Skip to main content Accessibility help
×
Home

ILL-POSEDNESS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH THE VELOCITY IN $L^{6}$ FRAMEWORK

  • Jiecheng Chen (a1) and Renhui Wan (a2)

Abstract

Ill-posedness for the compressible Navier–Stokes equations has been proved by Chen et al. [On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Revista Mat. Iberoam.31 (2015), 1375–1402] in critical Besov space $L^{p}$ $(p>6)$ framework. In this paper, we prove ill-posedness with the initial data satisfying

$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}_{0}-\bar{\unicode[STIX]{x1D70C}}\Vert _{{\dot{B}}_{p,1}^{\frac{3}{p}}}\leqslant \unicode[STIX]{x1D6FF},\quad \Vert u_{0}\Vert _{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\leqslant \unicode[STIX]{x1D6FF}. & & \displaystyle \nonumber\end{eqnarray}$$
To accomplish this goal, we require a norm inflation coming from the coupling term $L(a)\unicode[STIX]{x1D6E5}u$ instead of $u\cdot \unicode[STIX]{x1D6FB}u$ and construct a new decomposition of the density.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      ILL-POSEDNESS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH THE VELOCITY IN $L^{6}$ FRAMEWORK
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      ILL-POSEDNESS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH THE VELOCITY IN $L^{6}$ FRAMEWORK
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      ILL-POSEDNESS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH THE VELOCITY IN $L^{6}$ FRAMEWORK
      Available formats
      ×

Copyright

Footnotes

Hide All

The original version of this article was submitted without an identified corresponding author. A notice detailing this has been published and the error rectified in the online PDF and HTML copies.

*

Author for correspondence.

Footnotes

References

Hide All
1. Bahouri, H., Chemin, J.-Y. and Danchin, R., Fourier analysis and nonlinear partial differential equations, in Grundlehren der Mathematischen Wissenschaften (Springer, Heidelberg, 2011).
2. Bourgain, J. and Pavlović, N., Ill-posedness of the Navier–Stokes equations in a critical space in 3D, J. Func. Anal. 255 (2008), 22332247.
3. Charve, F. and Danchin, R., A global existence result for the compressible Navier–Stokes equations in the critical L p framework, Arch. Ration. Mech. Anal. 198 (2010), 233271.
4. Chemin, J.-Y. and Lerner, N., Flow of non-Lipschitz vector-fields and Navier–Stokes equations, J. Differential Equations 121 (1995), 314328.
5. Chen, Q., Miao, C. and Zhang, Z., Well-posedness in critical spaces for the compressible Navier–Stokes equations with density dependent viscosities, Rev. Mat. Iberoam. 26 (2010), 915946.
6. Chen, Q., Miao, C. and Zhang, Z., Global well-posedness for compressible Navier–Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math. 63 (2010), 11731224.
7. Chen, Q., Miao, C. and Zhang, Z., On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Rev. Mat. Iberoam. 31 (2015), 13751402.
8. Chikami, N. and Danchin, R., On the well-posedness of the full compressible Navier–Stokes system in critical Besov space, J. Differential Equations 258 (2015), 34353467.
9. Danchin, R., Global existence in critical spaces for compressible Navier–Stokes equations, Invent. Math. 141 (2000), 579614.
10. Danchin, R., Global existence in critical spaces for flows of compressible viscous and heat-conductive gases, Arch. Ration. Mech. Anal. 160 (2001), 139.
11. Danchin, R., Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations 26 (2001), 11831233.
12. Danchin, R., Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Comm. Partial Differential Equations 32 (2007), 13731397.
13. Danchin, R., A Lagrangian approach for the compressible Navier–Stokes equations, Ann. Inst. Fourier 64 (2014), 753791.
14. Feireisl, E., Dynamics of Viscous Vompressible Fluids (Oxford University Press, Oxford, 2004).
15. Fujita, H. and Kato, T., On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964), 269315.
16. Germain, P., Multipliers, paramultipliers, and weak-strong uniqueness for the Navier–Stokes equations, J. Differential Equations 226 (2006), 373428.
17. Germain, P., The second iterate for the Navier–Stokes equation, J. Funct. Anal. 255 (2008), 22482264.
18. Huang, X., Li, J. and Xin, Z., Global well-posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible Navier–Stokes equations, Comm. Pure Appl. Math. 65 (2012), 549585.
19. Kato, T. and Ponce, G., Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891907.
20. Lions, P.-L., Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models, Oxford Lecture Series in Mathematics and its Applications, Volume 10, (Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998).
21. Matsumura, A. and Nishida, T., The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), 337342.
22. Nash, J., Le problème de Cauchy pour les équations différentielles d’un fluide général, Bull. Soc. Math. France 90 (1962), 487497.
23. Sun, Y., Wang, C. and Zhang, Z., A Beale–Kato–Majda Blow-up criterion for the 3-D compressible Navier–Stokes equations, J. Math. Pures Appl. 95 (2011), 3647.
24. Sun, Y., Wang, C. and Zhang, Z., A Beale–Kato–Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal. 201 (2011), 727742.
25. Vasseur, A. F. and Yu, C., Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations, Invent. Math. 206 (2016), 935974.
26. Wang, B., Ill-posedness for the Navier–Stokes equations in critical Besov spaces , q -1 , Adv. Math. 208 (2015), 350372.
27. Wang, C., Wang, W. and Zhang, Z., Global well-posedness of compressible Navier–Stokes equations for some classes of large initial data, Arch. Ration. Mech. Anal. 213 (2014), 171214.
28. Xin, Z., Blowup of smooth solutions to the compressible Navier–Stokes equation with compact density, Comm. Pure Appl. Math. 51 (1998), 229240.
29. Yoneda, T., Ill-posedness of the 3D Navier–Stokes equations in a generalized Besov space near BMO -1 , J. Funct. Anal 258 (2010), 33763387.
30. Zhang, T., Global solutions of compressible Navier–Stokes equations with a density-dependent viscosity coefficient, J. Math. Phys 52 (2011), 043510.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

ILL-POSEDNESS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH THE VELOCITY IN $L^{6}$ FRAMEWORK

  • Jiecheng Chen (a1) and Renhui Wan (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed

This addendum applies to the following article(s):