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arXiv 2007.04918math.AP
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Zakharov-Kuznetsov方程大解的局部能量衰减

On local energy decay for large solutions of the Zakharov-Kuznetsov equation

Argenis Méndez, Claudio Muñoz, Felipe Poblete, Juan C. Pozo

备注：
45 pp., 4 figs
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摘要：

本文考虑Zakharov-Kutznesov（ZK）方程在$\mathbb R^d$中的存在性，其中有$d=2$和$3$ .这两个方程在$L^2(\mathbb R^d)$中是整体适定的。在本文中，我们证明了局部能量衰减的整体解决方案：如果$u(t)$是一个解决方案，ZK与$L^2(\mathbb R^d)$中的数据，然后\[ \liminf_{t\rightarrow \infty}\int_{Ω_d(t)}u^{2}({\bf x},t)\mathrm{d}{\bf x}=0, \]的适当区域的空间$Ω_d(t)\subseteq \mathbb R^d$周围的起源，增长无界的时间，不包含孤子区域。我们还证明了局部衰减的$H^1(\mathbb R^d)$解决方案。作为一个副产品，我们的结果扩展了由Gustavo Ponce和第二作者证明的KdV和四次KdV方程的衰变性质。还提供了连续的衰减率和其他强衰减结果。

英文摘要：

We consider the Zakharov-Kutznesov (ZK) equation posed in $\mathbb R^d$, with $d=2$ and $3$. Both equations are globally well-posed in $L^2(\mathbb R^d)$. In this paper, we prove local energy decay of global solutions: if $u(t)$ is a solution to ZK with data in $L^2(\mathbb R^d)$, then \[ \liminf_{t\rightarrow \infty}\int_{Ω_d(t)}u^{2}({\bf x},t)\mathrm{d}{\bf x}=0, \] for suitable regions of space $Ω_d(t)\subseteq \mathbb R^d$ around the origin, growing unbounded in time, not containing the soliton region. We also prove local decay for $H^1(\mathbb R^d)$ solutions. As a byproduct, our results extend decay properties for KdV and quartic KdV equations proved by Gustavo Ponce and the second author. Sequential rates of decay and other strong decay results are also provided as well.

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