Well-posedness for KdV-type equations with quadratic nonlinearity
Well-posedness for KdV-type equations with quadratic nonlinearity
复制标题
具有二次非线性的 KdV 型方程的适定性
DOI：
10.1007/s00028-019-00540-6
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Mamoru Okamoto
中科院分区：
文献类型：
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作者：
Hiroyuki Hirayama;Shinya Kinoshita;Mamoru Okamoto
我们考虑 KdV 型方程 $$\begin{aligned} \partial _tu + \frac{1}{3} \partial _x^3 u = c_1 u \partial _x^2u + c_2 (\partial _xu)^2, \quad u(0)=u_0 的柯西问题。 \end{aligned}$$Pilod (J Differ Equ 245(8):2055–2077, 2008) 表明，该柯西问题的流图在 Sobolev 空间中对于任何条件都无法进行二次微分。通过使用规范变换，我们指出如果初始数据位于有界基元内，则收缩映射定理适用于柯西问题。此外，我们证明柯西问题在有界原语中是局部适定的。
We consider the Cauchy problem of the KdV-type equation $$\begin{aligned} \partial _tu + \frac{1}{3} \partial _x^3 u = c_1 u \partial _x^2u + c_2 (\partial _xu)^2, \quad u(0)=u_0. \end{aligned}$$Pilod (J Differ Equ 245(8):2055–2077, 2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev spacefor anyif. By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are inwith bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed inwith bounded primitives.