论文标题
对聚焦2D修改的Zakharov-Kuznetsov方程的低规范爆炸解决方案的质量浓缩
Mass-concentration of low-regularity blow-up solutions to the focusing 2D modified Zakharov-Kuznetsov equation
论文作者
论文摘要
我们考虑了两个空间维度的聚焦修改后的Zakharov-Kuznetsov（MZK）方程。我们证明，在$ h^1（\ r^{2}）$ norm中在有限时间内爆炸的解决方案具有将其集中在其质量中的非平凡部分（更确切地说，至少，至少等于等于地面状态的质量）。对于$ h^s（\ r^2）$ norm in $ \ frac {17} {18} {18} <s <1 $的有限时间爆破解决方案，我们证明结果稍弱。此外，我们证明可以将更强的浓度结果扩展到$ \ frac {17} {18} {18} <s \ le 1 $，这是在解决方案的爆破速率上的额外假设下。此处使用的主要工具是$ i $ -METHOD和一个配置文件分解定理，用于$ h^1（\ r^{2}）$ functions的有限族。
We consider the focusing modified Zakharov-Kuznetsov (mZK) equation in two space dimensions. We prove that solutions which blow up in finite time in the $H^1(\R^{2})$ norm have the property that they concentrate a non-trivial portion of their mass (more precisely, at least the amount equal to the mass of the ground state) at blow-up time. For finite-time blow-up solutions in the $H^s(\R^2)$ norm for $\frac{17}{18} < s < 1$, we prove a slightly weaker result. Moreover, we prove that the stronger concentration result can be extended to the range $ \frac{17}{18} < s \le 1$ under an additional assumption on the upper bound of the blow-up rate of the solution. The main tools used here are the $I$-method and a profile decomposition theorem for a bounded family of $H^1(\R^{2})$ functions.
