论文标题
MKDV类型方程的耦合系统的尖锐适应性
Sharp well-posedness for a coupled system of mKdV type equations
论文作者
论文摘要
我们考虑与修改后的Korteweg-de Vries类型方程$$相关的初始值问题 \ partial_tv + \ partial_x^3v + \ partial_x（vw^2）= 0，\ \ v（x，0）= ϕ（x），， $$ $$ \partial_tw + α\partial_x^3w + \partial_x(v^2w) =0,\ \ w(x,0)=ψ(x),$$ and prove the local well-posedness results for given data in low regularity Sobolev spaces $H^{s}(\textrm{I}\!\textrm{R})\times h^{k}（\ textrm {i} \！\！\ textrm {r}）$，$ s，k> - \ frac12 $和$ | s -k | \ leq 1/2 $，对于$α\ neq 0,1 $。另外，我们证明：（i）在$ s <-1/2 $或$ k <-1/2 $或$ | s-k |> 2 $时，将初始数据带到解决方案的解决方案映射在原点上不可能为$ c^3 $; （ii）当（a）$ s-2k> 1 $或$ k <-1/2 $（b）$ k-2s> 1 $或$ s <-1/2 $（a）$ s-2k> 1 $或$ k <-1/2 $（a）$ s-2k> 1 $或$ k <-1/2 $; （c）$ s = k = -1/2 $; （iii）在某种意义上，我们无法减少三联估计的证据，证明了一些相关的双线性估计值（如Tao [19]），因此局部供应良好的结果是敏锐的。
We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations $$ \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,\ \ v(x,0)=ϕ(x), $$ $$ \partial_tw + α\partial_x^3w + \partial_x(v^2w) =0,\ \ w(x,0)=ψ(x),$$ and prove the local well-posedness results for given data in low regularity Sobolev spaces $H^{s}(\textrm{I}\!\textrm{R})\times H^{k}(\textrm{I}\!\textrm{R})$, $s,k> -\frac12$ and $|s-k|\leq 1/2$, for $α\neq 0,1$. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be $C^3$ at the origin, when $s<-1/2$ or $k<-1/2$ or $|s-k|>2$; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) $s-2k>1$ or $k<-1/2$ (b) $k-2s>1$ or $s<-1/2$; (c) $s=k=-1/2 $; (III) the local well-posedness result is sharp in a sense that we can not reduce the proof of the trilinear estimates, proving some related bilinear estimates (as in Tao [19]).
