Can one do mathematical physics research without much knowledge of physics
Historically, most of the mathematics was developed to solve physical problems, and a fair amount of physical intuition helps to understand the ideas behind the mathematical methods, interpret solutions and check that their mathematical form is consistent with subject-specific problem, and also to develop new methods and communicate them to the users. So in general some background in Physics improves your chances to do a successful research in Applied Mathematics, and in particular to carry out PhD program.

However, there are some areas of Maths where you do just fine without any Physics. For example, general analysis of PDEs can lead you into functional analysis, and I don't think that analysis of properties of Sobolev spaces relies hugely on physical intuition.