Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces
Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces
复制标题
DOI：
--
复制
发表时间：
2019-04
期刊：
影响因子：
--
通讯作者：
Tadahiro Oh;Oana Pocovnicu;N. Tzvetkov
中科院分区：
文献类型：
--
作者：
Tadahiro Oh;Oana Pocovnicu;N. Tzvetkov
我们研究随机初始数据低于 $L^2(\mathbb{T}^3)$ 的三维三次非线性波动方程 (NLW)。通过考虑随机线性解的二阶展开，我们证明了负 Sobolev 空间中重整化 NLW 的几乎确定的局部适定性。我们还证明了负 Sobolev 空间中未重整化的散焦三次 NLW 的新的不稳定性结果，这符合随机偏微分方程研究中所谓琐碎性的精神。更准确地说，通过使用给定的平滑确定性初始数据加上一定的截断随机初始数据来研究（未重整化）NLW，我们表明，当截断被删除时，对于任何确定性初始数据，解在分布意义上收敛到 $0$。
We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below $L^2(\mathbb{T}^3)$. By considering the second order expansion in terms of the random linear solution, we prove almost sure local well-posedness of the renormalized NLW in negative Sobolev spaces. We also prove a new instability result for the defocusing cubic NLW without renormalization in negative Sobolev spaces, which is in the spirit of the so-called triviality in the study of stochastic partial differential equations. More precisely, by studying (un-renormalized) NLW with given smooth deterministic initial data plus a certain truncated random initial data, we show that, as the truncation is removed, the solutions converge to $0$ in the distributional sense for any deterministic initial data.