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Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel–Lizorkin space \(F^{d+1}_{1, \infty }(\mathbb {R}^d)\)

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Abstract

An evidence of temporal discontinuity of the solution in \(F^s_{1, \infty }(\mathbb {R}^d)\) is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in related spaces are also discussed.

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Notes

  1. Before the possible blow-up time.

  2. In fact, [8] deals with \(u: [0, T] \rightarrow B^{1}_{\infty , 1}(\mathbb {R}^d)\).

  3. We may say that the velocity u(t) exists in \({F}^{s}_{1, \infty }(\mathbb {R}^d)\) for \(t \in [- T_1, T_1]\) (Remark 2).

  4. The function \(\Phi \) is defined at page 3.

References

  1. H. Bahouri, J.Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations, Springer, 2011.

  2. J. M. Bony, Calcul symbolique et propagation des singularitiés pour les équations aux dérivées partielles non linéaires, Ann. de l’Ecole Norm. Sup. 14(1981) 209–246.

    Article  MathSciNet  Google Scholar 

  3. J. Bourgain, D. Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math. 201(2015) 97–157.

    Article  MathSciNet  Google Scholar 

  4. J. Bourgain, D. Li, Strong ill-posedness of the incompressible Euler equation in integer \(C^{m}\) spaces, Geom. Funct. Anal. 25(2015) 1–86.

    Article  MathSciNet  Google Scholar 

  5. J.Y. Chemin, Perfect incompressible fluids, Clarendon Press, Oxford, 1981.

    Google Scholar 

  6. P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. 44(4)(2007) 603–621.

    Article  MathSciNet  Google Scholar 

  7. A. Majda, A. Bertozzi, Vorticity and incompressible flow, Cambridge University Press, 2002.

  8. H. Pak, Y. Park, Existence of solution for the Euler equations in a critical Besov space \(\textbf{B}^1_{\infty , 1}(\mathbb{R}^n)\), Comm. Partial Diff. Eq. 29(2004) 1149–1166.

  9. H. Pak, Y. Park, Persistence of the incompressible Euler equations in a Besov space \(\textbf{B}^{d+1}_{1, 1}(\mathbb{R}^d)\), Adv. Difference Equ. Article 153(2013). https://doi.org/10.1186/1687-1847-2013-153.

  10. H. Pak, J. Hwang, Persistence of the solution to the Euler equations in the end-point critical Triebel-Lizorkin space \(F^{d+1}_{1, \infty }(\mathbb{R}^d)\), arXiv:2302.13295.

  11. M. Vishik, Hydrodynamics in Besov spaces. Arch. Rational Mech. Anal. 145(1998) 197-214.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2019R1I1A3A01057195).

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  1. Department of Mathematics, Dankook University, 119 Dandae-ro, Dongnam-gu, Cheonan-si, Chungnam, 31116, Republic of Korea

    Hee Chul Pak

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Pak, H.C. Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel–Lizorkin space \(F^{d+1}_{1, \infty }(\mathbb {R}^d)\). J. Evol. Equ. 23, 79 (2023). https://doi.org/10.1007/s00028-023-00927-6

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