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Global well-posedness of the KP-I initial-value problem in the energy space

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Abstract

We prove that the KP-I initial-value problem

$$\begin{cases} \partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0 \,\text{ on }\,\mathbb{R}^2_{x,y}\times\mathbb{R}_t;\\ u(0)=\phi, \end{cases}$$

is globally well-posed in the energy space

$$\mathbf{E}^1(\mathbb{R}^2)=\big\{\phi:\mathbb{R}^2\to\mathbb{R}: \|\phi\|_{\mathbf{E}^1(\mathbb{R}^2)}\approx\|\phi\|_{L^2}+\|\partial_x\phi\|_{L^2}+\big\|\partial_x^{-1}\partial_y\phi\big\|_{L^2}<\infty\big\}.$$

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, WI, 53706, USA

    A.D. Ionescu

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    C.E. Kenig

  3. Department of Mathematics, University of California–Berkeley, Berkeley, CA, 94720, USA

    D. Tataru

Authors
  1. A.D. Ionescu
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  2. C.E. Kenig
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  3. D. Tataru
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Correspondence to A.D. Ionescu.

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Ionescu, A., Kenig, C. & Tataru, . Global well-posedness of the KP-I initial-value problem in the energy space . Invent. math. 173, 265–304 (2008). https://doi.org/10.1007/s00222-008-0115-0

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  • DOI: https://doi.org/10.1007/s00222-008-0115-0

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