论文标题
旋转修饰的kadomtsev-petviashvili方程的库奇问题的尖锐适应性
Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces
论文作者
论文摘要
我们考虑了旋转修饰的kadomtsev-petviashvili（rmkp）方程\ begin \ begin {align*} \ partial_ {x} \ left（u_ {t}-β\ partial_ {x} -partial_ {x} {x}^{3} u +\ partial_ {x}（u^{2}）\ right） +\ partial_ {y}^{2} u-γu= 0 \ end End {align*}在Anisotropic Sobolev space $当$β<0 $和$γ> 0时，$我们证明了Cauchy的问题在$ h^{s_ {1}，\> s_ {2}}}（\ Mathbb {r}^{2}} $中，带有$ S_我们的结果大大提高了R. M. Chen，Y. Liu，P。Z. Zhang的定理1.4（美国数学学会的交易，364（2012），3395---3425。关键的想法是，我们将频率空间分为常规区域和单数区域。 We further prove that the Cauchy problem for RMKP equation is ill-posed in $H^{s_{1},\>0}(\mathbb{R}^{2})$ with $s_{1}<-\frac{1}{2}$ in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not $ c^{3} $。当$β<0，γ> 0，$通过使用$ u^{p} $和$ v^{p} $ space，我们证明了cauchy问题在$ h^{ - \ frac {1} {1} {1} {2} {2} {2}，\> 0}，\> 0}（\ mathbb {\ mathbb {r}^2}^{2}）中。
We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation \begin{align*} \partial_{x}\left(u_{t}-β\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-γu=0 \end{align*} in the anisotropic Sobolev spaces $H^{s_{1},\>s_{2}}(\mathbb{R}^{2})$. When $β<0$ and $γ>0,$ we prove that the Cauchy problem is locally well-posed in $H^{s_{1},\>s_{2}}(\mathbb{R}^{2})$ with $s_{1}>-\frac{1}{2}$ and $s_{2}\geq 0$. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang( Transactions of the American Mathematical Society, 364(2012), 3395--3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in $H^{s_{1},\>0}(\mathbb{R}^{2})$ with $s_{1}<-\frac{1}{2}$ in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not $C^{3}$. When $β<0,γ>0,$ by using the $U^{p}$ and $V^{p}$ spaces, we prove that the Cauchy problem is locally well-posed in $H^{-\frac{1}{2},\>0}(\mathbb{R}^{2})$.
