Skip to main content
SpringerLink
Log in

Ill-posedness for the Cauchy problem of the two-dimensional compressible Navier-Stokes equations for an ideal gas

  • Published:
Journal of Elliptic and Parabolic Equations Aims and scope Submit manuscript

Abstract

We study the ill-posedness issue for the compressible viscous heat-conductive flows in two dimensions. In the scaling invariant spaces, negative regularity of the temperature causes an essential problem for the well-posedness, and the ill-posedness is obtained for all integrability indices except for the \(L^1\) case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bourgain, J., Pavlović, N.: Ill-posedness of the Navier-Stokes equations in a critical space in 3D. J. Funct. Anal. 255(9), 2233–2247 (2008)

    Article  MathSciNet  Google Scholar 

  2. Brezis, H., Cazenave, T.: A nonlinear heat equation with singular initial data. J. Anal. Math. 68, 277–304 (1996)

    Article  MathSciNet  Google Scholar 

  3. Chen, Q., Miao, C., Zhang, Z.: On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces. Rev. Mat. Iberoam. 31(4), 137–1402 (2015)

    Article  MathSciNet  Google Scholar 

  4. Chikami, N., Danchin, R.: On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces. J. Differ. Equ. 258(10), 3435–3467 (2015)

    Article  MathSciNet  Google Scholar 

  5. Danchin, R.: Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141(3), 579–614 (2000)

    Article  MathSciNet  Google Scholar 

  6. Danchin, R.: Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch. Ration. Mech. Anal. 160(1), 1–39 (2001)

    Article  MathSciNet  Google Scholar 

  7. Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm. Partial Differ. Equ. 26, 1183–1233 (2001)

    Article  MathSciNet  Google Scholar 

  8. Danchin, R.: Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density. Comm. Partial Differ. Equ. 32(9), 1373–1397 (2007)

    Article  MathSciNet  Google Scholar 

  9. Feichtinger, H.G.: Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, 1983. In: Radha, R., Krishna, M., Yhangavelu, S. (eds.) Proceeding of International Conference on Wavelets and Applications, pp. 1–56. New Delhi Allied Publishers, New Delhi (2003)

    Google Scholar 

  10. Feireisl, E.: Dynamics of viscous vompressible fluids. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2004)

    Google Scholar 

  11. Fujita, H.: On the blowing-up of solutions of the Cauchy problem for \(u_t=\Delta u+u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Ser I(13), 109–124 (1966)

    Google Scholar 

  12. Germain, P., Iwabuchi, T.: Self-similar solutions of the compressible Navier-Stokes equations. Arch. Rational Mech. Anal. (2021). https://doi.org/10.1007/s00205-021-01645-4

    Article  MathSciNet  MATH  Google Scholar 

  13. Haspot, B.: Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces. J. Differ. Equ. 251, 2262–2295 (2011)

    Article  MathSciNet  Google Scholar 

  14. Hoff, D., Zumbrun, K.: Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J. 44(2), 603–676 (1995)

    Article  MathSciNet  Google Scholar 

  15. Itaya, N.: On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluid. Kōdai Math. Sem. Rep. 23, 60–120 (1971)

    MathSciNet  MATH  Google Scholar 

  16. Iwabuchi, T., Ogawa, T.: Ill-posedness for the compressible Navier-Stokes equations under the barotropic condition in the limitting Besov spaces. J. Math. Soc. Jpn. https://mathsoc.jp/publication/JMSJ/pdf/JMSJ8159.pdf (in press)

  17. Iwabuchi, T.: Global solutions for the critical Burgers equation in the Besov spaces and the large time behavior. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(3), 687–713 (2015)

  18. Iwabuchi, T., Ogawa, T.: Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions. Trans. Am. Math. Soc. 367(4), 2613–2630 (2015)

    Article  Google Scholar 

  19. Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models. Oxford Lecture Series in Mathematics and its Applications 10, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1998)

    MATH  Google Scholar 

  20. Matsumura, A., Nishida, N.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)

    MathSciNet  MATH  Google Scholar 

  21. Matsumura, A., Nishida, N.: Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Comm. Math. Phys. 89(4), 445–464 (1983)

    Article  MathSciNet  Google Scholar 

  22. Nash, J.: Le probléme de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962)

    Article  MathSciNet  Google Scholar 

  23. Ogawa, T., Shimizu, S.: End-point maximal \(L^1\) -regularity for the Cauchy problem to a parabolic equation with variable coefficients. Math. Ann. 365(1–2), 661–705 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Tani, A.: The existence and uniqueness of the solution of equations describing compressible viscous fluid flow in a domain. Proc. Japan Acad. 52(7), 334–337 (1976)

    MathSciNet  MATH  Google Scholar 

  26. Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I. J. Funct. Anal. 207, 399–429 (2004)

    Article  MathSciNet  Google Scholar 

  27. Wang, B.: Ill-posedness for the Navier-Stokes equations in critical Besov spaces \(\dot{B}_{\infty, q}^{-1}\). Adv. Math. 268, 350–372 (2015)

    Article  MathSciNet  Google Scholar 

  28. Weissler, F.B.: Existence and non-existence of global solutions for a semilinear heat equation. Israel J. Math. 38, 29–40 (1981)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author was supported by JSPS Grant-in-Aid for Young Scientists (A) (No. 17H04824). The second author was supported by JSPS Grant-in-Aid, Scientific Research (S) (No. 19H05597) and JSPS Challenging Research (Pioneering) (No. 20K20284).

Author information

Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Tsukasa Iwabuchi & Takayoshi Ogawa

  2. Research Alliance Center of Mathematical Science/Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Takayoshi Ogawa

Authors
  1. Tsukasa Iwabuchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takayoshi Ogawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tsukasa Iwabuchi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iwabuchi, T., Ogawa, T. Ill-posedness for the Cauchy problem of the two-dimensional compressible Navier-Stokes equations for an ideal gas. J Elliptic Parabol Equ 7, 571–587 (2021). https://doi.org/10.1007/s41808-021-00136-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41808-021-00136-7

Keywords

Mathematics Subject Classification

Search

Navigation