AI 中文总结
研究带表面张力的三维Muskat问题，构造一族含参数小自相似解，通过多种估计结合压缩论证证明正时间下锥形奇异性即刻被圆化，确定了剖面的主导二次修正项。
AI 中文摘要
我们构造了带表面张力的三维单相Muskat问题的一族含参数的小自相似解，解的形式为$\u03B7_\u03B5(t,x) = t^{1/3}U_\u03B5(t^{-1/3}x)$，源自锥形初始数据$\u03B7_\u03B5(0,x)=\u03B5|x|$。这些剖面是锥形的线性毛细管正则化的扰动，我们确定了主导二次修正项。证明结合了线性相似算子的射线方向逆估计、二次项中有利的高高到低抵消，以及渐近锥形图上Dirichlet-Neumann算子的有限阶驯服估计；这些估计通过压缩论证得到解，并表明正时间下锥形奇异性会即刻被圆化。
英文摘要
We construct a one-parameter family of small self-similar solutions to the three-dimensional one-phase Muskat problem with surface tension. The solutions have the form $η_\varepsilon(t,x) = t^{1/3}U_\varepsilon(t^{-1/3}x)$ and emanate from the conical initial data $η_\varepsilon(0,x)=\varepsilon|x|$. The profiles are perturbations of the linear capillary regularization of the cone, and we identify the leading quadratic correction. The proof combines a raywise inverse estimate for the linear similarity operator, a favorable high--high-to-low cancellation in the quadratic term, and finite-order tame estimates for the Dirichlet--Neumann operator on asymptotically conical graphs. These estimates yield the solutions by a contraction argument and show that the conical singularity is instantaneously rounded for positive time.
Comments40 pages
