Year: 2015

On the spectral instability of parallel shear flows

ttn12 Uncategorized

This short note is to be published as the proceeding of a Laurent Schwartz PDE seminar talk that I gave last May at IHES, announcing our recent results (on channel flows and boundary layers), which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number $R \to \infty$. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows. In fact, the material in this note is only the first half of what I spoke on that day, skipping the steady case!

On wellposedness of Prandtl: a contradictory claim?

ttn12 Uncategorized

Yesterday, Nov 17, Xu and Zhang posted a preprint on the ArXiv, entitled “Well-posedness of the Prandtl equation in Sobolev space without monotonicity” (arXiv:1511.04850), claiming to prove what the title says. This immediately causes some concern or possible contrary to what has been known previously! Here, monotonicity is of the horizontal velocity component in the normal direction to the boundary. It’s well-known that monotonicity implies well-posedness of Prandtl (e.g., Oleinik in the 60s; see this previous post for Prandtl equations). It is then first proved by Gerard-Varet and Dormy that without monotonicity, the Prandtl equation is linearly illposed (and some followed-up works on the nonlinear case that I wrote with Gerard-Varet, and then with Guo). Is there a contradictory to what it’s known and this new preprint of Xu and Zhang? The purpose of this blog post is to clarify this.

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Stability of a collisionless plasma

ttn12 Kinetic theory, Plasma Physics ,

What is a plasma? A plasma is an ionized gas that consists of charged particles: positive ions and negative electrons. To describe the dynamics of a plasma, let {f^\pm(t,x,v)} be the (nonnegative) density distribution of ions and electrons, respectively, at time {t\ge 0}, position {x\in \Omega \subset \mathbb{R}^3}, and particle velocity (or momentum) {v\in \mathbb{R}^3}. The dynamics of a plasma is commonly modeled by the Vlasov equations

\displaystyle \frac{d}{dt} f^\pm (t,X(t), V(t)) = 0 \ \ \ \ \ (1)

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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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Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits

ttn12 Instabilities, New papers, Plasma Physics

Daniel Han-Kwan and I have just submitted a paper entitled: “Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits”, which is also available on arxiv: arXiv:1506.08537. In this paper, we study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit {\varepsilon \rightarrow 0}, with {\varepsilon} being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution {\mu} of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from {\mu} in arbitrary negative Sobolev norms within time of order {|\log \varepsilon|}. Second, we deduce the invalidity of the quasineutral limit in {L^2} in arbitrarily short time.

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The onset of instability in first-order systems

ttn12 Instabilities, New papers

Nicolas Lerner, Ben Texier and I have just submitted to arxiv our long paper on “The onset of instability in first-order systems”, in which we prove the Hadamard’s instability for first-order quasilinear systems that lose its hyperbolicity in positive times. Precisely, we consider the Cauchy problem for the following first-order systems of partial differential equations:

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Math 597F, Notes 8: unstable Orr-Sommerfeld solutions for stable profiles

ttn12 Math 597F: topics on boundary layers

We now turn to the delicate case: Orr-Sommerfeld solutions for stable profiles to Rayleigh. The results reported here are in a joint work with E. Grenier and Y. Guo, directing some tedious details of the proof to our paper. We consider the Orr-Sommerfeld problem:

\displaystyle Ray_\alpha(\phi) = \epsilon \Delta_\alpha ^2 \phi,

with zero boundary conditions on {\phi} and {\phi'}, in which {Ray_\alpha(\cdot)} denotes the Rayleigh operator and {\Delta_\alpha = \partial_z^2 - \alpha^2}.

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Math 597F, Notes 5: A few examples of 2D boundary layers

ttn12 Math 597F: topics on boundary layers

Let us give a few examples of boundary layer solutions to the Prandtl problem, derived in the last lecture. In 2D, we recall the Prandtl layer problem:

\displaystyle \left \{ \begin{aligned} u_t + u u_x + v u_z &= \mu u_{zz} - p_x, \qquad v = -\int_0^z u_x(x,\theta)\; d\theta \\ u_{\vert_{t=0}} &= u_0(x,z), \qquad u_{\vert_{z=0}} =0, \qquad \lim_{z\rightarrow \infty} u(t,x,z) = u^E(t,x), \end{aligned} \right. \ \ \ \ \ (1)

with the pressure gradient: {p_x = -u_t^E- u^Eu^E_x(t,x)}, where {u^E} denotes the tangential component of Euler flow on the boundary {y=0}. Here, the tangential velocity component {u} is an (only) unknown scalar function, and the normal velocity component {v} is defined through the divergence-free condition. A simplest example: in the case {u^E} is independent of {x} and {t}, any solution to the following heat problem

\displaystyle u_t = \mu u_{zz} , \qquad u_{\vert_{t=0}} = u_0(z), \qquad u_{\vert_{z=0}} =0, \qquad \lim_{z\rightarrow \infty} u(t,z) = u^E

gives a boundary layer solution of (1) in the form {[u,0]}. We call a solution of this form to be a shear flow.

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Math 597F, Notes 4: Prandtl boundary layer theory

ttn12 Math 597F: topics on boundary layers

With simple integration by parts, we were able to see in the last two lectures essentially the current “state of art” of the {L^2} convergence of Navier-Stokes to Euler. Embarrassingly, the inviscid limit problem is widely open as discussed. It is noted that the {L^2} energy norm is quite weak, and does not see in the inviscid limit the appearance of thin layers that might (and indeed will) occur near the boundary (for instance, the {L^2} norm of Kato’s layer is of order {\sqrt \nu}\to 0). We will have to work with a different, stronger norm. Regarding the significance of viscosity despite being arbitrarily small (e.g., viscosity of air at zero temperature is about {10^{-4}}, which seems to be neglectable), d’Alembert in the 18th century has already argued out that ideal flows can’t explain well many of the physics, and the viscosity plays a crucial role near the boundary; for instance, one of his conclusions, known as d’Alembert’s paradox, asserts that solid body emerged in stationary ideal flows feels no drag acting on it (in the layman words, birds can’t fly!). Not until the beginning of the 20th century, Prandtl then postulated a solution Ansatz that revolutionized the previous understanding of slightly viscous flows near a boundary, later known as Prandtl boundary layer theory. The theory gave birth to the field of aerodynamics, and is regarded as one of the greatest achievements in fluid dynamics in the last century. Below, I’ll derive the Prandtl boundary layers.

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Math 597F, Notes 3: Inviscid limit in the presence of a boundary

ttn12 Math 597F: topics on boundary layers

Most physicists don’t believe there is such an ideal fluid (i.e., no viscosity). It is clear however that the zero viscosity or infinite Reynolds number limit plays a central role in understanding turbulence, as seen in Kolmogorov’s theory, Onsager’s conjecture, and turbulent boundary layers. Hence, understanding the inviscid limit problem is of great practical and analytical importance. As expected in most singular perturbation problems, new phenomena will arise.

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Math 597F, Notes 2: Inviscid limit problem: absence of a boundary

ttn12 Math 597F: topics on boundary layers

In this lecture, I will briefly discuss the difficulty of the inviscid limit problem of Navier-Stokes. As it will be clear in the text, the issue is not due to the fact that the million-dollar regularity problem remains unsolved, but rather nature of the singular perturbation problem. Unless otherwise noted, throughout the course the solutions of both Euler and Navier-Stokes are assumed to be sufficiently smooth as one wishes (for instance, one works in the two-dimensional case or with local-in-time solutions, with which smooth solutions are known to exist).

Precisely, consider the incompressible NS equations in {\Omega = \mathbb{R}^n} or {\mathbb{T}^n} (periodic setting), with {n \ge 2},

\displaystyle \left \{ \begin{aligned} v_t + v \cdot \nabla v + \nabla p &= \nu \Delta v \\ \nabla \cdot v &=0 \\ v_{\vert_{t=0}} &= v_0(x), \end{aligned}\right. \ \ \ \ \ (1)

with small viscosity constant {\nu >0}. Unknowns in the equation are velocity field {v} and the pressure {p}. (note here that since {\nabla \cdot v =0}, one gets {\mathrm{div} (v\otimes v) = v \cdot \nabla v =( \sum_j v_j \partial_{x_j} v_k)}, with {v = (v_k)} denoting the vector field {v}). Assume that the initial data {v_0} and the corresponding solutions {v(x,t)} are sufficiently smooth. The most basic quantity associated with (1) is the total energy:
\displaystyle \int_\Omega |v(x,t)|^2 \; dx .

Let us calculate the rate of change of the total energy with respect to time:
\displaystyle \begin{aligned} \frac 12 \frac{d}{dt} \int_{\Omega} |v(x,t)|^2 \; dx &= \int_{\Omega} v_t \cdot v \; dx \\ &= \int_{\Omega} \Big( \nu \Delta v - \nabla p - v\cdot \nabla v \Big) \cdot v\; dx. \end{aligned}

We now apply the integration by parts (that is, using the divergence theorem and noting that there is no boundary contribution in our case as {\partial \Omega = \emptyset}). The middle term vanishes, since {\nabla \cdot v =0}. The last term is computed as follows:
\displaystyle \int_\Omega (v \cdot \nabla v) \cdot v\; dx = \int_\Omega (v\cdot \nabla ) \frac{|v|^2}{2} \; dx = - \int_{\Omega} \nabla \cdot v \frac{|v|^2}{2}\; dx = 0.

This calculation (we shall refer to it as the standard energy estimate!) yields the energy balance for NS solutions:
\displaystyle \begin{aligned} \frac 12 \frac{d}{dt} \int_{\Omega} |v(x,t)|^2 \; dx = - \nu \int_\Omega |\nabla v (x,t) |^2 \; dx. \end{aligned}

That is, smooth solutions of NS equations dissipate energy, and in particular, smooth solutions of Euler ({\nu =0}) conserve energy (i.e., constant in time).
A side remark: this latter fact turns out to be false for low-regularity Euler solutions. Precisely, Onsager (1949) conjectured that the {C^{\alpha}} Euler weak solutions conserve energy when {\alpha >1/3}, and in fact dissipate energy when {\alpha<1/3}. The first part of the conjecture was proved in 1994 by Eyink, and then by Constantin-E-Titi. The second part is essentially proved by the recent breakthrough of De Lellis and Székelyhidi Jr., and also Isett and Buckmaster (see, for instance this paper, or Isett’s PhD thesis, and the references therein), using convex integration techniques (introduced by Nash in his famous isometric embedding theorem, and later developed by Gromov in his study of h-principle). This {\frac 13} power is closely related to the theory of turbulence of Kolmogorov in 1941. I plan to expand this side remark on another blog post in the near future!

 

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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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MATH 597F: Overview of a new graduate topics course: Boundary Layers in Fluid Dynamics

ttn12 Math 597F: topics on boundary layers

This Spring 2015 semester, I teach a graduate topics course on “boundary layers in fluid dynamics“ at Penn State. The purpose of this topics course is to introduce a spectral analysis approach to analyze boundary layers and investigate the inviscid limit problem of Navier-Stokes equations. The problem of small viscosity limit or high Reynolds number (mathematically equivalent; see next lectures) has a very long story. Indeed, it is one of the most classical subjects in fluid dynamics. It interests most prominent physicists such as Lord Rayleigh, W. Orr, A. Sommerfeld, Heisenberg, W. Tollmien, H. Schlichting, among many others. It was already noted by Reynolds himself in his seminal experiment (1883) that the Reynolds number governs the transition from laminar to turbulent flows. The studies became active around 1930, motivated by the study of the boundary layer around wings. In airplanes design, it is crucial to study the boundary layer around the wing, and more precisely the transition between the laminar and turbulent regimes, and even more crucial to predict the point where boundary layer splits from the boundary. A large number of papers has been devoted to the estimation of the critical Rayleigh number of classical shear flows (Blasius profile, exponential suction/blowing profile, etc…). It was Heisenberg in 1924 who first estimated the critical Reynolds number of parallel shear flows. C. C Lin and then Tollmien around 1940s completed the picture with lower and upper stability branches, respectively for parallel flows and boundary layers. Most of the physical literature, together with many mathematical insights, on the subject is well documented by Drazin and Reid in their famous book on hydrodynamics instability.

Many substantial mathematical works follow to justify the formal asymptotic expansions used by the physicists, notably the work of Wasow in the 50s; see also his book on linear turning point theory. Despite many efforts, the linear stability theory has been mathematically incomplete. Recently, together with E. Grenier and Y. Guo, we provide a complete proof of the linear stability theory discovered by Heisenberg, C. C. Lin, and Tollmien; see our preprint on arxiv: arXiv:1402.1395. My ultimate plan is to present the spectral approach that we have developed to study boundary layers. Tentatively, I plan to cover

1. Derivation of fluid dynamics equations: Euler and Navier-Stokes equations.

2. The inviscid limit problem and an introduction to Prandtl boundary layers.

3. Singular perturbations: basic ODE theory.

4. Classical stability theory of shear flows: Orr-Sommerfeld equations.

5. Semigroups of linear operators, with applications to the linearized Navier-Stokes equations near a boundary layer.

6. Nonlinear stability theory: Arnold’s stability theorem, Guo-Strauss’ linear to nonlinear instability, Grenier’s nonlinear iterative scheme.

7. Time permitting, possible applications to stratified fluids, compressible flows, and thermal boundary layers.

 

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Hello World !

ttn12 Uncategorized

This is a test to see if latex works here. Consider the Poisson equation: -\Delta u = f or even more complicated equation: Navier-Stokes equations

$latex

\begin{aligned}

u_t + u \cdot \nabla u + \nabla p &= \nu \Delta u

\\

\nabla \cdot u & =0

\end{aligned}$

wow !