Formally, we expect the so-called Prandtl’s Ansatz:

where solves the Euler equations, and is the Prandtl boundary layer corrector, denoting the small viscosity or the inverse of the large Reynolds number.

The Prandtl boundary layers have been intensively studied in the mathematical literature. Notably, solutions to the Prandtl equations have been constructed for monotonic data (Oleinik ’60s, or recently Masmoudi-Wong or Alexandre at al.), or data with Gevrey or analytic regularity (Sammartino-Caflisch ’98 or recently Gerard-Varet and Masmoudi). The validity of the Prandtl’s Ansatz has been established by Sammartino-Caflisch ’98 for initial data with analytic regularity. The Ansatz, with a specific boundary layer profile, has also been recently justified for data with Gevrey regularity by Gerard-Varet, Maekawa and Masmoudi.

When only data with Sobolev regularity are assumed, Grenier proved in his CPAM 2000 paper that such an asymptotic expansion is false, up to a remainder of order in norm. The invalidity of the expansion is proved near boundary layers with an inflection point or more precisely near those that are spectrally unstable for the Rayleigh equations.

In this paper, we prove the Prandtl’s Ansatz is also false for Rayleigh’s stable shear flows, giving a proof of the Conjecture stated in Section 4 of Grenier-Guo-Toan. Such shear flows are stable for Euler equations, but not for Navier-Stokes equations: adding a small viscosity destabilizes the flow.

Roughly speaking, given an arbitrary stable boundary layer, the two main results in this paper are

Theorem 1 (Grenier-Toan: a rough statement)In the case of time-dependent boundary layers, we construct Navier-Stokes solutions, with arbitrarily small forcing of order , so that the Prandtl’s Ansatz is false near the boundary layer, up to a remainder of order in norm, being arbitrarily small.

Theorem 2 (Grenier-Toan: a rough statement)In the case of stationary boundary layers, we construct Navier-Stokes solutions, without forcing term, so that the Prandtl’s Ansatz is false, up to a remainder of order in norm.

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In this program, we study the linear problem

in which denotes the fluid vorticity and the fluid velocity. We solve the problem with no-slip boundary conditions on . We are aimed to derive uniform estimates in the inviscid limit . Observe that the fluid vorticity is unbounded, but localized, near the boundary, and therefore pointwise bounds on the Green function are needed to study the precise convolution with the boundary layer behavior.

As is a compact perturbation of the Laplacian (say in the usual space), is sectorial, has discrete unstable spectrum, and the corresponding semigroup can be described by the Dunford’s integral:

where is a contour on the right of the spectrum of .

In estimating the semigroup, we can move the contour across the discrete spectrum by adding corresponding projections on the eigenfunction. However, we cannot move the contour of integration across the Euler continuous spectrum (or equivalently, the phase velocity is near the range of and hence critical layers appear). In addition, there are unstable eigenvalues that exist near the critical layers and that vanish in the inviscid limit (Grenier-Guo-Toan, justifying the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among others).

One of the contributions of this paper is to carefully study the contour integral near the critical layers and thus to provide sharp bounds on the Navier-Stokes semigroup.

The first step is to study the resolvent solutions , or equivalently, solutions to the classical Orr-Sommerfeld equations (a ODEs), corresponding to each wavenumber and each phase velocity :

with zero boundary conditions on and . Here, . We need to study the Green function of the Orr-Sommerfeld problem.

When is away from the range of (or equivalently, is away from the continuous spectrum of Euler), solutions of Orr-Sommerfeld are regular and consist of two slow modes linked to the Rayleigh equations and two fast modes linked to the Airy equations . This is studied carefully in Grenier-Toan1.

When is near the range of , we have to deal with critical layers, points at which . The presence of critical layers greatly complicates the analysis of constructing Orr-Sommerfeld solutions and deriving uniform bounds for the corresponding Green function. Roughly speaking, there are two independent solutions to the Orr-Sommerfeld equations that are approximated by the Rayleigh solutions, whose solutions experience a singularity of the form . We thus need to analyze the smoothing effect of the Airy operator, and design precise function spaces to capture the singularity near the critical layers.

Finally, to capture the unbounded vorticity near the boundary, we study the semigroup in the boundary layer norms that were developed recently in Grenier-Toan2.

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More precisely, for incompressible Navier-Stokes flows with the no-slip boundary condition, the vorticity near the boundary is expected to behave as follows:

for small viscosity . That is, one expects the vorticity to become unbounded, of order , in the inviscid limit (for instance, this is indeed the case for data with analytic regularity).

The novelty of this paper is to introduce boundary layer norms that capture the precise boundary layer behavior of the linearized vorticity and to derive sharp semigroup bounds with respect to the boundary layer norms for the linearized Navier-Stokes around an unstable boundary layer. Such a result is possible, thanks to the precise estimates on the Green function for the classical Orr-Sommerfeld problem (see arXiv:1702.07924).

As an immediate application, we construct approximate solutions that exhibit an instability of the classical Prandtl’s layers.

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More precisely, let be the small viscosity and let be the linearized Navier-Stokes operator around a stationary boundary layer on the half-line , together with the classical no-slip boundary condition. Naturally, there are two cases: either is spectrally stable or spectrally unstable to the corresponding Euler operator . In this paper, we consider the unstable case, giving the existence of the maximal unstable eigenvalue , with .

Our first main result in this paper is roughly as follows:

Theorem 1There holds the sharp semigroup bound:

uniformly in time and uniformly in the inviscid limit . Here, denotes some exponentially weighted norm.

Certainly, the key difficulty in such a theorem is to derive sharp bounds in term of time growth and uniform estimates in the inviscid limit. Standard energy estimates yield precisely a semigroup bound of order , which is far from being sharp.

In fact, such a sharp semigroup bound is a byproduct of our much delicate study on the Orr-Sommerfeld problem, the resolvent equations of the linearized Navier-Stokes. In order to accurately capture the behavior of (unbounded) vorticity on the boundary, we are obliged to derive pointwise estimates on the corresponding Green function. We follow the seminal approach of Zumbrun-Howard developed in their study of stability of viscous shocks in the system of conservation laws.

That is to say, our second main result of this paper is to provide uniform bounds on the Green function of the classical Orr-Sommerfeld problem and derive pointwise bounds on the corresponding Green function of Navier-Stokes. This paper is the first in our program of deriving sharp semigroup bounds for Navier-Stokes around a boundary layer profile.

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describing the dynamics of wave density at wavenumber , . Here, denotes the positive coefficient of fluid viscosity, and is the collision term, describing pure resonant three-wave interactions:

with , with being the collision kernel. The integration is taken over resonant manifolds (of dimension), defined by the resonant conditions

with denoting the dispersion relation of the waves. For capillary waves, , the surface tension.

According to the Zahkarov’s weak turbulence theory, the kinetic equation admits nontrivial equilibria so that , which resembles the Kolmogorov spectrum of hydrodynamic turbulence describing the energy cascade. Such a stationary solution is often referred to as Kolmogorov-Zakharov spectra; see, for instance, the book of Nazarenko ’11. Several efforts have been made ever since its derivation in the 60’s, the fundamental question of existence and uniqueness of solutions to the kinetic equation remains unsolved. *The aim of this paper is to provide a (radial) solution to this very question.*

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Precisely, consider

with , , imposing the no-slip boundary condition on the moving plate: , with the moving speed . Here, denotes the inverse of the high Reynolds number.

The interest is to analyze the asymptotic limit as . We are in particular interested in the behavior near the boundary , about which Prandtl’s ansatz reads

in which solves the Euler equation (that is, Navier-Stokes with ), and solves the so-called Prandtl equation , which simply reads

for the Prandtl’s layers , plus appropriate boundary conditions to correct the mismatch of Euler and Navier-Stokes flows on the boundary.

Remarkably, the Prandtl’s equation has self-similar solutions (for instance, Blasius solutions), and the equation can be solved, either by Crocco’s transformation or von Mises’ transformation; see my previous post. Its much simplification to the real flows allows leads to tremendous applications and advances in science and engineering. Justification of the validity of Prandtl’s layers is needed.

In my aforementioned paper, we are able to prove that indeed there holds the Prandtl’s asymptotic expansion near parallel Euler flows, at least for a short plate. Precisely,

for , with small .

In particular, the stability of Prandtl’s expansion yields the validity of the inviscid limit for such a flow. It’s worth noting that it appears not possible, in general, to extend for a longer length of the plate, due to the boundary layer separation phenomenon. However, when the mismatch between Euler and Navier-Stokes flows is sufficiently small, most recently S. Iyer is able to extend our work for the expansion to be valid for all . In the above expansion, except the first Euler flow which is given, the Prandtl’s layers and next Euler flows solve a parabolic or elliptic equations, respectively, which are introduced to correct the error created from the previous step.

The delicacy in dealing with this asymptotic problem is not only at constructing the ansatz solutions, but also at deriving a stability estimate to control the remainder (that is, to confirm that the error indeed remains small in the inviscid limit). Let me briefly discuss the latter. Indeed, as in the unsteady case, the convection is extremely large: , which prevents us to solve the following linear equation for :

The usual energy or elliptic estimate fails, due to the large convection. However, the energy estimate roughly yields

noting that the norm of is in fact bounded.

To bound in term of , we still need to be able to inverse the linear operator! We then turn to the vorticity equation, which within the boundary layer reads

in which vorticity .

Our crucial observation is that the following operator is indeed positive

as long as remains positive. Thus, multiplying the vorticity equation with and noting that , we are able to derive the gradient estimate for in term of , and thus close the stability estimate, when .

Our next difficulty to overcome is the apparent loss of derivatives in the asymptotic expansions: precisely, there is a loss of three derivatives in due to the fact that is treated as a remainder in the Navier-Stokes equations, but not in the Prandtl’s, upon recalling . This three-derivative loss is serious, due to the possible lack of regularity of solutions to the elliptic problem in domains with corners. We treat this loss by adding yet another boundary layer, this time in near the corners, leaving some small and controllable errors in the momentum equations. The nonlinear iteration can be closed, combining the stability estimate, estimates, and a higher-order elliptic estimates.

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whose boundary oscillates at wavelength . Here, .

The inviscid limit problem of Navier-Stokes, with a boundary on which the classical no-slip condition is imposed, is essentially open due to various instabilities of Prandtl’s boundary layers (for instance, I discussed here). Near the boundary , the Prandtl’s ansatz reads

which experiences a large gradient . This prevents available analyses to obtain the inviscid limit: Navier-Stokes converges to Euler in certain norms, except the classical analyticity result by Sammartino-Caflisch ’98 (see also Maekawa ’14, where he obtains the inviscid limit, by allowing Sobolev regularity away from the boundary, but still analyticity needed near the boundary, or precisely Euler vorticity is assumed to be zero near the boundary). This large gradient is in fact the source of viscous instability (for instance, Grenier 2000, or Grenier-Guo-Nguyen ’16).

On the other hand, for Navier-Stokes with so-called Navier conditions: on the boundary (the shear stress is propositional to the tangential velocity), Iftimie-Sueur ’11 proved that the boundary layer ansatz reads

One observes that in this case, the Prandtl’s layers have smaller amplitude, and most importantly remains bounded. This boundedness allows Iftimie-Sueur ’11 to justify the inviscid limit; see also Masmoudi-Rousset ’12 for a strong compactness approach.

In this paper, we study the inviscid limit under the Navier boundary conditions on the oscillating boundary. One of our main motivations is that the Navier condition on the highly oscillating boundary becomes the classical no-slip condition (the case where the inviscid limit problem is unsolved) in the limit of . That is, the Navier condition is indeed a good approximation of the no-slip condition, when there is roughness on the boundary! This way, we wish to obtain the inviscid limit, avoiding the instability of Prandtl’s layers. In addition, it appears natural for , for otherwise the Prandtl’s layers would come into play.

Indeed, we prove in this paper the following theorem:

Theorem 1 (Rough statement)For and , arbitrarily large, the Navier-Stokes with viscosity on satisfies the asymptotic expansion

on , for any , in the limit of . In particular, the inviscid limit holds.

The key difficulty is again the large gradient , due to the appearance of boundary layers. Our first crucial observation is that we are able to construct boundary layers, which are essentially inviscid, whose vorticity is zero, leaving a small error in the momentum equation. This allows us to work with the vorticity equation, which reads

Since (inviscid) boundary layers have zero vorticity, is bounded, and hence standard energy estimates yield

As one roughly expects (for instance, energy estimates for velocity yield ), the above would yield at once a stability estimate for vorticity (!). However, making this rigorous turns out to be a delicate issue, mostly due to the large vorticity on the boundary! Indeed, the boundary condition for vorticity roughly reads

where , denoting the curvature of the oscillating boundary. The above energy estimate for vorticity fails due to the large vorticity on the boundary.

Our second crucial contribution is to design a weighted energy estimate for vorticity, and thereby derive a similar stability estimate. Finally, some delicate potential estimates are derived for the type of rough domains that we are considering, in particular to obtain a close sup norm on velocity. This paper also raises an interesting open problem: to establish the inviscid limit for the case when .

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in which denote the electrons density, the elementary charge, the electron temperature, and the electric potential.

More precisely, we consider a plasma consisting of electrons and one kind of ions, which are charged particles moving in an electromagnetic field. Let and be the corresponding density distribution functions for ions and electrons, respectively; here, represent particle velocity variables for ions and electrons belonging to (here or ), and denotes the space variable belonging to a periodic torus or an open set of with a boundary, and is the time. In absence of magnetic fields, the dynamics of the plasma is modeled by the following well-known system

where denote the electron and ion mass, the elementary charge (for the sake of simplicity we assume that the ion charge is equal to ). The electrostatic field is given by solving the usual Poisson equation, and the operator accounts for the collisions between the electrons (for example, a binary Boltzmann or Fokker-Planck operator).

Such a model has been widely used in plasma physics. Since the electron/ion mass ratio is small, the characteristic time scale of the dynamics of ions is significantly larger than that of electrons. As a consequence, if one addresses a model for the ion dynamics, it is very classical to use a fluid modeling for the electrons, assuming they have reached the thermal equilibrium; that is to say, the distribution function is a Maxwellian function with an electron temperature and a density given by the well-known *Maxwell-Boltzmann relation* is used for electrons density.

If we introduce non-dimensional parameter

and look at the scaled distribution functions

with being the characteristic electron density, we end up with the non-dimensional Vlasov-Poisson-Boltzmann system:

and the Poisson equation for the electric potential reads as

Here, denotes the Debye length. In this paper, under certain regularity assumption, we prove that the Maxwell-Boltzmann relation for electrons, keeping the dynamics of ions at the kinetic level, is obtained under the scaling assumption:

As , we obtain in the limit the electron density distribution

with being the inverse of the electron temperature. The Vlasov equation for ions remains the same, whereas the Poisson equation now reads

which is often referred to as the Poisson-Poincare equation.

The relaxation to the equilibrium of the form of a Maxwellian is precisely due to the presence of the collision operators, without which the equilibrium is of the form of a function of the particle energy . In this paper, we also prove that the reduced ions problem (that is, the Vlasov equation for ions, coupled with the Poisson-Poincare equation and the energy conservation) is wellposed globally in time.

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