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

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 ${\eta }_{\epsilon }(t,x)=t^{1\mathrm{/}3}{U}_{\epsilon }(t^{-1\mathrm{/}3}x)$ and emanate from the conical initial data ${\eta }_{\epsilon }(0,x)=\epsilon \mathrm{\mid }x\mathrm{\mid }$. 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.