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We consider the incompressible Euler equations on the (Eulerian) torus ${\mathbf{T}_E := ({\bf R}/{\bf Z})^d}$, which we write in divergence form as

$\displaystyle \partial_t u^i + \partial_j(u^j u^i) = - \eta^{ij} \partial_j p \ \ \ \ \ (1)$

$\displaystyle \partial_i u^i = 0, \ \ \ \ \ (2)$

where ${\eta^{ij}}$ is the (inverse) Euclidean metric. Here we use the summation conventions for indices such as ${i,j,l}$ (reserving the symbol ${k}$ for other purposes), and are retaining the convention from Notes 1 of denoting vector fields using superscripted indices rather than subscripted indices, as we will eventually need to change variables to Lagrangian coordinates at some point. In principle, much of the discussion in this set of notes (particularly regarding the positive direction of Onsager’s conjecture) could also be modified to also treat non-periodic solutions that decay at infinity if desired, but some non-trivial technical issues do arise non-periodic settings for the negative direction.

As noted previously, the kinetic energy

$\displaystyle \frac{1}{2} \int_{\mathbf{T}_E} |u(t,x)|^2\ dx = \frac{1}{2} \int_{\mathbf{T}_E} \eta_{ij} u^i(t,x) u^j(t,x)\ dx$

is formally conserved by the flow, where ${\eta_{ij}}$ is the Euclidean metric. Indeed, if one assumes that ${u,p}$ are continuously differentiable in both space and time on ${[0,T] \times \mathbf{T}}$, then one can multiply the equation (1) by ${u^l}$ and contract against ${\eta_{il}}$ to obtain

$\displaystyle \eta_{il} u^l \partial_t u^i + \eta_{il} u^l \partial_j (u^j u^i) = - \eta_{il} u^l \eta^{ij} \partial_j p = 0$

which rearranges using (2) and the product rule to

$\displaystyle \partial_t (\frac{1}{2} \eta_{ij} u^i u^j) + \partial_j( \frac{1}{2} \eta_{il} u^i u^j u^l ) + \partial_j (u^j p)$

and then if one integrates this identity on ${[0,T] \times \mathbf{T}_E}$ and uses Stokes’ theorem, one obtains the required energy conservation law

$\displaystyle \frac{1}{2} \int_{\mathbf{T}_E} \eta_{ij} u^i(T,x) u^j(T,x)\ dx = \frac{1}{2} \int_{\mathbf{T}_E} \eta_{ij} u^i(0,x) u^j(0,x)\ dx. \ \ \ \ \ (3)$

It is then natural to ask whether the energy conservation law continues to hold for lower regularity solutions, in particular weak solutions that only obey (1), (2) in a distributional sense. The above argument no longer works as stated, because ${u^i}$ is not a test function and so one cannot immediately integrate (1) against ${u^i}$. And indeed, as we shall soon see, it is now known that once the regularity of ${u}$ is low enough, energy can “escape to frequency infinity”, leading to failure of the energy conservation law, a phenomenon known in physics as anomalous energy dissipation.

But what is the precise level of regularity needed in order to for this anomalous energy dissipation to occur? To make this question precise, we need a quantitative notion of regularity. One such measure is given by the Hölder space ${C^{0,\alpha}(\mathbf{T}_E \rightarrow {\bf R})}$ for ${0 < \alpha < 1}$, defined as the space of continuous functions ${f: \mathbf{T}_E \rightarrow {\bf R}}$ whose norm

$\displaystyle \| f \|_{C^{0,\alpha}(\mathbf{T}_E \rightarrow {\bf R})} := \sup_{x \in \mathbf{T}_E} |f(x)| + \sup_{x,y \in \mathbf{T}_E: x \neq y} \frac{|f(x)-f(y)|}{|x-y|^\alpha}$

is finite. The space ${C^{0,\alpha}}$ lies between the space ${C^0}$ of continuous functions and the space ${C^1}$ of continuously differentiable functions, and informally describes a space of functions that is “${\alpha}$ times differentiable” in some sense. The above derivation of the energy conservation law involved the integral

$\displaystyle \int_{\mathbf{T}_E} \eta_{ik} u^k \partial_j (u^j u^i)\ dx$

that roughly speaking measures the fluctuation in energy. Informally, if we could take the derivative in this integrand and somehow “integrate by parts” to split the derivative “equally” amongst the three factors, one would morally arrive at an expression that resembles

$\displaystyle \int_{\mathbf{T}} \nabla^{1/3} u \nabla^{1/3} u \nabla^{1/3} u\ dx$

which suggests that the integral can be made sense of for ${u \in C^0_t C^{0,\alpha}_x}$ once ${\alpha > 1/3}$. More precisely, one can make

Conjecture 1 (Onsager’s conjecture) Let ${0 < \alpha < 1}$ and ${d \geq 2}$, and let ${0 < T < \infty}$.

• (i) If ${\alpha > 1/3}$, then any weak solution ${u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T} \rightarrow {\bf R})}$ to the Euler equations (in the Leray form ${\partial_t u + \partial_j {\mathbb P} (u^j u) = u_0(x) \delta_0(t)}$) obeys the energy conservation law (3).
• (ii) If ${\alpha \leq 1/3}$, then there exist weak solutions ${u \in C^0_t C^{0,\alpha}([0,T] \times \mathbf{T} \rightarrow {\bf R})}$ to the Euler equations (in Leray form) which do not obey energy conservation.

This conjecture was originally arrived at by Onsager by a somewhat different heuristic derivation; see Remark 7. The numerology is also compatible with that arising from the Kolmogorov theory of turbulence (discussed in this previous post), but we will not discuss this interesting connection further here.

The positive part (i) of Onsager conjecture was established by Constantin, E, and Titi, building upon earlier partial results by Eyink; the proof is a relatively straightforward application of Littlewood-Paley theory, and they were also able to work in larger function spaces than ${C^0_t C^{0,\alpha}_x}$ (using ${L^3_x}$-based Besov spaces instead of Hölder spaces, see Exercise 3 below). The negative part (ii) is harder. Discontinuous weak solutions to the Euler equations that did not conserve energy were first constructed by Sheffer, with an alternate construction later given by Shnirelman. De Lellis and Szekelyhidi noticed the resemblance of this problem to that of the Nash-Kuiper theorem in the isometric embedding problem, and began adapting the convex integration technique used in that theorem to construct weak solutions of the Euler equations. This began a long series of papers in which increasingly regular weak solutions that failed to conserve energy were constructed, culminating in a recent paper of Isett establishing part (ii) of the Onsager conjecture in the non-endpoint case ${\alpha < 1/3}$ in three and higher dimensions ${d \geq 3}$; the endpoint ${\alpha = 1/3}$ remains open. (In two dimensions it may be the case that the positive results extend to a larger range than Onsager's conjecture predicts; see this paper of Cheskidov, Lopes Filho, Nussenzveig Lopes, and Shvydkoy for more discussion.) Further work continues into several variations of the Onsager conjecture, in which one looks at other differential equations, other function spaces, or other criteria for bad behavior than breakdown of energy conservation. See this recent survey of de Lellis and Szekelyhidi for more discussion.

In these notes we will first establish (i), then discuss the convex integration method in the original context of the Nash-Kuiper embedding theorem. Before tackling the Onsager conjecture (ii) directly, we discuss a related construction of high-dimensional weak solutions in the Sobolev space ${L^2_t H^s_x}$ for ${s}$ close to ${1/2}$, which is slightly easier to establish, though still rather intricate. Finally, we discuss the modifications of that construction needed to establish (ii), though we shall stop short of a full proof of that part of the conjecture.

We thank Phil Isett for some comments and corrections.

These lecture notes are a continuation of the 254A lecture notes from the previous quarter.

We consider the Euler equations for incompressible fluid flow on a Euclidean space ${{\bf R}^d}$; we will label ${{\bf R}^d}$ as the “Eulerian space” ${{\bf R}^d_E}$ (or “Euclidean space”, or “physical space”) to distinguish it from the “Lagrangian space” ${{\bf R}^d_L}$ (or “labels space”) that we will introduce shortly (but the reader is free to also ignore the ${E}$ or ${L}$ subscripts if he or she wishes). Elements of Eulerian space ${{\bf R}^d_E}$ will be referred to by symbols such as ${x}$, we use ${dx}$ to denote Lebesgue measure on ${{\bf R}^d_E}$ and we will use ${x^1,\dots,x^d}$ for the ${d}$ coordinates of ${x}$, and use indices such as ${i,j,k}$ to index these coordinates (with the usual summation conventions), for instance ${\partial_i}$ denotes partial differentiation along the ${x^i}$ coordinate. (We use superscripts for coordinates ${x^i}$ instead of subscripts ${x_i}$ to be compatible with some differential geometry notation that we will use shortly; in particular, when using the summation notation, we will now be matching subscripts with superscripts for the pair of indices being summed.)

In Eulerian coordinates, the Euler equations read

$\displaystyle \partial_t u + u \cdot \nabla u = - \nabla p \ \ \ \ \ (1)$

$\displaystyle \nabla \cdot u = 0$

where ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$ is the velocity field and ${p: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$ is the pressure field. These are functions of time ${t \in [0,T)}$ and on the spatial location variable ${x \in {\bf R}^d_E}$. We will refer to the coordinates ${(t,x) = (t,x^1,\dots,x^d)}$ as Eulerian coordinates. However, if one reviews the physical derivation of the Euler equations from 254A Notes 0, before one takes the continuum limit, the fundamental unknowns were not the velocity field ${u}$ or the pressure field ${p}$, but rather the trajectories ${(x^{(a)}(t))_{a \in A}}$, which can be thought of as a single function ${x: [0,T) \times A \rightarrow {\bf R}^d_E}$ from the coordinates ${(t,a)}$ (where ${t}$ is a time and ${a}$ is an element of the label set ${A}$) to ${{\bf R}^d}$. The relationship between the trajectories ${x^{(a)}(t) = x(t,a)}$ and the velocity field was given by the informal relationship

$\displaystyle \partial_t x(t,a) \approx u( t, x(t,a) ). \ \ \ \ \ (2)$

We will refer to the coordinates ${(t,a)}$ as (discrete) Lagrangian coordinates for describing the fluid.

In view of this, it is natural to ask whether there is an alternate way to formulate the continuum limit of incompressible inviscid fluids, by using a continuous version ${(t,a)}$ of the Lagrangian coordinates, rather than Eulerian coordinates. This is indeed the case. Suppose for instance one has a smooth solution ${u, p}$ to the Euler equations on a spacetime slab ${[0,T) \times {\bf R}^d_E}$ in Eulerian coordinates; assume furthermore that the velocity field ${u}$ is uniformly bounded. We introduce another copy ${{\bf R}^d_L}$ of ${{\bf R}^d}$, which we call Lagrangian space or labels space; we use symbols such as ${a}$ to refer to elements of this space, ${da}$ to denote Lebesgue measure on ${{\bf R}^d_L}$, and ${a^1,\dots,a^d}$ to refer to the ${d}$ coordinates of ${a}$. We use indices such as ${\alpha,\beta,\gamma}$ to index these coordinates, thus for instance ${\partial_\alpha}$ denotes partial differentiation along the ${a^\alpha}$ coordinate. We will use summation conventions for both the Eulerian coordinates ${i,j,k}$ and the Lagrangian coordinates ${\alpha,\beta,\gamma}$, with an index being summed if it appears as both a subscript and a superscript in the same term. While ${{\bf R}^d_L}$ and ${{\bf R}^d_E}$ are of course isomorphic, we will try to refrain from identifying them, except perhaps at the initial time ${t=0}$ in order to fix the initialisation of Lagrangian coordinates.

Given a smooth and bounded velocity field ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$, define a trajectory map for this velocity to be any smooth map ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ that obeys the ODE

$\displaystyle \partial_t X(t,a) = u( t, X(t,a) ); \ \ \ \ \ (3)$

in view of (2), this describes the trajectory (in ${{\bf R}^d_E}$) of a particle labeled by an element ${a}$ of ${{\bf R}^d_L}$. From the Picard existence theorem and the hypothesis that ${u}$ is smooth and bounded, such a map exists and is unique as long as one specifies the initial location ${X(0,a)}$ assigned to each label ${a}$. Traditionally, one chooses the initial condition

$\displaystyle X(0,a) = a \ \ \ \ \ (4)$

for ${a \in {\bf R}^d_L}$, so that we label each particle by its initial location at time ${t=0}$; we are also free to specify other initial conditions for the trajectory map if we please. Indeed, we have the freedom to “permute” the labels ${a \in {\bf R}^d_L}$ by an arbitrary diffeomorphism: if ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is a trajectory map, and ${\pi: {\bf R}^d_L \rightarrow{\bf R}^d_L}$ is any diffeomorphism (a smooth map whose inverse exists and is also smooth), then the map ${X \circ \pi: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is also a trajectory map, albeit one with different initial conditions ${X(0,a)}$.

Despite the popularity of the initial condition (4), we will try to keep conceptually separate the Eulerian space ${{\bf R}^d_E}$ from the Lagrangian space ${{\bf R}^d_L}$, as they play different physical roles in the interpretation of the fluid; for instance, while the Euclidean metric ${d\eta^2 = dx^1 dx^1 + \dots + dx^d dx^d}$ is an important feature of Eulerian space ${{\bf R}^d_E}$, it is not a geometrically natural structure to use in Lagrangian space ${{\bf R}^d_L}$. We have the following more general version of Exercise 8 from 254A Notes 2:

Exercise 1 Let ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$ be smooth and bounded.

• If ${X_0: {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is a smooth map, show that there exists a unique smooth trajectory map ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ with initial condition ${X(0,a) = X_0(a)}$ for all ${a \in {\bf R}^d_L}$.
• Show that if ${X_0}$ is a diffeomorphism and ${t \in [0,T)}$, then the map ${X(t): a \mapsto X(t,a)}$ is also a diffeomorphism.

Remark 2 The first of the Euler equations (1) can now be written in the form

$\displaystyle \frac{d^2}{dt^2} X(t,a) = - (\nabla p)( t, X(t,a) ) \ \ \ \ \ (5)$

which can be viewed as a continuous limit of Newton’s first law ${m^{(a)} \frac{d^2}{dt^2} x^{(a)}(t) = F^{(a)}(t)}$.

Call a diffeomorphism ${Y: {\bf R}^d_L \rightarrow {\bf R}^d_E}$ (oriented) volume preserving if one has the equation

$\displaystyle \mathrm{det}( \nabla Y )(a) = 1 \ \ \ \ \ (6)$

for all ${a \in {\bf R}^d_L}$, where the total differential ${\nabla Y}$ is the ${d \times d}$ matrix with entries ${\partial_\alpha Y^i}$ for ${\alpha = 1,\dots,d}$ and ${i=1,\dots,d}$, where ${Y^1,\dots,Y^d:{\bf R}^d_L \rightarrow {\bf R}}$ are the components of ${Y}$. (If one wishes, one can also view ${\nabla Y}$ as a linear transformation from the tangent space ${T_a {\bf R}^d_L}$ of Lagrangian space at ${a}$ to the tangent space ${T_{Y(a)} {\bf R}^d_E}$ of Eulerian space at ${Y(a)}$.) Equivalently, ${Y}$ is orientation preserving and one has a Jacobian-free change of variables formula

$\displaystyle \int_{{\bf R}^d_F} f( Y(a) )\ da = \int_{{\bf R}^d_E} f(x)\ dx$

for all ${f \in C_c({\bf R}^d_E \rightarrow {\bf R})}$, which is in turn equivalent to ${Y(E) \subset {\bf R}^d_E}$ having the same Lebesgue measure as ${E}$ for any measurable set ${E \subset {\bf R}^d_L}$.

The divergence-free condition ${\nabla \cdot u = 0}$ then can be nicely expressed in terms of volume-preserving properties of the trajectory maps ${X}$, in a manner which confirms the interpretation of this condition as an incompressibility condition on the fluid:

Lemma 3 Let ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$ be smooth and bounded, let ${X_0: {\bf R}^d_L \rightarrow {\bf R}^d_E}$ be a volume-preserving diffeomorphism, and let ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ be the trajectory map. Then the following are equivalent:

• ${\nabla \cdot u = 0}$ on ${[0,T) \times {\bf R}^d_E}$.
• ${X(t): {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is volume-preserving for all ${t \in [0,T)}$.

Proof: Since ${X_0}$ is orientation-preserving, we see from continuity that ${X(t)}$ is also orientation-preserving. Suppose that ${X(t)}$ is also volume-preserving, then for any ${f \in C^\infty_c({\bf R}^d_E \rightarrow {\bf R})}$ we have the conservation law

$\displaystyle \int_{{\bf R}^d_L} f( X(t,a) )\ da = \int_{{\bf R}^d_E} f(x)\ dx$

for all ${t \in [0,T)}$. Differentiating in time using the chain rule and (3) we conclude that

$\displaystyle \int_{{\bf R}^d_L} (u(t) \cdot \nabla f)( X(t,a)) \ da = 0$

for all ${t \in [0,T)}$, and hence by change of variables

$\displaystyle \int_{{\bf R}^d_E} (u(t) \cdot \nabla f)(x) \ dx = 0$

which by integration by parts gives

$\displaystyle \int_{{\bf R}^d_E} (\nabla \cdot u(t,x)) f(x)\ dx = 0$

for all ${f \in C^\infty_c({\bf R}^d_E \rightarrow {\bf R})}$ and ${t \in [0,T)}$, so ${u}$ is divergence-free.

To prove the converse implication, it is convenient to introduce the labels map ${A:[0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_L}$, defined by setting ${A(t): {\bf R}^d_E \rightarrow {\bf R}^d_L}$ to be the inverse of the diffeomorphism ${X(t): {\bf R}^d_L \rightarrow {\bf R}^d_E}$, thus

$\displaystyle A(t, X(t,a)) = a$

for all ${(t,a) \in [0,T) \times {\bf R}^d_L}$. By the implicit function theorem, ${A}$ is smooth, and by differentiating the above equation in time using (3) we see that

$\displaystyle D_t A(t,x) = 0$

where ${D_t}$ is the usual material derivative

$\displaystyle D_t := \partial_t + u \cdot \nabla \ \ \ \ \ (7)$

acting on functions on ${[0,T) \times {\bf R}^d_E}$. If ${u}$ is divergence-free, we have from integration by parts that

$\displaystyle \partial_t \int_{{\bf R}^d_E} \phi(t,x)\ dx = \int_{{\bf R}^d_E} D_t \phi(t,x)\ dx$

for any test function ${\phi: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$. In particular, for any ${g \in C^\infty_c({\bf R}^d_L \rightarrow {\bf R})}$, we can calculate

$\displaystyle \partial_t \int_{{\bf R}^d_E} g( A(t,x) )\ dx = \int_{{\bf R}^d_E} D_t (g(A(t,x)))\ dx$

$\displaystyle = \int_{{\bf R}^d_E} 0\ dx$

and hence

$\displaystyle \int_{{\bf R}^d_E} g(A(t,x))\ dx = \int_{{\bf R}^d_E} g(A(0,x))\ dx$

for any ${t \in [0,T)}$. Since ${X_0}$ is volume-preserving, so is ${A(0)}$, thus

$\displaystyle \int_{{\bf R}^d_E} g \circ A(t)\ dx = \int_{{\bf R}^d_L} g\ da.$

Thus ${A(t)}$ is volume-preserving, and hence ${X(t)}$ is also. $\Box$

Exercise 4 Let ${M: [0,T) \rightarrow \mathrm{GL}_d({\bf R})}$ be a continuously differentiable map from the time interval ${[0,T)}$ to the general linear group ${\mathrm{GL}_d({\bf R})}$ of invertible ${d \times d}$ matrices. Establish Jacobi’s formula

$\displaystyle \partial_t \det(M(t)) = \det(M(t)) \mathrm{tr}( M(t)^{-1} \partial_t M(t) )$

and use this and (6) to give an alternate proof of Lemma 3 that does not involve any integration in space.

Remark 5 One can view the use of Lagrangian coordinates as an extension of the method of characteristics. Indeed, from the chain rule we see that for any smooth function ${f: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$ of Eulerian spacetime, one has

$\displaystyle \frac{d}{dt} f(t,X(t,a)) = (D_t f)(t,X(t,a))$

and hence any transport equation that in Eulerian coordinates takes the form

$\displaystyle D_t f = g$

for smooth functions ${f,g: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$ of Eulerian spacetime is equivalent to the ODE

$\displaystyle \frac{d}{dt} F = G$

where ${F,G: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}}$ are the smooth functions of Lagrangian spacetime defined by

$\displaystyle F(t,a) := f(t,X(t,a)); \quad G(t,a) := g(t,X(t,a)).$

In this set of notes we recall some basic differential geometry notation, particularly with regards to pullbacks and Lie derivatives of differential forms and other tensor fields on manifolds such as ${{\bf R}^d_E}$ and ${{\bf R}^d_L}$, and explore how the Euler equations look in this notation. Our discussion will be entirely formal in nature; we will assume that all functions have enough smoothness and decay at infinity to justify the relevant calculations. (It is possible to work rigorously in Lagrangian coordinates – see for instance the work of Ebin and Marsden – but we will not do so here.) As a general rule, Lagrangian coordinates tend to be somewhat less convenient to use than Eulerian coordinates for establishing the basic analytic properties of the Euler equations, such as local existence, uniqueness, and continuous dependence on the data; however, they are quite good at clarifying the more algebraic properties of these equations, such as conservation laws and the variational nature of the equations. It may well be that in the future we will be able to use the Lagrangian formalism more effectively on the analytic side of the subject also.

Remark 6 One can also write the Navier-Stokes equations in Lagrangian coordinates, but the equations are not expressed in a favourable form in these coordinates, as the Laplacian ${\Delta}$ appearing in the viscosity term becomes replaced with a time-varying Laplace-Beltrami operator. As such, we will not discuss the Lagrangian coordinate formulation of Navier-Stokes here.