[Paper Review] The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality

Shuanglin Shao|ArXiv.org|Aug 31, 2008

Advanced Mathematical Physics Problems27 references3 citations

TL;DR

This paper establishes a linear profile decomposition for the Airy equation in $L^2$ with real or complex initial data, using refined Strichartz estimates and concentration-compactness principles. It proves the existence of maximizers for the Airy-Strichartz inequality by analyzing concentration profiles under symmetries, resolving the lack of Galilean and pseudo-conformal symmetries via a novel adaptation of profile decomposition techniques.

ABSTRACT

In this paper, we establish the linear profile decomposition for the Airy equation with complex or real initial data in $L^2$, respectively. As an application, we obtain a dichotomy result on the existence of maximizers for the symmetric Airy-Strichartz inequality.

Motivation & Objective

• To develop a linear profile decomposition for the Airy equation with $L^2$ initial data, addressing the absence of Galilean and pseudo-conformal symmetries present in Schrödinger equations.
• To establish a dichotomy on the existence of maximizers for the symmetric Airy-Strichartz inequality via profile decomposition and concentration-compactness.
• To prove the existence of extremal functions (maximizers) for the Airy-Strichartz inequality in the critical $L^6_{t,x}$ norm with scaling-critical regularity.
• To adapt techniques from Schrödinger equation profile decompositions to the Airy equation by compensating for missing symmetries through refined analysis of frequency and space localization.

Proposed method

• Utilizes the refined Strichartz inequality of Kenig-Ponce-Vega as a foundational estimate to control the $L^6_{t,x}$ norm of solutions.
• Applies the concentration-compactness principle to handle the lack of compactness in the Strichartz inequality, decomposing sequences into profiles and a remainder.
• Implements a profile decomposition framework where solutions are expressed as superpositions of rescaled, translated, and modulated profiles, with almost orthogonal behavior in Strichartz and $L^2$ norms.
• Analyzes asymptotic behavior under scaling and modulation parameters ($h_n^j$, $\xi_n^j$), distinguishing cases based on whether $h_n^j \xi_n^j$ converges or diverges.
• Uses the almost orthogonality of profiles in the $L^6_{t,x}$ norm to derive a contradiction when multiple profiles contribute significantly, isolating a single maximizer.
• Applies the limiting profile argument in the case $|h_n^j \xi_n^j| \to \infty$, relating the Airy Strichartz norm to the Schrödinger Strichartz norm via a change of variables and asymptotic equivalence.

Experimental results

Research questions

• RQ1 Does a linear profile decomposition exist for the Airy equation in $L^2$ despite the absence of Galilean and pseudo-conformal symmetries?
• RQ2 Can the existence of maximizers for the Airy-Strichartz inequality be established using profile decomposition techniques?
• RQ3 What is the structure of extremal sequences for the Airy-Strichartz inequality, and how do they relate to known maximizers of the Schrödinger equation?
• RQ4 How does the lack of symmetries affect the concentration-compactness approach in the Airy equation context?

Key findings

• The linear profile decomposition for the Airy equation in $L^2$ is established for both real and complex initial data, decomposing bounded sequences into profiles and a remainder term.
• A dichotomy is proven: either a maximizer exists for the Airy-Strichartz inequality, or the extremal sequence concentrates in a way that reduces to the Schrödinger case.
• When $h_n^j \xi_n^j \to \xi^{j_0} \in \mathbb{R}$, the maximizer $\phi^{j_0}$ satisfies $\|D^{1/6}e^{-t\partial_x^3}\phi^{j_0}\|_{L^6_{t,x}} = S_{\text{airy}}^{\mathbb{C}}$ and $\|\phi^{j_0}\|_{L^2} = 1$, proving existence of an Airy maximizer.
• When $|h_n^j \xi_n^j| \to \infty$, the limiting profile satisfies $S_{\text{airy}}^{\mathbb{C}} = 3^{-1/6} S_{\text{schr}}^{\mathbb{C}}$, and $\phi^{j_0}$ is a maximizer for the Schrödinger Strichartz inequality.
• The proof shows that $S_{\text{airy}}^{\mathbb{C}} = 3^{-1/6} S_{\text{schr}}^{\mathbb{C}}$ under the divergent frequency case, linking the Airy and Schrödinger extremal problems.
• The existence of a maximizer is guaranteed in the critical case $q=r=6$, $\alpha=1/6$, by showing that the extremal sequence concentrates into a single profile with unit $L^2$ norm and optimal Strichartz norm.