The Euler equations in a critical case of the generalized Campanato space

Chae, Dongho; Wolf, Jörg

doi:10.1016/j.anihpc.2020.06.006

Chae, Dongho ^1,² ; Wolf, Jörg ¹

¹ a Department of Mathematics, Chung-Ang University, Dongjak-gu Heukseok-ro 84, Seoul 06974, Republic of Korea
² b School of Mathematics, Korea Institute for Advanced Study, Dongdaemun-gu Hoegi-ro 85, Seoul 02455, Republic of Korea

Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 201-241

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

In this paper we prove local in time well-posedness for the incompressible Euler equations in $R^{n}$ for the initial data in $L_{1 (1)}^{1} (R^{n})$ , which corresponds to a critical case of the generalized Campanato spaces $L_{q (N)}^{s} (R^{n})$ . The space is studied extensively in our companion paper [9], and in the critical case we have embeddings $B_{\infty, 1}^{1} (R^{n}) ↪ L_{1 (1)}^{1} (R^{n}) ↪ C^{0, 1} (R^{n})$ , where $B_{\infty, 1}^{1} (R^{n})$ and $C^{0, 1} (R^{n})$ are the Besov space and the Lipschitz space respectively. In particular $L_{1 (1)}^{1} (R^{n})$ contains non- $C^{1} (R^{n})$ functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to $L_{1 (1)}^{1} (R^{n})$ , for which the solution to the Euler equations blows up in finite time.

MR Zbl

DOI : 10.1016/j.anihpc.2020.06.006

Classification : 35Q31, 76B03, 76D03
Keywords: Euler equation, Generalized Campanato space, Local well-posedness

Affiliations des auteurs :

Chae, Dongho ^{1, 2} ; Wolf, Jörg ¹

¹ a Department of Mathematics, Chung-Ang University, Dongjak-gu Heukseok-ro 84, Seoul 06974, Republic of Korea
² b School of Mathematics, Korea Institute for Advanced Study, Dongdaemun-gu Hoegi-ro 85, Seoul 02455, Republic of Korea

@article{AIHPC_2021__38_2_201_0,
     author = {Chae, Dongho and Wolf, J\"org},
     title = {The {Euler} equations in a critical case of the generalized {Campanato} space},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {201--241},
     year = {2021},
     publisher = {Elsevier},
     volume = {38},
     number = {2},
     doi = {10.1016/j.anihpc.2020.06.006},
     mrnumber = {4211985},
     zbl = {1458.35308},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2020.06.006/}
}

TY  - JOUR
AU  - Chae, Dongho
AU  - Wolf, Jörg
TI  - The Euler equations in a critical case of the generalized Campanato space
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 201
EP  - 241
VL  - 38
IS  - 2
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2020.06.006/
DO  - 10.1016/j.anihpc.2020.06.006
LA  - en
ID  - AIHPC_2021__38_2_201_0
ER  -

%0 Journal Article
%A Chae, Dongho
%A Wolf, Jörg
%T The Euler equations in a critical case of the generalized Campanato space
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 201-241
%V 38
%N 2
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2020.06.006/
%R 10.1016/j.anihpc.2020.06.006
%G en
%F AIHPC_2021__38_2_201_0

Chae, Dongho; Wolf, Jörg. The Euler equations in a critical case of the generalized Campanato space. Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 201-241. doi: 10.1016/j.anihpc.2020.06.006

Bibliographie
Cité par

[1] Bahouri, H.; Chemin, J.-Y.; Danchin, R. Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011 | MR | Zbl | DOI

[2] Bardos, C.; Titi, E. Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations, Discrete Contin. Dyn. Syst. Ser., Volume 3 (2010) no. 2, pp. 185-197 | MR | Zbl

[3] Beale, J.T.; Kato, T.; Majda, A. Remarks on the breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys., Volume 94 (1984), pp. 61-66 | MR | Zbl | DOI

[4] Bourdaud, G. $L^{p}$ -estimates for certain non-regular pseudo-differential operators, Commun. Partial Differ. Equ., Volume 7 (1982), pp. 1023-1033 | MR | Zbl | DOI

[5] Bourgain, J.; Li, D. Strong illposedness of the incompressible Euler equations in integer $C^{m}$ spaces, Geom. Funct. Anal., Volume 25 (2015), pp. 1-86 | MR | Zbl | DOI

[6] Bourgain, J.; Li, D. Strong illposedness of the incompressible Euler equations in borderline Sobolev spaces, Invent. Math., Volume 201 (2015), pp. 97-157 | MR | Zbl | DOI

[7] Campanato, S. Proprieti di una famiglia di spazi funzionali, Ann. Sc. Norm. Super. Pisa, Volume 18 (1964), pp. 137-160 | MR | Zbl | Numdam

[8] Chae, D. Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., Volume 38 (2004) no. 3–4, pp. 339-358 | MR | Zbl

[9] Chae, D.; Wolf, J. Transport equation in generalized Campanato spaces, 2019 | arXiv | Zbl

[10] Chemin, J.-Y. Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998 | MR | Zbl | DOI

[11] Cheskidov, A.; Shvydkoy, R. Ill-posedness of the basic equations of fluid dynamics in Besov spaces, Proc. Am. Math. Soc., Volume 138 (2010) no. 3, pp. 1059-1067 | MR | Zbl | DOI

[12] Constantin, P. On the Euler equations of incompressible fluids, Bull. Am. Math. Soc. N.S., Volume 44 (2007) no. 4, pp. 603-621 | MR | Zbl | DOI

[13] Elgindi, T.M.; Masmoudi, N. $L^{\infty}$ ill-posedness for a class of equations arising in hydrodynamics, Arch. Ration. Mech. Anal., Volume 235 (2020) no. 3, pp. 1979-2025 | MR | Zbl | DOI

[14] Giaquinta, M. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, 1983 | MR | Zbl

[15] Kato, T. Nonstationary flows of viscous and ideal fluids in $R^{3}$ , J. Funct. Anal., Volume 9 (1972), pp. 296-305 | MR | Zbl | DOI

[16] Kato, T.; Ponce, G. Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., Volume 41 (1988) no. 7, pp. 891-907 | MR | Zbl | DOI

[17] Kozono, H.; Taniuchi, Y. Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Commun. Math. Phys., Volume 214 (2000) no. 1, pp. 191-200 | MR | Zbl | DOI

[18] Lemarié-Rieusset, P.G. Euler equations and real harmonic analysis, Arch. Ration. Mech. Anal., Volume 204 (2012), pp. 355-386 | MR | Zbl | DOI

[19] Lions, P.L. Mathematical Topics in Fluid Mechanics, vol. 1, Oxford University Press, New York, 1996 | Zbl | MR

[20] Majda, A.; Bertozzi, A. Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[21] Misioek, 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., Volume 370 (2018) no. 7, pp. 4709-4730 | MR | Zbl | DOI

[22] Pak, H.C.; Park, Y.J. Existence of solution for the Euler equations in a critical Besov space $B_{\infty, 1}^{1} (R^{n})$ , Commun. Partial Differ. Equ., Volume 29 (2004), pp. 1149-1166 | MR | Zbl | DOI

[23] Taylor, M.E. Pseudodifferential operators, paradifferential operators, and layer potentials, AMS, Volume 81 (2000) | Zbl

[24] Triebel, H. Theory of Function Spaces, Monographs in Mathematics, vol. 84, Birkhäuser, Basel, 1992 | MR | Zbl

[25] Vishik, M. Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., Volume 145 (1998), pp. 197-214 | MR | Zbl | DOI

[26] Vishik, M. Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. Éc. Norm. Supér., Volume 32 (1999) no. 6, pp. 769-812 | MR | Zbl | Numdam | DOI

[27] Yudovich, V. Nonstationary flow of an ideal incompressible liquid, Zh. Vych. Mat., Volume 3 (1963), pp. 1032-1066 | MR | Zbl

Cité par Sources :