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Keywords:
Navier-Stokes equation; Euler equation; ill-posedness; Besov space
Navier-Stokes equation; Euler equation; ill-posedness; Besov space
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 $t = 0$.
References:
[1] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften 343. Springer, Berlin (2011). DOI 10.1007/978-3-642-16830-7 | MR 2768550 | Zbl 1227.35004
[2] Bourgain, J., Li, D.: Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces. Invent. Math. 201 (2015), 97-157. DOI 10.1007/s00222-014-0548-6 | MR 3359050 | Zbl 1320.35266
[3] Bourgain, J., Li, D.: Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces. Geom. Funct. Anal. 25 (2015), 1-86. DOI 10.1007/s00039-015-0311-1 | MR 3320889 | Zbl 1480.35316
[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. DOI 10.1016/j.jmaa.2019.123493 | MR 4007200 | Zbl 1426.35057
[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. DOI 10.1090/S0002-9939-09-10141-7 | MR 2566571 | Zbl 1423.76085
[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. DOI 10.1016/j.jfa.2018.07.004 | MR 3926133 | Zbl 1412.35218
[7] Kato, T., Lai, C. Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56 (1984), 15-28. DOI 10.1016/0022-1236(84)90024-7 | MR 0735703 | Zbl 0545.76007
[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. DOI 10.1215/S0012-7094-87-05526-8 | MR 0904939 | Zbl 0649.76011
[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. DOI 10.1016/j.nonrwa.2023.103941 | MR 4604351 | Zbl 1529.35358
[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. DOI 10.1007/s00208-015-1213-0 | MR 3451386 | Zbl 1336.35280
[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. DOI 10.1090/tran/7101 | MR 3812093 | Zbl 1388.35160
[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. DOI 10.1081/PDE-200033764 | MR 2097579 | Zbl 1091.76006
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