Density nonlinearities and a field theory for the dynamics of simple fluids

We study the role of the Jacobian arising from a constraint enforcing the nonlinear relationg=V, where ,g, andV are the mass density, the momentum density, and the local velocity field, respectively, in the field-theoretic formulation of the nonlinear fluctuating hydrodynamics of simple fluids. By investigating the Jacobian directly and by developing a field-theoretic formulation without the constraint, we find that no changes in dynamics result as compared to the previous formulation developed by Das and Mazenko (DM). In particular, the cutoff mechanism discovered by DM is shown to be a consequence of the 1/ nonlinearity in the problem, not of the constraint. The consequences of this result for the static properties of the system are also discussed.     

Renormalized perturbation theory for a toy model of fluctuating nonlinear hydrodynamics of supercooled liquids

We study the field theoretic renormalized perturbation theory for a toy model of fluctuating nonlinear hydrodynamics (FNH) of compressible liquids. The toy model contains a density-like and a momentum-like variable without any spatial dependence. We present a detailed derivation of a set of coupled equations among correlation and response functions for these variables. In particular, we focus on how the static limit of the correlation and response functions can be achieved in the renormalized perturbation theory. Numerical methods of solving these equations at a given order of the loop expansion are explained and the results for the one-loop theory are given in detail. The simple nature of the toy model enables us to compare the static limit obtained from the exact solution with that of the one-loop order. This shows explicitly the range of validity of the one-loop theory in the field theoretic formulation.     

Global Well-Posedness of Strong Solutions of Doi Model with Large Viscous Stress

We study models of dilute rigid rod-like polymer solutions. We establish the global well-posedness of the Doi model for large data and for arbitrarily large viscous stress parameter. The main ingredient in the proof is the fact that the viscous stress adds dissipation to high derivatives of velocity.