Article
| Title: | Ill-posedness for the Navier-Stokes and Euler equations in Besov spaces (English) |
| Author: | Yu, Yanghai |
| Author: | Liu, Fang |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 69 |
| Issue: | 6 |
| Year: | 2024 |
| Pages: | 757-767 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We construct a new initial data to prove the ill-posedness of both Navier-Stokes and Euler equations in weaker Besov spaces in the sense that the solution maps to these equations starting from $u_0$ are discontinuous at $t = 0$. (English) |
| Keyword: | Navier-Stokes equation |
| Keyword: | Euler equation |
| Keyword: | ill-posedness |
| Keyword: | Besov space |
| MSC: | 35Q30 |
| MSC: | 35Q31 |
| MSC: | 47J06 |
| DOI: | 10.21136/AM.2024.0089-24 |
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| Date available: | 2024-12-13T18:59:30Z |
| Last updated: | 2024-12-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152668 |
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| Reference: | [1] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations.Grundlehren der Mathematischen Wissenschaften 343. Springer, Berlin (2011). Zbl 1227.35004, MR 2768550, 10.1007/978-3-642-16830-7 |
| Reference: | [2] Bourgain, J., Li, D.: Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces.Invent. Math. 201 (2015), 97-157. Zbl 1320.35266, MR 3359050, 10.1007/s00222-014-0548-6 |
| Reference: | [3] Bourgain, J., Li, D.: Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces.Geom. Funct. Anal. 25 (2015), 1-86. Zbl 1480.35316, MR 3320889, 10.1007/s00039-015-0311-1 |
| Reference: | [4] Cheskidov, A., Dai, M.: Discontinuity of weak solutions to the 3D NSE and MHD equations in critical and supercritical spaces.J. Math. Anal. Appl. 481 (2020), Article ID 13493, 16 pages. Zbl 1426.35057, MR 4007200, 10.1016/j.jmaa.2019.123493 |
| Reference: | [5] Cheskidov, A., Shvydkoy, R.: Ill-posedness of the basic equations of fluid dynamics in Besov spaces.Proc. Am. Math. Soc. 138 (2010), 1059-1067. Zbl 1423.76085, MR 2566571, 10.1090/S0002-9939-09-10141-7 |
| Reference: | [6] Guo, Z., Li, J., Yin, Z.: Local well-posedness of the incompressible Euler equations in $B_{\infty,1}^1$ and the inviscid limit of the Navier-Stokes equations.J. Funct. Anal. 276 (2019), 2821-2830. Zbl 1412.35218, MR 3926133, 10.1016/j.jfa.2018.07.004 |
| Reference: | [7] Kato, T., Lai, C. Y.: Nonlinear evolution equations and the Euler flow.J. Funct. Anal. 56 (1984), 15-28. Zbl 0545.76007, MR 0735703, 10.1016/0022-1236(84)90024-7 |
| Reference: | [8] Kato, T., Ponce, G.: On nonstationary flows of viscous and ideal fluids in $L^p_s(\Bbb R^2)$.Duke Math. J. 55 (1987), 487-499. Zbl 0649.76011, MR 0904939, 10.1215/S0012-7094-87-05526-8 |
| Reference: | [9] Li, J., Yu, Y., Zhu, W.: Ill-posedness for the Euler equations in Besov spaces.Nonlinear Anal., Real World Appl. 74 (2023), Article ID 103941, 9 pages. Zbl 1529.35358, MR 4604351, 10.1016/j.nonrwa.2023.103941 |
| Reference: | [10] Misiołek, G., Yoneda, T.: Local ill-posedness of the incompresssible Euler equations in $C^1$ and $B^1_{\infty,1}$.Math. Ann. 363 (2016), 243-268. Zbl 1336.35280, MR 3451386, 10.1007/s00208-015-1213-0 |
| Reference: | [11] Misiołek, G., Yoneda, T.: Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces.Trans. Am. Math. Soc. 370 (2018), 4709-4730. Zbl 1388.35160, MR 3812093, 10.1090/tran/7101 |
| Reference: | [12] Pak, H. C., Park, Y. J.: Existence of solutions for the Euler equations in a critical Besov space $B^1_{\infty,1}(\Bbb R^n)$.Commun. Partial Differ. Equations 29 (2004), 1149-1166. Zbl 1091.76006, MR 2097579, 10.1081/PDE-200033764 |
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