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Archive for January, 2015

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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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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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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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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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 !

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

\begin{aligned}    u_t + u \cdot \nabla u + \nabla p &= \nu \Delta u    \\    \nabla \cdot u & =0    \end{aligned}

wow !

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