Almost sure local well-posedness for a derivative nonlinear wave equation
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866915314055249920 |
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| author | Bringmann, Bjoern |
| author_facet | Bringmann, Bjoern |
| contents | We study the derivative nonlinear wave equation \( - \partial_{tt} u + Δu = |\nabla u|^2 \) 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1809_00220 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Almost sure local well-posedness for a derivative nonlinear wave equation Bringmann, Bjoern Analysis of PDEs 35L05, 35L15, 35L71 We study the derivative nonlinear wave equation \( - \partial_{tt} u + Δu = |\nabla u|^2 \) 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. |
| title | Almost sure local well-posedness for a derivative nonlinear wave equation |
| topic | Analysis of PDEs 35L05, 35L15, 35L71 |
| url | https://arxiv.org/abs/1809.00220 |