Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

We study the derivative nonlinear wave equation ∂ttu + u = |δu|2 on R1+3. The deterministic theory is determined by the Lorentz-critical regularity sL = 2, and both local well-posedness above sL as well as ill-posedness below sL are known. In this paper, we show the local existence of solutions for randomized initial data at the super-critical regularities s 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.