Self-similar solutions of the three-dimensional Muskat problem with surface tension

Abstract

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 $\eta_\varepsilon(t,x) = t^{1/3}U_\varepsilon(t^{-1/3}x)$ and emanate from the conical initial data $\eta_\varepsilon(0,x)=\varepsilon|x|$. 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.