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

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.