Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case
Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case
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KdV-Burgers 方程的尖锐不适定性和适定性结果：实线情况
DOI：
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发表时间：
2009
期刊：
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通讯作者：
Stéphane Vento
中科院分区：
文献类型：
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作者：
L. Molinet;Stéphane Vento
通过证明KdV-Burgers方程在$H^{-1}(R) $上是适定的，且解映射从$H^{-1}(R) $解析到$C([0,T];H^{-1}(R))$，而在$H^ s(R) $上是病态的，只要$ s 0 $足够小，我们完成了关于Sobolev空间中局部柯西问题的已知结果。据我们所知，这是色散-耗散方程的第一个这种类型的结果。我们在这里开发的框架对于证明其他色散-耗散模型的类似结果应该非常有用
We complete the known results on the local Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $ H^{-1}(R) $ with a solution-map that is analytic from $H^{-1}(R) $ to $C([0,T];H^{-1}(R))$ whereas it is ill-posed in $ H^s(R) $, as soon as $ s 0 $ small enough. As far as we know, this is the first result of this type for a dispersive-dissipative equation. The framework we develop here should be very useful to prove similar results for other dispersive-dissipative models