Compressible Navier--Stokes Flow in Schrödinger-Type Variables
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866914517749858304 |
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| author | Beattie, James R. Sokolova, Max Negandhi, Khush Ripperda, Bart |
| author_facet | Beattie, James R. Sokolova, Max Negandhi, Khush Ripperda, Bart |
| contents | Fluid equations are nonlinear, dissipative, and non-Hamiltonian, which makes their relation to Schrödinger evolution and quantum algorithms nontrivial. We derive an exact Eulerian Cole-Hopf-type reformulation of isothermal compressible Navier-Stokes (NS) flow in Schrödinger-type amplitude variables. To our knowledge, this gives the first exact Cole-Hopf-type Schrödinger-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 Schrödinger-type equations with nonlinear self-consistent potentials. We show that the mixed density-compressive amplitude $Ψ_α=ρ^αΘ^{1-2α}$, where $ρ$ is the density, $Θ$ is the compressive amplitude, and $α\neq 0,\,1/2$, satisfies a nonlinear Schrödinger-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 Schrödinger-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27088 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Compressible Navier--Stokes Flow in Schrödinger-Type Variables Beattie, James R. Sokolova, Max Negandhi, Khush Ripperda, Bart Fluid Dynamics High Energy Astrophysical Phenomena Plasma Physics Fluid equations are nonlinear, dissipative, and non-Hamiltonian, which makes their relation to Schrödinger evolution and quantum algorithms nontrivial. We derive an exact Eulerian Cole-Hopf-type reformulation of isothermal compressible Navier-Stokes (NS) flow in Schrödinger-type amplitude variables. To our knowledge, this gives the first exact Cole-Hopf-type Schrödinger-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 Schrödinger-type equations with nonlinear self-consistent potentials. We show that the mixed density-compressive amplitude $Ψ_α=ρ^αΘ^{1-2α}$, where $ρ$ is the density, $Θ$ is the compressive amplitude, and $α\neq 0,\,1/2$, satisfies a nonlinear Schrödinger-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 Schrödinger-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. |
| title | Compressible Navier--Stokes Flow in Schrödinger-Type Variables |
| topic | Fluid Dynamics High Energy Astrophysical Phenomena Plasma Physics |
| url | https://arxiv.org/abs/2604.27088 |