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Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces

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Abstract

For the \(d\)-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space \(H^s(\mathbb R^d)\), \(s>s_c:=d/2+1\). The borderline case \(s=s_c\) was a folklore open problem. In this paper we consider the physical dimension \(d=2\) and show that if we perturb any given smooth initial data in \(H^{s_c}\) norm, then the corresponding solution can have infinite \(H^{s_c}\) norm instantaneously at \(t>0\). In a companion paper [1] we settle the 3D and more general cases. The constructed solutions are unique and even \(C^{\infty }\)-smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems.

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Notes

  1. The \(L^{\infty }\) end-point Kato-Ponce inequality (conjectured in [14]) and several new Kato-Ponce type inequalities are proved in recent [2].

  2. Counter examples for the case \(s<d/p+1\) was also considered therein.

  3. Similar results also hold for vorticity functions which are odd in \(x_1\), or odd in both \(x_1\) and \(x_2\).

  4. Actually it is easy to show that \(u\) is log-Lipschitz.

  5. In the actual perturbation argument, we need to divide it by a suitable power of \(\Vert D \phi \Vert _{\infty }\).

  6. Note that the perturbation \(\beta (x)\) therein can be chosen to be odd in \(x_1\) and \(x_2\).

  7. One needs to inductively shrink the \(\delta _i\) further (so that Lemma 6.5 can be applied) and re-choose the profiles \(h_j\) if necessary.

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Acknowledgments

We thank the anonymous referees for very helpful suggestions. J. Bourgain was supported in part by NSF No. 1301619. D. Li was supported in part by NSF under agreement No. DMS-1128155. D. Li was also supported by an Nserc discovery grant. The first author thanks the UC Berkeley math department, where part of the work was done, for its hospitality. The second author thanks the Institute for Advanced Study for its hospitality and providing excellent work conditions.

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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08544, USA

    Jean Bourgain

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Dong Li

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  1. Jean Bourgain
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Correspondence to Dong Li.

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Bourgain, J., Li, D. Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces. Invent. math. 201, 97–157 (2015). https://doi.org/10.1007/s00222-014-0548-6

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