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

By A. Ionescu, Carlos E. Kenig, D. Tataru. 2007

146 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f45d91c70beee8fbd87ae900fd8b030b51c6cdbf

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In Applied Mathematics

In Control and Systems Engineering

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Citations in the corpus

Global well-posedness of Korteweg-de Vries equation in H^-3/4 (R) (2008, background): “From the definition of Xk, we see that for any l ∈ Z+ and fk ∈ Xk (see also [8]), ∞ ∑”
Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton (2010, methodology): “[4] put forward a method to establish global well-posedness in the energy space for the Cauchy problem associated to the KP-I equation.”
Well-posedness for the fifth-order KdV equation in the energy space (2012, methodology): “Following [15], we state some important properties of the F s α(T ) spaces.”
Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation (2008, background): “Structures of this kind of spaces were introduced, for instance, in [11], [ 4 ] and [3] for the BO equation.”
Improvement of the energy method for strongly non resonant dispersive equations and applications (2014, methodology): “This type of method was first introduced by Koch and Tzvetkov [2003] (see also [Kenig and Koenig 2003] for some improvements) in the framework of Strichartz’s spaces and then by Koch and Tataru [2007] (see also [Ionescu et al. 2008]) in the framework of Bourgain’s spaces.”
Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces (2008, methodology): “This difference equation has less symmetries, but some special symmetries for real-valued solutions in L2. We then follow the methods in [ 13 ] to”