论文标题
索波列夫（Sobolev）索波列夫（Sobolev
Resonant Decompositions and Global Well-posedness for 2D Zakharov-Kuznetsov Equation in Sobolev spaces of Negative Indices
论文作者
论文摘要
$ \ mathbb {r}^2 $上的Zakharov-Kuznetsov方程的Cauchy问题显示为全球范围的$ H^{s} $提供的初始日期，提供了$ h^{s} $提供$ s> - \ frac {1} {1} {13} $。由于保护法在$ l^2 $以下的Sobolev空间中无效，因此我们使用Colliander，Kekel，Staffilani，Takaoka和Tao引入的$ i $ -Method构建了几乎保守的数量。与KDV方程相反，主要困难是处理问题的多维和多线性设置，这些相互作用很重要。证明依赖于双线性strichartz估计和非线性织机 - 惠特尼不平等。
The Cauchy problem for Zakharov-Kuznetsov equation on $\mathbb{R}^2$ is shown to be global well-posed for the initial date in $H^{s}$ provided $s>-\frac{1}{13}$. As conservation laws are invalid in Sobolev spaces below $L^2$, we construct an almost conserved quantity using multilinear correction term following the $I$-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In contrast to KdV equation, the main difficulty is to handle the resonant interactions which are significant due to the multidimensional and multilinear setting of the problem. The proof relies upon the bilinear Strichartz estimate and the nonlinear Loomis-Whitney inequality.
