Existence results for viscous polytropic fluids with vacuum. J. Differ. Equ. 228, 377–411 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Cho, Y., Kim, H.: On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscr. Math. 120, 91–129 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Choe, H.J., Kim, H.: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190, 504–523 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Coutand, D., Lindblad, H., Shkoller, S.: A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum. Commun. Math. Phys. 296, 559–587 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Coutand, D., Shkoller, S.: Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum. Arch. Ration. Mech. Anal. 206, 515–616 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Danchin, R.: Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141, 579–614 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Ding, S., Wen, H., Yao, L., Zhu, C.: Global spherically symmetric classical solution to compressible Navier-Stokes equations with large initial data and vacuum. SIAM J. Math. Anal. 44, 1257–1278 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall Inc, Englewood Cliffs, N.J. (1964)

    MATH  Google Scholar 

  13. Han, Q.: A basic course in partial differential equations, vol. 120 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011

  14. Hoff, D.: Global existence for \(1\)D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Am. Math. Soc. 303, 169–181 (1987)

    MATH  Google Scholar 

  15. Hoff, D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Ration. Mech. Anal. 132, 1–14 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hoff, D., Serre, D.: The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51, 887–898 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Hoff, D., Smoller, J.: Non-formation of vacuum states for compressible Navier-Stokes equations. Commun. Math. Phys. 216, 255–276 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Huang, X., Li, J.: Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations. Arch. Ration. Mech. Anal. 227, 995–1059 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang, X., Li, J., Xin, Z.: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Commun. Pure Appl. Math. 65, 549–585 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jang, J., Masmoudi, N.: Well and ill-posedness for compressible Euler equations with vacuum. J. Math. Phys. 53, 115625, 11 (2012)

  21. Jang, J., Masmoudi, N.: Well-posedness of compressible Euler equations in a physical vacuum. Commun. Pure Appl. Math. 68, 61–111 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jiang, S., Zhang, P.: On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Commun. Math. Phys. 215, 559–581 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Kanel, J.I.: The Cauchy problem for equations of gas dynamics with viscosity. Sibirsk. Mat. Zh. 20, 293–306, 463 (1979)

  24. Kazhikhov, A.V.: On the Cauchy problem for the equations of a viscous gas. Sibirsk. Mat. Zh. 23, 60–64, 220 (1982)

  25. Kazhikhov, A.V., Shelukhin, V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Prikl. Mat. Meh. 41, 282–291 (1977)

    MathSciNet  Google Scholar 

  26. Lions, P.-L.: Existence globale de solutions pour les équations de Navier–Stokes compressibles isentropiques. C. R. Acad. Sci. Paris Sér. I Math. 316, 1335–1340 (1993)

  27. Lions, P.-L.: Limites incompressible et acoustique pour des fluides visqueux, compressibles et isentropiques. C. R. Acad. Sci. Paris Sér. I Math. 317, 1197–1202 (1993)

  28. Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 2, vol. 10 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998. Compressible models, Oxford Science Publications

  29. Majda, A.J., Bertozzi, A.L.: Vorticity and incompressible flow, vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002

  30. 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 Math. Sci. 55, 337–342 (1979)

  31. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  32. Matsumura, A., Nishida, T.: Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Nash, J.: Le problème de Cauchy pour les équations différentielles d'un fluide général. Bull. Soc. Math. Fr. 90, 487–497 (1962)

    Article  MATH  Google Scholar 

  34. Salvi, R., Straskraba, I.: Global existence for viscous compressible fluids and their behavior as \(t\rightarrow \infty \). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40, 17–51 (1993)

  35. Serre, D.: Solutions faibles globales des équations de Navier–Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér. I Math. 303, 639–642 (1986)

  36. Serre, D.: Sur l'équation monodimensionnelle d'un fluide visqueux, compressible et conducteur de chaleur. C. R. Acad. Sci. Paris Sér. I Math., 303, 703–706 1986

  37. Serrin, J.: On the uniqueness of compressible fluid motions. Arch. Ration. Mech. Anal. 3, 271–288 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xin, Z.: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun. Pure Appl. Math. 51, 229–240 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xin, Z., Yan, W.: On blowup of classical solutions to the compressible Navier-Stokes equations. Commun. Math. Phys. 321, 529–541 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Xin, Z., Yuan, H.: Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations. J. Hyperbolic Differ. Equ. 3, 403–442 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their careful reading, helpful suggestions and valuable comments, which helped us a lot to improve the presentation of this manuscript, particularly regarding the proofs of Lemmas 2.3 and 4.3. The research of Li was supported partially by the National Natural Science Foundation of China (Nos. 11461161007, 11671384, 11871047, and 11225012,), and the “Capacity Building for Sci-Tech Innovation - Fundamental Scientific Research Funds 007175304800 and 025185305000/182”. The research of Wang was supported by Grant Nos. 231668 and 250070 from the Research Council of Norway. The research of Xin was supported partially by the Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants CUHK-14305315 and CUHK-4048/13P, NSFC/RGC Joint Research Scheme N-CUHK443/14, and Focused Innovations Scheme from The Chinese University of Hong Kong.

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Li, HL., Wang, Y. & Xin, Z. Non-existence of Classical Solutions with Finite Energy to the Cauchy Problem of the Compressible Navier–Stokes Equations. Arch Rational Mech Anal 232, 557–590 (2019). https://doi.org/10.1007/s00205-018-1328-z

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  • DOI: https://doi.org/10.1007/s00205-018-1328-z

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