The author is not familiared with the current theories and the specific model for dark matter, nor the PDE aspect of mathematical theories related to the dark matter problem, so I apologize in advance if the opinions are out of date or the question is mathematically ill-posed currently, and welcome any advise and improvement!
Some backgrouds: Originally dark matter was posed to explain the flattening of spiral rotational curve observed in galaxies, which does not match the density distribution measured from luminosity, modelled by Boltzmann equation (and its deviations). See for example [Chandrasekhar S.;Principles of Stellar Dynamics]. There is also an interesting stability theory which explained the formation of spiral arms [C.LIN,F.SHU;On the spiral structure of disk galaxies]. To quote from Poincare's celebrated "Les Méthodes Nouvelles de la Mécanique Céleste":
'Le but final de la Mécanique céleste est de résoudre cette grande question de savoir si la loi de Newton explique à elle seule tous les phénomènes astronomiques.'
Motivated by this, if one considers a simplified 2-dimensional system, with axially-symmetric velocity field $u_i(x,t)$ given by rotation curves observed, and $\rho_i(x,t)$ the correponding density distribution measured. My question may be formulated:
Question: How to find the functional $F(u_i,\rho_i,\frac{\partial u_i}{\partial x},\frac{\partial \rho_i}{\partial x},\dots,x,t)=0$ which hold for all $i$, such that the functional $F$ coincide with newtonian gravitational law locally in $x$, and is generic in the functional space?
A first impression is to do as Kepler did: trial and error. But from a mathematical point of view, there are still many unexplored area in differential\difference equations which may lead to the well-posedness (or ill-posedness) of the above question. Inverse problems in PDEs seem to deal with the case where the differential equations underlining the problem are given, and to find the coefficients. But is there any progress on finding the underlying differential equation itself?
