Nonlinear dispersive regularization of inviscid gas dynamics

Abstract

Ideal gas dynamics can develop shock-like singularities with discontinuous density. Viscosity typically regularizes such singularities and leads to a shock structure. On the other hand, in 1d, singularities in the Hopf equation can be non-dissipatively smoothed via KdV dispersion. Here, we develop a minimal conservative regularization of 3d ideal adiabatic flow of a gas with polytropic exponent $γ$. It is achieved by augmenting the Hamiltonian by a capillarity energy $β(ρ) (\nabla ρ)^2$. The simplest capillarity coefficient leading to local conservation laws for mass, momentum, energy and entropy using the standard Poisson brackets is $β(ρ) = β_*/ρ$ for constant $β_*$. This leads to a Korteweg-like stress and nonlinear terms in the momentum equation with third derivatives of $ρ$, which are related to the Bohm potential and Gross quantum pressure. Just like KdV, our equations admit sound waves with a leading cubic dispersion relation, solitary and periodic traveling waves. As with KdV, there are no steady continuous shock-like solutions satisfying the Rankine-Hugoniot conditions. Nevertheless, in 1d, for $γ= 2$, numerical solutions show that the gradient catastrophe is averted through the formation of pairs of solitary waves which can display approximate phase-shift scattering. Numerics also indicate recurrent behavior in periodic domains. These observations are related to an equivalence between our regularized equations (in the special case of constant specific entropy potential flow in any dimension) and the defocussing nonlinear Schrodinger equation (cubically nonlinear for $γ= 2$), with $β_*$ playing the role of $\hbar^2$. Thus, our regularization of gas dynamics may be viewed as a generalization of both the single field KdV & NLS equations to include the adiabatic dynamics of density, velocity, pressure & entropy in any dimension.