Compressible Navier--Stokes Flow in Schr\"odinger-Type Variables

TL;DR

This paper derives an exact reformulation of compressible Navier-Stokes equations into Schr"odinger-type equations using amplitude variables, enabling new analytical and computational approaches.

Contribution

It presents the first exact Cole-Hopf-type Schr"odinger-variable reformulation of compressible Navier-Stokes flow, connecting fluid dynamics with quantum-inspired equations.

Findings

Exact two-dimensional reformulation verified against direct simulations.

Reformulation exposes operator structures for potential quantum algorithms.

Extension to three dimensions maintains core structure with additional analogues.

Abstract

Fluid equations are nonlinear, dissipative, and non-Hamiltonian, which makes their relation to Schr\"odinger evolution and quantum algorithms nontrivial. We derive an exact Eulerian Cole-Hopf-type reformulation of isothermal compressible Navier-Stokes (NS) flow in Schr\"odinger-type amplitude variables. To our knowledge, this gives the first exact Cole-Hopf-type Schr\"odinger-variable reformulation of compressible NS flow. In two dimensions, a Helmholtz decomposition separates the velocity into compressive and vortical potentials, whose logarithmic transforms yield two scalar imaginary-time Schr\"odinger-type equations with nonlinear self-consistent potentials. We show that the mixed density-compressive amplitude $\Psi_\alpha=\rho^\alpha\Theta^{1-2\alpha}$, where $\rho$ is the density, $\Theta$ is the compressive amplitude, and $\alpha\neq 0,\,1/2$, satisfies a nonlinear Schr\"odinger-type equation with a vector-potential-coupled Laplacian. The transformed system is exactly equivalent to compressible NS and is nonlocal only through Helmholtz and Poisson projections. In three dimensions, the density-carrying equation retains the same vector-potential-coupled structure, while the solenoidal sector admits a compressible analogue of Ohkitani's incompressible NS Cole-Hopf formulation. Unlike unitary hydrodynamic Schr\"odinger-flow representations, the present equations are imaginary-time heat or drift-diffusion equations with self-consistent potentials, but they remain an exact change of variables for compressible NS. A two-dimensional Kelvin-Helmholtz unstable shear-layer calculation verifies the transformed equations against a direct compressible NS simulation. The formulation exposes operator structures that may be useful for reduced flow descriptions, quantum algorithms for operator evolution, and quantum partial differential equation solvers.