On the ill-posedness of the Prandtl equation

Abstract

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data, or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with non-degenerate critical points. Interestingly, the strong instability is due to vicosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.

A few more typos corrected. Definition of L_eps added on page 12. Confusing semigroup notation corrected

Cited by in corpus (94)

• Almost global existence for the Prandtl boundary layer equations
• Gevrey Stability of Prandtl Expansions for 2D Navier-Stokes
• Mathematics and Turbulence: where do we stand?
• On the ill-posedness of the Prandtl equations in three space dimensions
• Formal Derivation and Stability Analysis of Boundary Layer Models in MHD
• A note on the Prandtl boundary layers
• Ill-posedness of the hydrostatic Euler and singular Vlasov equations
• Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations
• On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain
• The inviscid limit of Navier-Stokes for analytic data on the half-space
• On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations
• Stability issues in the quasineutral limit of the one-dimensional Vlasov-Poisson equation
• Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points
• Sublayer of Prandtl boundary layers
• Energy dissipation caused by boundary layer instability at vanishing viscosity
• A Kato type Theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body
• On the Euler+Prandtl expansion for the Navier-Stokes equations
• Vanishing viscosity and surface tension limits of incompressible viscous surface waves
• Well-posedness of boundary layer equations for time-dependent flow of non-Newtonian fluids
• Optimal Prandtl expansion around concave boundary layer
• Global-in-$x$ Stability of Steady Prandtl Expansions for 2D Navier-Stokes Flows
• Optimal Gevrey stability of hydrostatic approximation for the Navier-Stokes equations in a thin domain
• On the supercritically diffusive magneto-geostrophic equations
• Linear instability analysis on compressible Navier-Stokes equations with strong boundary layer
• Long time well-posdness of Prandtl system with small and analytic initial data
• Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit
• Improved well-posedness for the Triple-Deck and related models via concavity
• Global existence and decay of solutions to Prandtl system with small analytic data
• Justification of Prandtl Ansatz for MHD boundary layer
• 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
• On the zero-viscosity limit of the Navier-Stokes equations in the half-space
• On the Well-posedness of Reduced $3D$ Primitive Geostrophic Adjustment Model with Weak Dissipation
• On the Steady Prandtl Boundary Layer Expansions
• Hydrostatic approximation of the 2D MHD system in a thin strip with a small analytic data
• Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces
• Local-in-Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods
• Some asymptotic profiles for the viscous Moore-Gibson-Thompson equation in the $L^q$ framework
• Well-posedness in Gevrey function space for the three-dimensional Prandtl equations
• The van Dommelen and Shen singularity in the Prandtl equations
• Hölder Continuous Solutions to the Three-dimensional Prandtl System
• Boundary Layer Expansions for the Stationary Navier-Stokes Equations
• Nonuniqueness of generalised weak solutions to the primitive and Prandtl equations
• On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class
• MHD boundary layers theory in Sobolev spaces without monotonicity. I. well-posedness theory
• Well-posedness of the Prandtl equation without any structural assumption
• Layer separation of the 3D incompressible Navier-Stokes equation in a bounded domain
• Verification of Prandtl boundary layer ansatz for the steady electrically conducting fluids with a moving physical boundary
• On the $L^{\infty }$ stability of Prandtl expansions in Gevrey class
• On the inviscid limit of the Navier-Stokes equations
• Vanishing viscosity limit of navier-stokes equations in gevrey class
• The Inviscid Limit of the Navier-Stokes Equations with Kinematic and Navier Boundary Conditions
• On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions
• The inviscid limit for the Navier-Stokes equations with data analytic only near the boundary
• Global small analytic solutions of MHD boundary layer equations
• Well-posedness of the MHD boundary layer system in Gevrey function space without Structural Assumption
• Green function of Orr Sommerfeld equations away from critical layers
• Logarithmic lattice models for flows with boundaries
• Viscous-Inviscid Interactions in a Boundary-Layer Flow Induced by a Vortex Array
• Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate
• The 3D inviscid limit problem with data analytic near the boundary
• Regularity and Expansion for Steady Prandtl Equations
• Long time well-posdness of the Prandtl equations in Sobolev space
• Stability of the boundary layer expansion for the 3D plane parallel MHD flow
• 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
• Boundary Layer Analysis for the Fast Horizontal Rotating Fluids
• Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow
• On Analytic Solutions of the Prandtl Equations with Robin Boundary Condition in Half Space
• Blow-up criterion for the 2-D Prandtl equation
• Well-posedness of thermal layer equations for inviscid compressible flows
• Critical exponent and sharp lifespan estimates for semilinear third-order evolution equations
• The vanishing viscosity limit in the presence of a porous medium
• Well-Posedness of the Nonlinear Unsteady Prandtl Equations with Robin Boundary Condition in Weighted Sobolev Spaces
• Well-posedness in Sobolev spaces of the two-dimensional MHD Boundary layer equations without viscosity
• Local-in-time Well-posedness of Boundary Layer System for the Full Incompressible MHD Equations by Energy Methods
• On the hydrostatic approximation of the Navier-Stokes equations in a thin strip
• Global-in-time Stability of 2D MHD boundary Layer in the Prandtl-Hartmann Regime
• Singularities and unsteady separation for the inviscid two-dimensional Prandtl's system
• Local-in-time well-posedness for Compressible MHD boundary layer
• Boundary layers interactions in the plane parallel incompressible flows
• Derivation of an intermediate viscous Serre-Green-Naghdi equation
• Prandtl-Batchelor flows on an annulus
• Global hydrostatic approximation of hyperbolic Navier-Stokes system with small Gevrey class data
• On the "viscous incompressible fluid + rigid body" system with Navier conditions
• Real analytic local well-posedness for the Triple Deck
• Hydrostatic approximation of the 2D primitive equations in a thin strip
• Vanishing viscosity limit to vortex sheet for the isentropic compressible circularly symmetric 2D flow
• Local well-posedness of Prandtl equations for compressible flow in two space variables
• Interaction of a vortex induced by a rotating cylinder with a plane
• Zero-viscosity limit of the Navier-Stokes equations with the Navier friction boundary condition
• Full Regularity and Well-Posedness of the Nonlinear Unsteady Prandtl Equations with Robin or Dirichlet Boundary Condition
• Symmetrical Prandtl boundary layer expansions of steady Navier-Stokes equations on bounded domain
• A Note on the Well-posedness of Prandtl Equations in Dimension Two