In this paper, we present an error estimate for the filtered Lie splitting scheme applied to the Zakharov system, characterized by solutions exhibiting very low regularity across all dimensions. Our findings are derived from the application of multilinear estimates established within the framework of discrete Bourgain spaces. Specifically, we demonstrate that when the solution $(E,z,z_t) \in H^{s+r+1/2}\times H^{s+r}\times H^{s+r-1}$, the error in $H^{r+1/2}\times H^{r}\times H^{r-1}$ is $\mathcal{O}(\tau^{s/2})$ for $s\in(0,2]$, where $r=\max(0,\frac d2-1)$. To the best of our knowledge, this represents the first explicit error estimate for the splitting method based on the original Zakharov system, as well as the first instance where low regularity error estimates for coupled equations have been considered within the Bourgain framework. Furthermore, numerical experiments confirm the validity of our theoretical results.

翻译：本文针对解在所有维度上均表现出极低正则性的Zakharov系统，提出了应用于该系统的滤波Lie分裂格式的误差估计。我们的研究结果基于在离散Bourgain空间框架下建立的多线性估计。具体而言，我们证明当解$(E,z,z_t) \in H^{s+r+1/2}\times H^{s+r}\times H^{s+r-1}$时，其在$H^{r+1/2}\times H^{r}\times H^{r-1}$空间中的误差为$\mathcal{O}(\tau^{s/2})$，其中$s\in(0,2]$，$r=\max(0,\frac d2-1)$。据我们所知，这是基于原始Zakharov系统的分裂方法的首个显式误差估计，也是在Bourgain框架下首次考虑耦合方程的低正则性误差估计。此外，数值实验验证了我们理论结果的有效性。