How to only blend two textures partly within one material?

I have two textures, the first one like a white round in the middle, and the other part is all black. The second is a general dirty texture. The final result what i want is, the white round part is still white, the black part is still black, but do a blend of two texture at the black-and-white junction part, and make this transition gradient. Is that possible to get it only with material stuff?

The simply lerp or multiply seems not enough for this request. It is glad to know any suggestions! Thanks!

I would try vertex painting.