论文标题
关于Zakharov-Kuznetsov方程的大型解决方案的局部能量衰减
On local energy decay for large solutions of the Zakharov-Kuznetsov equation
论文作者
论文摘要
我们考虑Zakharov-Kutznesov（ZK）方程式以$ \ Mathbb r^d $，$ d = 2 $和$ 3 $。这两个方程式在$ l^2（\ Mathbb r^d）$中均具有良好的功能。在本文中，我们证明了全球解决方案的局部能量衰减：如果$ u（t）$是$ l^2（\ mathbb r^d）$的ZK解决方案，则\ [\ liminf_ {\ liminf_ { x}，t）\ mathrm {d} {\ bf x} = 0，\]对于空间的合适区域$ω_d（t）\ subseteq \ subseteq \ mathbb r^d $周围的原点周围，及时无绑定，不及时生长，不包含soliton区域。我们还证明了$ h^1（\ mathbb r^d）$解决方案的本地衰减。作为副产品，我们的结果扩展了Gustavo Ponce和第二作者证明的KDV和四分之一KDV方程的衰减属性。还提供了衰减和其他强衰减结果的顺序速率。
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.
