A note on the Prandtl boundary layers

Abstract

This note concerns a nonlinear ill-posedness of the Prandtl equation and an invalidity of asymptotic boundary-layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear ill-posedness results established by Gérard-Varet and Dormy [2], and an analysis in Guo and Tice [5]. We show that the asymptotic boundary-layer expansion is not valid for non-monotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well-posedness and prove that the nonlinear Prandtl equation is not well-posed in this sense near non-stationary and non-monotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well-posed.

incorporated referee's comments. to appear CPAM

References in corpus (1)

• On the ill-posedness of the Prandtl equation

Cited by in corpus (27)

• On the ill-posedness of the Prandtl equations in three space dimensions
• On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations
• Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations
• Global-in-$x$ Stability of Steady Prandtl Expansions for 2D Navier-Stokes Flows
• A well-posedness theory for the Prandtl equations in three space variables
• Well-posedness in Gevrey function space for 3D Prandtl equations without Structural Assumption
• Justification of Prandtl Ansatz for MHD boundary layer
• Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces
• Boundary Layer Expansions for the Stationary Navier-Stokes Equations
• Well-posedness of The Prandtl Equation in Sobolev Spaces
• MHD boundary layers theory in Sobolev spaces without monotonicity. I. well-posedness theory
• On the inviscid limit of the Navier-Stokes equations
• On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions
• Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate
• Regularity and Expansion for Steady Prandtl Equations
• Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow
• Separation of the two-dimensional unsteady Prandtl boundary layers under an adverse pressure gradient
• Well-Posedness of the Nonlinear Unsteady Prandtl Equations with Robin Boundary Condition in Weighted Sobolev Spaces
• Local-in-time Well-posedness of Boundary Layer System for the Full Incompressible MHD Equations by Energy Methods
• Well-posedness of thermal layer equations for inviscid compressible flows
• Global-in-time Stability of 2D MHD boundary Layer in the Prandtl-Hartmann Regime
• Asymptotic Analysis of Transport Equation in Bounded Domains
• Full Regularity and Well-Posedness of the Nonlinear Unsteady Prandtl Equations with Robin or Dirichlet Boundary Condition
• Boundary layers interactions in the plane parallel incompressible flows
• Symmetrical Prandtl boundary layer expansions of steady Navier-Stokes equations on bounded domain
• Local well-posedness of Prandtl equations for compressible flow in two space variables
• Remarks on the Prandtl Equation