fluid dynamics

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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A roadmap to nonuniqueness of L^p weak solutions to Euler

ttn12 fluid dynamics, Instabilities

In this post, I discuss a rather classical roadmap to obtain the non-uniqueness of {L^p} weak solutions to the classical incompressible Euler equations; namely, focusing on the two-dimensional case, which reads in the vorticity formulation for vorticity function {\omega}:

\displaystyle \partial_t \omega + v \cdot \nabla \omega = 0, \qquad v = \nabla^\perp \Delta^{-1}\omega \ \ \ \ \ (1)

on {\mathbb{R}^2}, with initial vorticity in {L^p} (hence, vorticity remains in {L^p} for all times). It’s known, going far back to Yudovich ’63, that weak solutions with bounded vorticity are unique, leaving open the question of uniqueness of solutions whose vorticity is only in {L^p} for {p\in [1,\infty)}. This blog post is to discuss the possible quick roadmap to proving nonuniqueness arising from the instability nature of fluid models, focusing on the Euler equations (1).

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Notes on the large time of Euler equations and inviscid damping

ttn12 fluid dynamics , ,

Consider the classical incompressible Euler equations in two dimension; namely written in the vorticity formulation, the transport equation for the unknown scalar vorticity {\omega = \omega(t,x,y)},

\displaystyle \partial_t \omega + (u \cdot \nabla) \omega =0

posed on a spatial domain {\Omega \subset \mathbb{R}^2}, where the velocity field {u\in \mathbb{R}^2} is obtained through the Biot-Savart law {u = \nabla^\perp \Delta^{-1} \omega}  (with convention (a_1,a_2)^\perp = (a_2,-a_1)). By construction, the velocity field {u} is incompressible: {\nabla \cdot u=0}. When dealing with domains with a boundary, {\Delta^{-1} } is defined together with a Dirichlet boundary condition that corresponds to the no-penetration condition of fluids {u\cdot n =0} on the boundary.

The global well-posedness theory of 2D Euler is classical: (1) smooth initial data give rise to solutions that remain smooth for all times (e.g. the Beale-Kato-Majda criterium holds, as vorticity is uniformly bounded for all times; in fact, being transported along the volume-preserving flow, all {L^p} norms of vorticity are conserved), and (2) weak solutions with bounded vorticity are unique (see Yudovich ’63).

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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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Mourre’s theory and local decay estimates, with some applications to linear damping in fluids

ttn12 fluid dynamics , , , ,

In his famous 1981 paper, Mourre gave a sufficient condition for a self-adjoint operator {H} to assure the absence of its singular continuous spectrum. More precisely, consider a self-adjoint operator {H} on a Hilbert space {\mathcal{H}} (e.g., {L^2} with the usual norm), and assume that there is a self-adjoint operator {A}, called a conjugate operator of {H} on an interval {I\subset \mathbb{R}}, so that

\displaystyle P_I i[H,A] P_I \ge \theta_I P_I + P_I K P_I \ \ \ \ \ (1)

for some positive constant {\theta_I} and some compact operator {K} on {\mathcal{H}}, where {P_I} denotes the spectral projection of {H} onto {I}, the commutator {[H,A] = HA - AH}, and the inequality is understood in the sense of self-adjoint operators.

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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

ttn12 boundary layers, fluid dynamics, Instabilities, Uncategorized , , ,

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

ttn12 boundary layers, fluid dynamics, Uncategorized , ,

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

ttn12 boundary layers, fluid dynamics, New papers , ,

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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Prandtl’s layer expansions for steady Navier-Stokes

ttn12 fluid dynamics, New papers

In 1904, Prandtl conjectured that slightly viscous flows can be decomposed into the inviscid flows away from the boundary and a so-called Prandtl’s layer near the boundary. While various instabilities indicate the failure of the conjecture for unsteady flows (for instance, see Grenier 2000), recently with Y. Guo, we are able to prove that the conjecture holds for certain steady Navier-Stokes flows; see our paper which is to appear on Annals of PDEs.

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Math 505, Mathematical Fluid Mechanics: Notes 2

ttn12 fluid dynamics, Math 505: fluid dynamics

I go on with some basic concepts and classical results in fluid dynamics [numbering is in accordance with the previous notes]. Throughout this section, I consider compressible barotropic ideal fluids with the pressure law {p = p(\rho)} or incompressible ideal fluids with constant density {\rho = \rho_0} (and hence, the pressure is an unknown function in the incompressible case).

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Madelung version of Schrödinger: a link between classical and quantum mechanics

ttn12 fluid dynamics, Schrodinger

This week I am at the Wolfgang Pauli Institute (WPI) in Vienna for the summer school on “Schrödinger equations”. Several interesting talks on or related to Schrödinger, including those of Y. Brenier on Madelung equations, F. Golse on mean field and classical limits of N-body quantum system, P. Germain on the derivation of the kinetic wave equation, C. Bardos on Maxwell-Boltzmann relation for electrons, F. Nier on Bosonic mean field dynamics, among others (still two days to go!). I also spoke on the Grenier’s iterative scheme, as discussed in my previous blog.

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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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Math 597F, Notes 1: Euler and Navier-Stokes equations

ttn12 fluid dynamics, Math 597F: topics on boundary layers

This is the first lecture of my Math 597F topics course. In this lecture, I will derive the partial differential equations, known as Euler and Navier-Stokes equation, that are widely used to model the dynamics of a fluid.  To begin, let {\rho(x,t) \in \mathbb{R}, v(x,t)\in \mathbb{R}^n} be the density function and velocity vector field of the fluid at a position {x\in \mathbb{R}^n} and a time {t}. Consider a fluid particle with initial position {x}. Its trajectory {X(t;x)} as time evolves is governed by the following ODE equation:

\displaystyle \frac{d}{dt}X(t; x ) = v (X(t;x), t), \qquad X(0; x) = x,

in which {x} serves as a parameter. An important quantity is the Jacobian of {X(t;x)} with respect to {x}: define

\displaystyle J(t;x): = \mbox{det} \nabla_x X(t;x).

Here the subscript denotes partial derivatives with respect to {x}.

Lemma 1 {\frac{d}{dt} J(t;x) = (\mathrm{div}_x v ) J(t;x) .}

Proof: Let {X(t;x) = (X_1, \cdots, X_n)} be a particle trajectory. Using the ODE equation for a particle trajectory, we compute

\displaystyle \begin{aligned} \frac{d}{dt}J(t;x) &= \sum_k \mbox{det} (\nabla_x X_1, \cdots, \frac{d}{dt}\nabla_x X_k, \cdots ,\nabla_x X_n ) \\ &= \sum_k \mbox{det} (\nabla_x X_1, \cdots, \sum_\ell \nabla_x X_\ell \frac{\partial}{\partial {x_\ell}} v_k, \cdots ,\nabla_x X_n ) \\ &= \sum_k \frac{\partial}{\partial {x_k}} v_k \mbox{det} (\nabla_x X_1, \cdots, \nabla_x X_k, \cdots ,\nabla_x X_n ) \\& = (\mathrm{div}_x v )J (t;x). \end{aligned}

\Box

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