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Global well-posedness of the KP-I initial-value problem in the energy space. (English) Zbl 1188.35163
Summary: We prove that the KP-I initial-value problem $$\cases \partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0 &\text{on}\,\Bbb{R}^2_{x,y}\times\Bbb{R}_t;\\ u(0)=\varphi, \endcases$$ is globally well-posed in the energy space $$\bold{E}^1(\Bbb{R}^2)=\big\{\phi:\Bbb{R}^2\to\Bbb{R}: \|\varphi\|_{\bold{E}^1(\Bbb{R}^2)}\approx\|\varphi\|_{L^2}+\|\partial_x\varphi\|_{L^2}+\big\|\partial_x^{-1}\partial_y\varphi\big\|_{L^2}<\infty\big\}.$$

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35B45A priori estimates for solutions of PDE
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References:
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