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

2018/09/01 by Bringmann, Bjoern · 2 citations
#35L05 #35L15 #35L71 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.00220

Abstract

We study the derivative nonlinear wave equation \( - ∂_tt u + Δu = |∇ u|² \) on \( ℝ¹⁺³ \). 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 \( s≥ 1.984\). 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.

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• Invariant Gibbs dynamics for the nonlinear Schrödinger equations on the disc
• Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation

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