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

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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

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     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},
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     doi = {10.1016/j.anihpc.2020.06.006},
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     zbl = {1458.35308},
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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

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