We prove a local well posedness result for the modified Korteweg-de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an as-ymptotic description of small solutions as t $\rightarrow$ +$\infty$ and construct solutions with a prescribed blow up behavior as t $\rightarrow$ 0.

中文翻译：

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修正 Korteweg-de Vries 方程的自相似动力学

我们在设计为包含自相似解的临界空间中证明了修正 Korteweg-de Vries 方程的局部适定性结果。因此，我们可以围绕自相似解来研究这个方程的流动：特别是，我们将小解的渐近描述为 t $\rightarrow$ +$\infty$ 并构造具有指定爆炸的解行为为 t $\rightarrow$ 0。