Rigidity of acute angled corners for one phase Muskat interfaces

We consider the one-phase Muskat problem modeling dynamics of free boundary a single fluid in porous media. In stable regime, we prove local well-posedness for interfaces that are general curves and can have singularities. particular, acute angle corners or cusps. Moreover, show isolated corners/cusps on interface must be rigid, meaning corner is preserved finite time, there no rotation at tip, particle tip remains velocity points vertically downward.