Boundary layers

The inviscid limit problem for Navier-Stokes equations

ttn12 boundary layers, fluid dynamics , , ,

A longstanding open problem is to establish the inviscid limit of classical solutions to the incompressible Navier-Stokes equations for smooth initial data on a domain with boundaries. The question is of great physical and mathematical interest, and it deeply links to the transition to turbulence in fluids that may possibly take place faster than expected due to the presence of a boundary. In this article, I shall give a quick overview of this subject, and then highlight some recent works with my former student, Trinh T. Nguyen, (currently a Van Vleck Assistant Professor at University of Wisconsin, Madison), whose main results establish the inviscid limit for smooth data that are only required to be analytic locally near the boundary. This may be the best possible type of positive results that one can hope for, given the known violent instabilities at the boundary, which I shall discuss below. Before getting on, this picture should already hint at the great delicacy in studying boundary layers (source internet):

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Generator functions and their applications

ttn12 fluid dynamics, Kinetic theory, New papers , , , ,

The Cauchy-Kovalevskaya theorem is a classical convenient tool to construct analytic solutions to partial differential equations, which allows one to view and treat them as if they are ordinary differential equations:

\displaystyle u_t = F(t,x,u,u_x)

for unknown functions {u(t,x)} in {t\ge 0} and {x\in \mathbb{R}^d} (or some spatial domain). Roughly, if {F(t,x,u,w)} is locally analytic near a point {(0,x_0,u_0,w_0)} then the PDE has a unique solution {u(t,x)} which is analytic near {(0,x_0)}, as established by Cauchy (1842) and generalized by Kovalevskaya as part of her dissertation (1875). The theorem has found many applications such as in fluid dynamics and kinetic theory where it is used to provide existence of analytic solutions including those that are obtained at certain asymptotic limits (e.g., inviscid limit, Boltzmann-to-fluid limit, Landau damping, inviscid damping,…). There are modern formulations of the abstract theorem: see, for instance, Asano, Baouendi and Goulaouic, Caflisch, Nirenberg, and Safonov. In this blog post, I present generator functions, as an alternative approach to the use of the Cauchy-Kovalevskaya theorem, recently introduced in my joint work with E. Grenier (ENS Lyon), and discuss the versatility and simplicity of their use to applications. In addition to providing existence of analytic solutions, I will also mention another use of generator functions to capture some physics that would be otherwise missed for analytic data (namely, an analyticity framework to capture physical phenomena that are not seen for analytic data!).

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Invalidity of Prandtl’s boundary layers

ttn12 boundary layers, fluid dynamics, Instabilities, New papers , ,

I’ve just submitted this paper with Grenier (ENS Lyon) which studies Prandtl’s boundary layer asymptotic expansions for incompressible fluids on the half-space in the inviscid limit. In 1904, Prandtl introduced his well known boundary layers in order to describe the transition from Navier-Stokes to Euler equations in the inviscid limit.

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Green function for linearized Navier-Stokes around a boundary layer profile: near critical layers

ttn12 boundary layers, fluid dynamics, Instabilities , , ,

Emmanuel Grenier and I have just submitted this 84-page! long paper, also posted on arxiv (arXiv:1705.05323). This work is a continuation and completion of the program (initiated in Grenier-Toan1 and Grenier-Toan2) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a stationary boundary layer profile.

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Sublayer of Prandtl boundary layers

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The aim of this paper (arXiv:1705.04672), with E. Grenier, is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: {\nu \rightarrow 0}.  In his CPAM2000 paper, Grenier proved that there exists no Prandtl’s asymptotic expansion involving one Prandtl’s boundary layer with thickness of order {\sqrt\nu}, which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order {\nu^{3/4}}. In this paper, we point out how the stability of the classical Prandtl’s layer is linked to the stability of this sublayer. In particular, we prove that the two layers cannot both be nonlinearly stable in {L^\infty}.  That is, either the Prandtl’s layer or the boundary sublayer is nonlinearly unstable in the sup norm.

Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms

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I’ve just uploaded this paper, with E. Grenier, on the arXiv (arXiv:1703.00881), entitled Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms, aiming a better understanding of the classical Prandtl’s boundary layers. Indeed, one of the key difficulties in dealing with boundary layers is the creation of (unbounded) vorticity in the inviscid limit.

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Green function for linearized Navier-Stokes around boundary layers: away from critical layers

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I’ve just submitted this new paper with E. Grenier (ENS de Lyon) on arxiv (scheduled to announce next Tuesday 1:00GMT), in which we construct the Green function for the classical Orr-Sommerfeld equations and derive sharp semigroup bounds for linearized Navier-Stokes equations around a boundary layer profile. This is part of the long program to understand the stability of classical Prandtl’s layers appearing in the inviscid limit of incompressible Navier-Stokes flows.

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Grenier’s nonlinear iterative scheme

ttn12 boundary layers, fluid dynamics, Instabilities ,

In his paper [Grenier, CPAM 2000], Grenier introduced a nonlinear iterative scheme to prove the instability of Euler and Prandtl equations. Recently, the scheme is also proved to be decisive in the study of water waves: [Ming-Rousset-Tzvetkov, SIAM J. Math. Anal., 2015], and plasma physics: [Han-Kwan & Hauray, CMP 2015] or my recent paper with Han-Kwan (see also my previous blog discussions). I am certain that it can be useful in other contexts as well. In this blog post, I’d like to give a sketch of the scheme to prove instability.

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