论文标题
Muskat方程的终点Sobolev理论
Endpoint Sobolev theory for the Muskat equation
论文作者
论文摘要
本文致力于针对二维Muskat方程具有关键规律性的解决方案的研究。我们证明，在$ l^2 $函数的端点Sobolev空间中，Cauchy问题在$ l^2 $中均得到很好的影响。相对于方程式的缩放，该结果是最佳的。一个众所周知的困难是，一个人无法定义流量图，以使寿命在此关键Sobolev空间的有限子集上从下面界定。为了克服这一点，我们使用以前的作品中引入的加权分数拉普拉斯人来估算取决于初始数据本身的规范的解决方案。我们的证明是第一个为Muskat方程确定了无效的结构，从而可以补偿大斜率的抛物线行为的退化。
This paper is devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. We prove that the Cauchy problem is well-posed on the endpoint Sobolev space of $L^2$ functions with three-half derivative in $L^2$. This result is optimal with respect to the scaling of the equation. One well-known difficulty is that one cannot define a flow map such that the lifespan is bounded from below on bounded subsets of this critical Sobolev space. To overcome this, we estimate the solutions for a norm which depends on the initial data themselves, using the weighted fractional Laplacians introduced in our previous works. Our proof is the first in which a null-type structure is identified for the Muskat equation, allowing to compensate for the degeneracy of the parabolic behavior for large slopes.
