On 2D Navier-Stokes free boundary: nonnegative density and small viscosity contrast

2025/07/12 by Francisco Gancedo, Gancedo, Francisco, Eduardo García-Juárez +3 · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.09333

openalex publication_date 2025/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is concerned with the evolution of two incompressible, immiscible fluids in two dimensions governed by the inhomogeneous Navier-Stokes equations. We prove global-in-time well-posedness, establishing the preservation of the natural C^1+γ Hölder regularity of the free boundary, for 0<γ<1. This is the first result that allows for nonnegative density driven by a low-regularity initial velocity, while also remaining valid in the presence of a small viscosity jump.

Citations

• Global-in-time well-posedness for the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient
• Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density
• On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations
• Free boundary regularity of vacuum states for incompressible viscous flows in unbounded domains
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• Compressible Navier-Stokes equations with ripped density
• Some free boundary problem for two phase inhomogeneous incompressible flow
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• The incompressible navier-stokes equations in vacuum
• Global regularity of 2D density patches for inhomogeneous Navier-Stokes
• Almost Exponential Decay of Periodic Viscous Surface Waves without Surface Tension
• Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density
• The Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay
• A Lagrangian approach for the incompressible Navier-Stokes equations with variable density
• Decay of viscous surface waves without surface tension

Cited by

• Well-posedness for 2D non-homogeneous incompressible fluids with general density-dependent odd viscosity

Related

• Global Regularity of 2D Navier-Stokes Free Boundary with Small Viscosity Contrast
• Global solutions to 2-d inhomogeneous navier-stokes system with general velocity
• Global regularity of 2D density patches for inhomogeneous Navier-Stokes
• Global-in-time well-posedness for the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient
• Global regularities of two-dimensional density patch for inhomogeneous incompressible viscous flow with general density