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
Mots-clés : Euler equation, Generalized Campanato space, Local well-posedness

Affiliations des auteurs :

Chae, Dongho ^{1, 2} ; Wolf, Jörg ¹

@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},
     publisher = {Elsevier},
     volume = {38},
     number = {2},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.06.006},
     mrnumber = {4211985},
     zbl = {1458.35308},
     language = {en},
     url = {http://archive.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  - http://archive.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 http://archive.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. http://archive.numdam.org/articles/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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

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

Cité par Sources :