2025/06/21 by Dormy, Emmanuel, Gerard-Varet, David
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn)
paper · doi:10.48550/arxiv.2506.17749
We provide a mathematical analysis of the `diffusion-free' boundary conditions recently introduced by Lin and Kerswell for the numerical treatment of inertial waves in a fluid contained in a rotating sphere. We consider here the full setting of the nonlinear Navier-Stokes equation in a general bounded domain Ω of ℝd, d=2 or 3. We show that diffusion-free boundary conditions Δu ⋅ τ\vert∂ Ω = 0, u ⋅ n\vert∂ Ω = 0 when d=2, Δu × n\vert∂ Ω = 0, u ⋅ n\vert∂ Ω = 0 when d=3, allow for a satisfactory well-posedness theory of the full Navier-Stokes equations (global in time for d=2, local for d=3). Moreover, we perform a boundary layer analysis in the limit of vanishing viscosity ν→ 0. We establish that the amplitude of the boundary layer flow is in this case of order ν, i.e. much lower than in the case of standard Dirichlet or even stress-free conditions. This confirms analytically that this choice of boundary conditions may be used to reduce diffusive effects in numerical studies relying on the Navier-Stokes equation to approach nearly inviscid solutions.