Almost sure local well-posedness for a derivative nonlinear wave equation

TL;DR

This paper proves the local existence of solutions for a derivative nonlinear wave equation with randomized initial data at super-critical regularities, overcoming the lack of probabilistic smoothing effects through innovative decomposition and refined estimates.

Contribution

It introduces an adaptive iterative decomposition method and refined Strichartz estimates to establish local well-posedness below the Lorentz-critical regularity for the first time.

Findings

Solutions exist locally for initial data at regularities s ≥ 1.984

Overcomes the absence of probabilistic smoothing effects

Employs refined Strichartz estimates and decomposition techniques

Abstract

We study the derivative nonlinear wave equation \( - \partial_{tt} u + \Delta u = |\nabla u|^2 \) on \( \mathbb{R}^{1+3} \). The deterministic theory is determined by the Lorentz-critical regularity \( s_L = 2 \), and both local well-posedness above \( s_L \) as well as ill-posedness below \( s_L \) are known. In this paper, we show the local existence of solutions for randomized initial data at the super-critical regularities \( s\geq 1.984\). In comparison to the previous literature in random dispersive equations, the main difficulty is the absence of a (probabilistic) nonlinear smoothing effect. To overcome this, we introduce an adaptive and iterative decomposition of approximate solutions into rough and smooth components. In addition, our argument relies on refined Strichartz estimates, a paraproduct decomposition, and the truncation method of de Bouard and Debussche.