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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

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.

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