Global Well-Posedness and Blow-Up for the fifth order $L^2-$critical KP-I equation

TL;DR

This work analyzes the fifth-order L^2-critical KP-I equation, establishing local well-posedness in anisotropic energy spaces for $s>\tfrac{3}{2}$ and deriving global well-posedness results when the initial $L^2$ norm is below the ground-state threshold, with a detailed use of sharp anisotropic Sobolev inequalities and virial identities. It further investigates transverse blow-up by constructing invariant regions and leveraging virial-type identities to obtain finite-time blow-up for localized data, while also proving global well-posedness for small data on the cylinder $\mathbb{R}\times\mathbb{T}$ via refined Strichartz estimates. The paper thus provides a comprehensive treatment of global dynamics, blow-up phenomena, and cylinder behavior for the $L^2$-critical fifth-order KP-I hierarchy, including both the standard and modified equations.

Abstract

In the current paper, we investigate the fifth order modified KP-I eqaution, namely \begin{equation*} \partial_t u-\partial_{x}^{5}u-\partial_{x}^{-1}\partial_{y}u+\partial_{x}(u^3)=0. \end{equation*} This equation is $L^2$ critical and we prove on $\mathbb{R}\times\mathbb{R}$ that it is globally well posed in the natural energy space if the $L^2$ norm of the initial data is less the $L^2$ norm of the ground state associated to this equation. We also find a subspace of the natural energy space associated to this equation where we have local well-posedness, nevertheless if the initial data is sufficiently localized we obtain blow-up. On $\mathbb{R}\times \mathbb{T},$ we prove global well-posedness in the energy space for small data.

Theorems & Definitions (56)

• Proposition 1
• Theorem 1.1
• Definition 1.1
• Lemma 1
• Corollary 1
• Proposition 2
• Theorem 1.2
• Theorem 1.3
• Proposition 3
• Lemma 2