Finite time blowup for an averaged three-dimensional Navier-Stokes equation

Introduction

The purpose of this paper is to formalise the “supercriticality” barrier for the (infamous) global regularity problem for the Navier-Stokes equation, using a blowup solution to a certain averaged version of Navier-Stokes equation to demonstrate that any proposed positive solution to the regularity problem which does not use the finer structure of the nonlinearity cannot possibly be successful. This barrier also suggests a possible route to provide a negative answer to this problem, that is to say it suggests a program for constructing a blowup solution to the true Navier-Stokes equations.

The barrier is not particularly sensitive to the precise formulationSee for an analysis of the relationship between different formulations of the Navier-Stokes regularity problem in three dimensions. It is likely that our main results also extend to higher dimensions than three, although we will not pursue this matter here. of the regularity problem, but to state the results in the cleanest fashion we will take the homogeneous global regularity problem in the Euclidean setting in three spatial dimensions as our formulation:

in a distributional sense at least (actually, at the $H^{10}_{\operatorname{df}}$ level of regularity it is not difficult to justify (1.6) in the classical sense for mild solutions).

The distinction between smooth finite energy solutions and $H^{10}_{\operatorname{df}}$ mild solutions is essentially non-existent (at leastFor data which is only in $H^{10}_{\operatorname{df}}$, there is a technical distinction between the two solution concepts, due to a lack of unlimited time regularity at the initial time $t=0$ that is ultimately caused by the non-local effects of the divergence-free condition $\nabla\cdot u=0$, requiring one to replace the notion of a smooth solution with that of an almost smooth solution; see for details. However, in this paper we will only concern ourselves with Schwartz initial data, so that this issue does not arise. for Schwartz initial data), and the reader may wish to conflate the two notions on a first reading. More rigorously, we can reformulate Conjecture 1.1 as the following logically equivalent conjecture:

Conjecture 1.1 and Conjecture 1.2 are equivalent.

We use the results from , although this equivalence is essentially classical and was previously well known to experts.

If we take the inner product of (1.6) with $u$ and integrate in time using (1.2), we arrive atOne has to justify the integration by parts of course, but this is routine under the hypothesis of a mild solution; we omit the (standard) details. the fundamental energy identity

for any mild solution to the Navier-Stokes equation.

If one was unaware of the supercritical nature of the Navier-Stokes equation, one might attempt to obtain a positive solution to Conjecture 1.1 or Conjecture 1.2 by combining (1.7) (or equivalently, (1.2)) with various harmonic analysis estimates for the inhomogeneous heat equation

(or, in integral form, $u(t)=e^{t\Delta}u_{0}+\int_{0}^{t}e^{(t-t^{\prime})\Delta}F(t^{\prime})\ dt^{\prime}$), together with harmonic analysis estimates for the Euler bilinear operator $B$, a simple example of which is the estimate

for some absolute constant $C$. Such an approach succeeds for instance if the initial data $u_{0}$ is sufficiently smallOne can of course also consider other perturbative regimes, in which the solution $u$ is expected to be close to some other special solution than the zero solution. There is a vast literature in these directions, see e.g. and the references therein. in a suitable critical norm (see for an essentially optimal result in this direction), or if the dissipative operator $-\Delta$ is replaced by a hyperdissipative operator $(-\Delta)^{\alpha}$ for some $\alpha\geq 5/4$ (see ) or with very slightly less hyperdissipative operators (see ). Unfortunately, standard scaling heuristics (see e.g. [40, §2.4]) have long indicated to the experts that the energy estimate (1.7) (or (1.2)), together with the harmonic analysis estimates available for the heat equation and for the Euler bilinear operator $B$, are not sufficient by themselves to affirmatively answer Conjecture 1.1. However, these scaling heuristics are not formalised as a rigorous barrier to solvability, and the above mentioned strategy to solve the Navier-Stokes global regularity problem continues to be attempted on occasion.

The most conclusive way to rule out such a strategy would of course be to demonstrateIt is a classical fact that mild solutions to a given initial data are unique, see e.g. [41, Theorem 5.4(iii)]. a mild solution to the Navier-Stokes equation that develops a singularity in finite time, in the sense that the $H^{10}_{\operatorname{df}}$ norm of $u(t)$ goes to infinity as $t$ approaches a finite time $T_{*}$. Needless to say, we are unable to produce such a solution. However, we will in this paper obtain a finite time blowup (mild) solution to an averaged equation

We pause to mention some previous blowup results in this direction. If one drops the cancellation requirement (1.2), so that one no longer has the energy identity (1.7), then blowup solutions for various Navier-Stokes type equations have been constructed in the literature. For instance, in finite time blowup for a “cheap Navier-Stokes equation” $\partial_{t}u=\Delta u+\sqrt{-\Delta}(u^{2})$ (with $u$ now a scalar field) was constructed in the one-dimensional setting, with the results extended to higher dimensions in . As remarked in that latter paper, it is essential to the methods of proof that no energy identity is available. In a slightly different direction, finite time blowup was established in for a complexified version of the Navier-Stokes equations, in which the energy identity was again unavailable (or more precisely, it is available but non-coercive). These models are not exactly of the type (1.9) considered in this paper, but are certainly very similar in spirit.

Further models of Navier-Stokes type, which obey an energy identity, were introduced by Plecháç and Şverák , , by Katz and Pavlovic , and by Hou and Lei ; of these three, the model in is the most relevant for our work and will be discussed in detail in Section 1.2 below. These models differ from each other in several respects, but interestingly, in all three cases there is substantial evidence of blowup in five and higher dimensions, but not in three or four dimensions; indeed, for all three of the models mentioned above there are global regularity results in three dimensions, even in the presence of blowup results for the corresponding inviscid model. Numerical evidence for blowup for the Navier-Stokes equations is currently rather scant (except in the infinite energy setting, see , ); the blowup evidence is much stronger in the case of the Euler equations (see for a recent result in this direction, and for a survey), but it is as yet unclearHowever, in , finite time blowup for a three-dimensional “partially viscous” Navier-Stokes type model, in which some but not all of the fields are subject to a viscosity term, was established. whether these blowup results have direct implications for Navier-Stokes in the three-dimensional setting, due to the relatively significant strength of the dissipation.

Finally, we mention work , , establishing finite time blowup for supercritical fractal Burgers equations; such equations are not exactly of Navier-Stokes type, being scalar one-dimensional equations rather than incompressible vector-valued three-dimensional ones, but from a scaling perspective the results are of the same type, namely a demonstration of blowup whenever the norms controlled by the conservation and monotonicity laws are all supercritical.

Finally, we will averageIn an earlier version of this manuscript, no averaging over dilations was assumed, but it was pointed out to us by the referee that the non-degeneracy condition (3.24) failed if one did not introduce dilation averaging. over the dilation operators

almost surely for any natural numbers $k_{1},k_{2},k_{3}$ and some finite $C$. To phrase this definition without probabilistic notation, we have

for some probability space $(\Omega,\mu)$ and some measurable maps $R_{i,\cdot}:\Omega\to{\operatorname{SO}}(3)$, $\lambda_{i,\cdot}:\Omega\to(0,+\infty)$ and $m_{i,\cdot}(D):\Omega\to\mathcal{M}_{0}$, where $\mathcal{M}_{0}$ is given the Borel $\sigma$-algebra coming from the seminorms $\|\|_{k}$, and one has

By the rotation symmetry $\langle B({\operatorname{Rot}}_{R}u,{\operatorname{Rot}}_{R}v),{\operatorname{Rot}}_{R}w\rangle=\langle B(u,v),w\rangle$, we may eliminate one of the three rotation operators ${\operatorname{Rot}}_{R_{i,\omega}}$ in (1.13) if desired, and similarly for the dilation operator. By some Fourier analysis (related to the fractional Leibniz rule) it should also be possible to eliminate one of the Fourier multipliers $m_{i,\omega}(D)$. However, we will not attempt to do so here.

Because we have not imposed any symmetry or anti-symmetry hypotheses on the averaging measure $\mu$, rotations $R_{j}$, and Fourier multipliers $m_{j}(D)$, the analogue

We are now ready to state the main result of the paper.

One can also rewrite the averaged Navier-Stokes equation (1.9) in a form more closely resembling (1.1), namely

where $T$ is an averaged version of the convection operator $(u\cdot\nabla)u$, defined by $T=\frac{1}{2}(T_{12}+T_{21})$ where

for $ij=12,21$. We can also ensure that the inviscid form of the averaged Navier-Stokes equation conserves helicity, as well as total momentum, angular momentum, and vorticity; see Remark 4.3 below.

2. Overview of proof

To construct the stable blowup solution, we were motivated by the work on regularity and blowup of the system of ODE

for some Schwartz function $\psi$ with Fourier transform vanishing near the origin. We remark that the analogue of the energy identity (1.7) in this setting is the identity

valid whenever $X_{n}$ exhibits sufficient decay as $n\to\pm\infty$ (we do not formalise this statement here).

We will defer for now the technical issue (which we regard as being of secondary importance) of transferring blowup results from dyadic Navier-Stokes models to averaged Navier-Stokes models, and focus on the question of whether blowup solutions may be constructed for ODE systems such as (1.17).

Blowup solutions for the equation (1.17) are known to exist for sufficiently small $\alpha$; specifically, for $\alpha<1/4$ this was (essentially) established in , while for $\alpha<1/3$ this was established in , with global regularity established in the critical and subcritical regimes $\alpha\geq 1/2$. If a blowup solution could be constructedThe results in can be however adapted to establish a version of Theorem 1.5 in six and higher dimensions, while the results in give a version in five and higher dimensions (and just barely miss the four-dimensional case); this can be done by adapting the arguments in this paper (and using the above-cited blowup results as a substitute for the lengthier ODE analysis in this paper), and we leave the details to the interested reader. Interestingly, the results in , on a somewhat different Navier-Stokes type model also indicate blowup in five and higher dimensions, while giving global regularity instead in lower dimensions; similarly for a third Navier-Stokes model introduced in . with the value $\alpha=2/5$, then this would be a dyadic analogue of Theorem 1.5. Unfortunately for our purposes, for the values $\lambda=2^{1/\alpha},\alpha=2/5$, global regularity was established in (for non-negative initial data $X_{n}(0)$), by carefully identifying a region of phase space that is invariant under forward evolution of (1.17), and which in particular prevents the energy $X_{n}$ from concentrating too strongly at a single value of $n$. However, the argument in is sensitive to the specific numerical value of $\lambda$ (and also relies heavily on the assumption of initial non-negativity), and does not rule out the possibility of blowup at $\alpha=2/5$ for some variant of the system (1.17).

From multiplying (1.17) by $X_{n}$, we arrive at the energy transfer equations

One can fix this problem by suitably modifying the model equation (1.17). One rather drastic (and not particularly satisfactory) way to do this is to forcibly (i.e., exogenously) shut off most of the nonlinear interactions, so that only one pair $X_{n},X_{n+1}$ of adjacent modes experiences a nonlinear (but energy-conserving) interaction at any given time. Specifically, one can consider a truncated-nonlinearity ODE

Our strategy, then, is to design a system of ODE similar to (1.17) that can endogenously simulate the exogenous truncations $1_{n-1=n(t)}$, $1_{n=n(t)}$ of (1.21). As shown in , this cannot be done for the scalar equation (1.17), at least when $\lambda$ is equal to $2$. However, by replacing (1.17) with a vector-valued generalisation, in which one has four scalar functions $X_{1,n}(t),X_{2,n}(t),X_{3,n}(t),X_{4,n}(t)$ associated to each scale $n$, rather than a single scalar function $X_{n}(t)$, it turns out to be possible to use quadratic interactions of the same strength as the terms $\lambda^{n-1}X_{n-1}^{2},\lambda^{n}X_{n}X_{n+1}$ appearing in (1.17) to induce such a simulation, while still respecting the energy identity. The precise system of ODE used is somewhat complicated (see Section 6), but it can be described as a sequence of “quadratic circuits” connected in series, with each circuit built out of a small number of “quadratic logic gates”, each corresponding to a certain type of basic quadratic nonlinear interaction. Specifically, we will combine together some “pump” gates that transfer energy from one mode to another (and which are the only gate present in (1.17)) with “amplifier” gates (that use one mode to ignite exponential growth in another mode) and “rotor” gates (that use one mode to rotate the energy between two other modes). By combining together these gates with carefully chosen coupling constants (a sort of “quadratic engineering” task somewhat analogous to the more linear circuit design tasks in electrical engineering), we can set up a transfer of energy from scale $n$ to scale $n+1$ which can be made arbitrarily abrupt, in that the duration of the time interval separating the regime in which most of the energy is at scale $n$, and most of the energy is at scale $n+1$, can be made as small as desired. Furthermore, this transfer is delayed somewhat from the time at which the scale $n$ first experiences a large influx of energy. The combination of the delay in energy transfer and the abruptness of that transfer means that the process of transferring energy from scale $n$ to scale $n+1$ is not itself interrupted (up to negligible errors) by the process of transferring energy from scale $n+1$ to $n+2$, and this permits us (after a lengthy bootstrap argument) to construct a blowup solution to this equation, which resembles the blowup solution for the truncated ODE (1.21).

This almost finishes the proof of Theorem 1.5, except that the dyadic model equation involves the dyadic Laplacian instead of the Euclidean Laplacian. However, it turns out that the analysis of the dyadic system of ODE can be adapted to the case of non-dyadic dissipation, by using local energy inequalities as a substitute for the exact ODE that appear in the dyadic model. While this complicates the analysis slightly, the effect is ultimately negligible due to the perturbative nature of the dissipation.

3. A program for establishing blowup for the true Navier-Stokes equations?

To summarise the strategy of proof of Theorem 1.5, a solution to a carefully chosen averaged version

of the Euler equations is constructed which behaves like a “von Neumann machine” (that is, a self-replicating machine) in the following sense: at a given time $t_{n}$, it evolves as a sort of “quadratic computer”, made out of “quadratic logic gates”, which is “programmed” so that after a reasonable period of time $t_{n+1}-t_{n}$, it abruptly “replicates” into a rescaled version of itself (being $1+\epsilon_{0}$ times smaller, and about $(1+\epsilon_{0})^{5/2}$ times faster), while also erasing almost completely the previous iteration of this machine. This replication process is stable with respect to perturbations, and in particular can survive the presence of a supercritical dissipation if the initial scale of the machine is sufficiently small.

This suggests an ambitious (but not obviously impossible) program (in both senses of the word) to achieve the same effect for the true Navier-Stokes equations, thus obtaining a negative answer to Conjecture 1.1. Define an ideal (incompressible, inviscid) fluid to be a divergence-free vector field $u$ that evolves according to the true Euler equations

Somewhat analogously to how a quantum computer can be constructed from the laws of quantum mechanics (see e.g. ), or a Turing machine can be constructed from cellular automata such as Conway’s “Game of Life” (see e.g. ), one could hope to design logic gates entirely out of ideal fluid (perhaps by using suitably shaped vortex sheets to simulate the various types of physical materials one would use in a mechanical computer). If these gates were sufficiently “Turing complete”, and also “noise-tolerant”, one could then hope to combine enough of these gates together to “program” a von Neumann machine consisting of ideal fluid that, when it runs, behaves qualitatively like the blowup solution used to establish Theorem 1.5. Note that such replicators, as well as the related concept of a universal constructor, have been built within cellular automata such as the “Game of Life”; see e.g. .

Once enough logic gates of ideal fluid are constructed, it seems that the main difficulties in executing the above program are of a “software engineering” nature, and would be in principle achievable, even if the details could be extremely complicated in practice. The main mathematical difficulty in executing this “fluid computing” program would thus be to arrive at (and rigorously certify) a design for logical gates of inviscid fluid that has some good noise tolerance properties. In this regard, ideas from quantum computing (which faces a unitarity constraint somewhat analogous to the energy conservation constraint for ideal fluids, albeit with the key difference of having a linear evolution rather than a nonlinear one) may prove to be useful.

A significant (but perhaps not insuperable) obstacle to this program is that in addition to the conservation of energy, the Euler equations obey a number of additional conservation laws, such as conservation of helicity, with vortex lines also being transported by the flow; see e.g. . This places additional limitations on the type of fluid gates one could hope to construct; however, as these conservation laws are indefinite in sign, it may still be possible to design computational gates that respect all of these laws.

It is worth pointing out, however, that even if this program is successful, it would only demonstrate blowup for a very specific type of initial data (and tiny perturbations thereof), and is not necessarily in contradiction with the belief that one has global regularity for most choices of initial data (for some carefully chosen definition of “most”, e.g. with overwhelming (but not almost sure) probability with respect to various probability distributions of initial data). However, we do not have any new ideas to contribute on how to address this latter question, other than to state the obvious fact that deterministic methods alone are unlikely to be sufficient to resolve the problem, and that stochastic methods (e.g. those based on invariant measures) are probably needed.

4. Acknowledgments

I thank Nets Katz for helpful discussions, Zhen Lei and Gregory Seregin for help with the references, and the anonymous referee for a careful reading and pointing out an error in a previous version of this manuscript. The author is supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1266164.

Notation

If $P$ is a mathematical statement, we use $1_{P}$ to denote the quantity $1$ when $P$ is true and when $P$ is false.

Averaging the Euler bilinear operator

In this section we show that certain bilinear operators, which are spatially localised variants of the “cascade operators” introduced in , can be viewed as averaged Euler bilinear operators.

We now formalise the class of local cascade operators we will be working with. For technical reasons, we will use the integer powers $(1+\epsilon_{0})^{n}$ of $1+\epsilon_{0}$ for some sufficiently small $\epsilon_{0}>0$ as our dyadic range of scales, rather than the more traditional powers of two, $2^{n}$. Roughly speaking, the reason for this is to ensure that any triangle of side lengths that are of comparable size, in the sense that they all between $(1+\epsilon_{0})^{n-O(1)}$ and $(1+\epsilon_{0})^{n+O(1)}$ for some $n$, are almost equilateral; this lets us avoid some degeneracies in the tensor symbol implicit in (1.3) that would otherwise complicate the task of expressing certain bilinear operators as averages of the Euler bilinear operator $B$ (specifically, the smallness of $\epsilon_{0}$ is needed to establish the non-degeneracy condition (3.24) below).

Note from the Plancherel theorem that one has

We did not impose that the $\psi_{i}$ were divergence free, but one could easily do so via Leray projections if desired, in which case the operators $C(u,v)$ defined via duality in (3.1) can be expressed more directly as

and similarly for any local cascade operator $C$ one has

under the additional restriction that $\lambda$ is an integer power of $1+\epsilon_{0}$.

Theorem 1.5 is then an immediate consequence of the following two results.

Let $\epsilon_{0}>0$ be a sufficiently small absolute constant. Then every local cascade operator (with dyadic scale parameter $\epsilon_{0}$) is an averaged Euler bilinear operator.

Theorem 3.3 is the main technical result of this paper, and its proof will occupy the subsequent sections of this paper. In this section we establish Theorem 3.2. This will be done by a somewhat lengthy series of averaging arguments and Fourier decompositions, together with some elementary three-dimensional geometry, with the result ultimately following from a certain non-degeneracy property of the trilinear form $\Lambda$ defined in (1.4); the arguments are unrelated to those in the rest of the paper, and readers may wish to initially skip this section and move on to the rest of the argument.

Henceforth $\epsilon_{0}>0$ will be assumed to be sufficiently small (e.g. $\epsilon_{0}=10^{-10}$ will suffice). In this section, the implied constants in the $O()$ notation are not permitted to depend on $\epsilon_{0}$.

It will be convenient to complexify the problem in order to freely use Fourier-analytic tools at later stages of the argument. To this end, we introduce the following notation.

and that one has the integrability conditions

Suppose we can show that every local cascade operator $C$ is a complex average of the Euler bilinear operator $B$ in the sense of the above definition. The multipliers $m_{j,\omega}(D)$ for $j=1,2,3$ appearing in the expansion (3.4) are not required to be real, but we can decompose them as $m_{j,\omega,1}(D)+im_{j,\omega,2}(D)$ where $m_{j,\omega,1}(D),m_{j,\omega,2}(D)$ are real (and with the seminorms of $m_{j,\omega,1}(D),m_{j,\omega,2}(D)$ bounded by a multiple of the corresponding seminorm of $m_{j,\omega}(D)$). Thus we can decompose the right-hand side of (3.4) as the sum of $2^{3}=8$ pieces, each of which is of the same form as the original right-hand side up to a power of $i$, and with all the $m_{j,\omega}(D)$ appearing in each piece being a real Fourier multiplier. As the left-hand side of (3.4) is real (as are the inner products on the right-hand side), we may eliminate all the terms on the right-hand side involving odd powers of $i$ by taking real parts. The power of $i$ in each of the four remaining terms is now just a sign $\pm 1$ and can be absorbed into the $m_{1,\omega}(D)$ factor; by concatenating together four copies of $(\Omega,\mu)$ we may now obtain an expansion of the form (3.4) in which all the $m_{j,\omega}(D)$ are real. Finally, by multiplying $m_{1,\omega}(D)$ by a normalising constant we may take $(\Omega,\mu)$ to be a probability space rather than a finite measure space. Combining all these manipulations, we conclude Theorem 3.2. Thus, it will suffice to show that every local cascade operator is a complex average of the Euler bilinear operator $B$.

2. Second step: frequency localisation

By again using $m_{1,\omega}(D)$ to absorb scalar factors, we see that if $C$ is a complex average of $C^{\prime}$, then any complex scalar multiple of $C$ is a complex average of $C^{\prime}$; also, by concatenating finite measure spaces together we see from Definition 3.4 that if $C_{1},C_{2}$ are both complex averages of $C^{\prime}$, then $C_{1}+C_{2}$ is an complex average of $C^{\prime}$. Thus the space of averages of the Euler bilinear operator is closed under finite linear combinations, and so it will suffice to show that every basic local cascade operator is a complex average of the Euler bilinear operator.

By decomposing the $\psi_{j}$, $j=1,2,3$ in (3.4) into finitely many (complex-valued) pieces, we may replace the basic local cascade operator with the complexified basic local cascade operator $C$ defined by

see Figure 1. The exact normalisation in (3.7) is somewhat arbitrary, but the vanishing (3.8) is convenient for technical reasons; also, it is necessary to ensure that $\xi^{0}_{1},\xi^{0}_{2},\xi^{0}_{3}$ have distinct magnitudes in order to avoid a certain degeneracy later in the argument (namely, the failure of (3.24) below).

Once we perform this normalisation, we will have no further need of averaging over dilations, and will rely purely on Fourier and rotation averaging to obtain the required representation of the cascade operator $C$.

3. Third step: forcing frequency comparability

thus $\eta(|\xi_{1}|,|\xi_{2}|,|\xi_{3}|)$ is only non-vanishing when $\xi_{1},\xi_{2},\xi_{3}$ have comparable magnitude.

Note that $\eta(N_{1},N_{2},N_{3})=\eta(1,e^{\log(N_{2}/N_{1})},e^{\log(N_{3}/N_{1})})$, and that $(x,y)\mapsto\eta(1,e^{x},e^{y})$ is a smooth compactly supported function. By FourierOne could also use Mellin inversion here if desired. inversion, we thus have a representation of the form

then $B_{\eta}$ is a complex average of $B$ (note that $\|D^{it}\|_{k}$ grows polynomially in $t$ for each $k$). From (1.3) and Fubini’s theorem (working first with Schwartz $u,v,w$ to justify all the exchange of integrals, and then taking limits) we see thatNote that we do not define $\eta(|\xi_{1}|,|\xi_{2}|,|\xi_{3}|)$ when one of $\xi_{1},\xi_{2},\xi_{3}$ vanishes, but this is only occurs on a set of measure zero and so there is no difficulty defining the integral.

It thus suffices to show that $C$ is a complex average of $B_{\eta}$.

is clearly a complex average of $B_{\eta}$, and so it suffices to show that $C$ is a complex average of $B_{\eta,\rho}$.

4. Fourth step: localising to a single frequency scale

Suppose for now that we can show that $C_{0}$ is a complex average of $B_{\eta,\rho,0}$ (without the use of dilation operators), thus

for some $m_{j,\omega}$, $R_{j}$ ($j=1,2,3$), and $(\Omega,\mu)$ as in Definition 3.4. From the definition of $C_{0}$ (and the support hypotheses on $\psi_{1},\psi_{2},\psi_{3}$), we see that we may smoothly localise each $m_{j,\omega}$ to the ball $B(\xi_{j}^{0},O(\epsilon_{0}^{3}))$ without loss of generality (and without destroying the fact that the $m_{i,\omega}(D)$ are Fourier multipliers of order that obey (3.5)). If we then define

is equal to $-\pi i\langle C_{n_{1}}(u,v),w\rangle$ when $n_{1}=n_{2}=n_{3}$, and vanishing otherwise if $\epsilon_{0}$ is small enough (thanks to the support properties of $m_{i,\omega,n}$, $\eta$ and $\rho$). Summing, we see that

(as before, one can work first with Schwartz $u,v,w$, and then take limits), thus demonstrating that $C$ is a complex average of $B_{\eta,\rho}$ as desired (absorbing the $\frac{1}{-\pi i}$ factor into $m_{1,\omega}$). Thus, to finish the proof of Theorem 3.2, it suffices to show that $C_{0}$ is a complex average of $B_{\eta,\rho,0}$.

5. Fifth step: extracting the symbol

We have reduced matters to the task of obtaining a representation (3.10) for $\langle C_{0}(u,v),w\rangle$. By (3.9) and Plancherel’s theorem, we may expand $\langle C_{0}(u,v),w\rangle$ as

for all $\xi_{j}\in B(\xi_{j}^{0},\epsilon_{0}^{3})$ and $X_{j}\in\xi_{j}^{\perp}$, $j=1,2,3$. Indeed, if one applies (3.12) with $(X_{1},X_{2},X_{3})=(\hat{u}(\xi_{1}),\hat{v}(\xi_{2}),\hat{w}(\xi_{3}))$, contracts the resulting tensor against $\overline{\hat{\psi}_{1}(\xi_{1})}\otimes\overline{\hat{\psi}_{2}(\xi_{2})}\otimes\overline{\hat{\psi}_{3}(\xi_{3})}$ and then integrates in $\xi_{1},\xi_{2},\xi_{3}$ (absorbing the $\overline{\hat{\psi}_{i}}$ and $F$ factors into the $m_{j,\omega}$ terms, after first breaking $F$ into $27$ components), we obtain the desired decomposition (3.10) (after replacing $\Omega$ with the disjoint union of $27$ copies of $\Omega$ to accommodate the contributions from the various components of $F$). As before, one may wish to first work with Schwartz $u,v,w$ to justify the interchanges of integrals, and then take limits at the end of the argument.

6. Sixth step: simplifying the weights

It remains to obtain the decomposition (3.12). We will restrict attention to those rotations $R_{j,\omega}$ which almost fix $\xi_{j}^{0}$ in the sense that

for $j=1,2,3$. With this restriction, the weight $\varphi(\frac{1}{\epsilon_{0}^{2}}(R_{1,\omega}\xi_{1}-\xi_{1}^{0}))\eta(|\xi_{1}|,|\xi_{2}|,|\xi_{3}|)$ is equal to one (for $\epsilon_{0}$ small enough), and so (3.12) simplifies to

denote the set of sextuples $(R_{1},R_{2},R_{3},\xi_{1},\xi_{2},\xi_{3})$ where $R_{j}\in{\operatorname{SO}}(3)$ with $|R_{j}\xi_{j}^{0}-\xi_{j}^{0}|<\epsilon_{0}^{2}/4$ for $j=1,2,3$, and $\xi_{i}\in B(\xi_{i}^{0},2\epsilon_{0}^{3})$ for $i=1,2,3$ with

For $\epsilon_{0}$ small enough, we see from the implicit function theorem that this is a smooth manifold (of dimension $15$), and that for any choice of $\xi_{j}\in B(\xi_{j}^{0},2\epsilon_{0}^{3})$ for $j=1,2,3$, the slice

Suppose that we can find a smooth function

whenever $\xi_{j}\in B(\xi_{j}^{0},\epsilon_{0}^{3})$ and $X_{j}\in\xi_{j}^{\perp}$, $j=1,2,3$, where $d\sigma(R_{1},R_{2},R_{3})$ is surface measure on $\Sigma_{\xi_{1},\xi_{2},\xi_{3}}$. By a change of variables, this can be rewritten as

By a Fourier expansion and another smooth truncation, we may thus write

7. Seventh step: restricting to rotations around fixed axes

It remains to find a smooth function $F^{\prime}$ for which one has the required representation (3.15). Observe from (3.8) and the implicit function theorem (for $\epsilon_{0}$ small enough) that if $\xi_{j}\in B(\xi_{j}^{0},\epsilon_{0}^{3})$ for $j=1,2,3$, one can find rotations $R_{j,\xi_{1},\xi_{2},\xi_{3}}\in{\operatorname{SO}}(3)$ for $j=1,2,3$ with

(where $\|\|$ denotes the operator norm) is an open submanifold of $\Sigma_{\xi_{1},\xi_{2},\xi_{3}}$. Also, if we use the ansatz

for $X_{j}\in\xi_{j}^{\perp}$, $j=1,2,3$. Thus, if we can find a smooth function

for any $\xi_{j}\in B(\xi_{j}^{0},\epsilon^{3})$ and $X_{j}\in\xi_{j}^{\perp}$. Averaging this over all $S\in{\operatorname{SO}}(3)$ with $\|S-I\|\leq\epsilon^{2}/8$, and inverting the tensored rotation operator $R_{1,\xi_{1},\xi_{2},\xi_{3}}\otimes R_{2,\xi_{1},\xi_{2},\xi_{3}}\otimes R_{3,\xi_{1},\xi_{2},\xi_{3}}$, we obtain a representation of the desired form (3.15). Thus it suffices to find a smooth function $F^{\prime\prime}$ with the representation (3.19).

8. Eighth step: parameterising in terms of rotation angles

for all eight choices of sign patterns $(\sigma_{1},\sigma_{2},\sigma_{3})\in\{-1,+1\}^{3}$, and some smooth functions

9. Ninth step: Fourier inversion and checking a non-degeneracy condition

By (3.21), (1.4) and decomposing $R_{\eta_{j}}^{\gamma_{j}}n$ into a complex linear combination of $e^{-i\gamma_{j}}$ and $e^{i\gamma_{j}}$, we see that for fixed $\eta_{1},\eta_{2},\eta_{3}$, we may expand

for all $(\eta_{1},\eta_{2},\eta_{3})\in\Gamma$ and all choices of signs $(\sigma_{1},\sigma_{2},\sigma_{3})\in\{-1,+1\}^{3}$.

For this, we finally need to use the precise form of $\Lambda$. From (3.21), (1.4) we can write $\Theta_{\eta_{1},\eta_{2},\eta_{3}}(\gamma_{1},\gamma_{2},\gamma_{3})$ as

where $u_{i}:=\eta_{i}/|\eta_{i}|$. Expanding

(the minus sign arising here from the $i$ in the denominator in (3.25)). Similarly with $\sigma_{2},\gamma_{2}$ replaced by $\sigma_{1},\gamma_{1}$ respectively. Inserting these expansions and comparing with (3.23), we conclude that

But by (3.17), $\eta_{j}=\xi_{j}^{0}+O(\epsilon_{0}^{3})$, which from (3.7) implies that

As $-\sigma_{1}+\frac{1}{\sqrt{2}}\sigma_{2}+\frac{1}{\sqrt{2}}\sigma_{1}\sigma_{2}\sigma_{3}$ is bounded away from zero for $\sigma_{1},\sigma_{2},\sigma_{3}\in\{-1,+1\}$, the non-degeneracy claim (3.24) follows for $\epsilon_{0}$ small enough. This concludes the proof of Theorem 3.2.

The averaging over dilation operators was only needed to place the base frequencies $\xi^{0}_{1},\xi^{0}_{2},\xi^{0}_{3}$ in a location where the non-degeneracy condition (3.24) held. This condition in fact holds for generic $\xi^{0}_{1},\xi^{0}_{2},\xi^{0}_{3}$, and so even without the use of averaging over dilations it should be the case that most local cascade operators are expressible as averaged Euler operators. As there is some freedom to select the local cascade operators in Theorem 3.3, this should still be enough to establish a slightly stronger version of Theorem 1.5 in which one does not use any averaging over dilations. We will however not pursue this matter here.

Reduction to an infinite-dimensional ODE

We now begin the proof of Theorem 3.3. We fix $0<\epsilon_{0}<1$; henceforth we allow all implied constants in the $O()$ notation to depend on $\epsilon_{0}$. We suppose that Theorem 3.3 failed, so that one can always constructThis hypothesis of global existence is technically convenient so that we may assume some a priori regularity on our solution, namely $H^{10}$. Alternatively, one could develop an $H^{10}$ local well-posedness theory for (3.3), and unconditionally construct a mild $H^{10}$ solution that blows up in a finite time by a minor modification of the arguments in this paper; we leave the details of this variant of the argument to the interested reader. global mild solutions to any initial value problem of the form (3.3) with $C$ a local cascade operator and $u_{0}$ a Schwartz divergence-free vector field.

As in Definition 3.1, we define the rescaled functions

for $(i_{1},i_{2},i_{3},\mu_{1},\mu_{2},\mu_{3})\in S$. From Definition 3.1 we see that $C$ is indeed a local cascade operator (it is a sum of $|S|=7m^{3}$ basic local cascade operators), and (4.2) ensures that $C$ is symmetric. Clearly

for all $i_{1},i_{2},i_{3}\in\{1,\dots,m\}$ and $(\mu_{1},\mu_{2},\mu_{3})\in S$.

We will select initial data $u_{0}$ of the formOur analysis is in fact somewhat stable, and will also apply if $u_{0}$ is a sufficiently small perturbation of $\psi_{1,n_{0}}$ in the $H^{10}_{\operatorname{df}}$ norm, thus creating blowup for a non-empty open set of initial data in smooth topologies, although this open set is rather small and is also quite far from the origin (due to the large nature of $n_{0}$). We leave the details of this modification to the interested reader.

for all $n<n_{0}$, $i=1,\dots,m$, and $t\geq 0$.

As $u$ is a mild solution to (3.3), we have

then from the a priori regularity $u\in C^{0}_{t}H^{10}_{x}$ we obtain (4.6) from the Plancherel identity. Taking inner products of (4.14) with $\psi_{i,n}$, we have

or in differentiated form (using (4.6) to justify the calculations)

In particular this shows that $u_{i,n}$ is continuously differentiable in time (in the $L^{2}_{x}$ topology, say), which implies that the $X_{i,n}$ are continously differentiable.

It is unfortunate that the $\psi_{i,n}$ are not eigenfunctions of the Laplacian $\Delta$, otherwise $u_{i,n}$ would be always be a scalar multiple of $\psi_{i,n}$ (that is, $u_{i,n}=X_{i,n}\psi_{i,n}$), and the equation (3.3) would collapse to a system of ODE in the $X_{i,n}$ variables. However, it is still possible to get good control on the dynamics even without the eigenfunction property. To do this, we use the local energies $E_{i,n}$ from (4.5). From Cauchy-Schwarz we have

and from Plancherel and the $C^{0}_{t}H^{10}_{x}$ bound on $u$ we have (4.7) for all $0<T<\infty$.

By taking inner products of (4.15) with $u_{i,n}$, and noting that

we obtain the local energy inequality (4.11). Indeed, one could use Fourier analysis to place an additional dissipation term of $8\pi^{2}(1+\epsilon_{0})^{2n}E_{i,n}$ on the right-hand side of (4.11), but we will not need to use this term here (it is too small to be of much use, since we are in the regime where dissipation can be treated as a negligible perturbation).

If instead, if we take inner products of (4.15) with $\psi_{i,n}$, and note that

while from (4.8) we see that $E_{i,n}-\frac{1}{2}X_{i,n}^{2}$ vanishes at time zero. The claim (4.12) then follows from (4.16) and the fundamental theorem of calculus.

Finally, we prove (4.13). For $i_{3}=1,\dots,m$ and $n<n_{0}$, we see from (4.11), (4.8), (4.9) and the fundamental theorem of calculus that

for any $t\geq 0$. Summing this for $i=1,\dots,m$ and $n<n_{0}$, and using (4.6), (4.7) to ensure all summations and integrals are absolutely convergent, we conclude that

By (4.3), all the terms here can be grouped into terms that sum to zero, except for those terms with $n=n_{0}-1$, $(\mu_{1},\mu_{2},\mu_{3})\in\{(1,0,0),(0,1,0)\}$; thus

By the constraint on $(\mu_{1},\mu_{2},\mu_{3})$, two of the terms $X_{i_{1},n_{0}-1-\mu_{3}+\mu_{1}}$, $X_{i_{2},n_{0}-1-\mu_{3}+\mu_{2}}$, $X_{i_{3},n_{0}-1}$ may be bounded by $\sum_{n<n_{0}}\sum_{i=1}^{m}E_{i,n}$, and the remaining term may be controlled by (4.6), leading to the bound

for all $0\leq t\leq T$ and some finite quantity $C_{T,n_{0}}$ depending on $T,n_{0}$ (and on the quantity in (4.6)). By Gronwall’s inequality, we conclude that $\sum_{n<n_{0}}\sum_{i=1}^{m}E_{i,n}(t)=0$ for all $t\geq 0$, giving (4.13). ∎

The above lemma shows that (3.3) almost collapses into an ODE system for the $X_{i,n}$. As a first approximation, the reader may wish to ignore the role of the energies $E_{i,n}$ (or identify them with $\frac{1}{2}X_{i,n}^{2}$), and pretend that (4.10) is replaced by either the inviscid equation

in the analysis that follows. Note that the viscous equation generalises the dyadic Katz-Pavlovic equation (1.17) (with $\lambda=(1+\epsilon_{0})^{5/2}$ and $\alpha=2/5$), which corresponds to a simple case in which $m=1$.

Theorem 3.3 now follows from the following ODE result:

We will prove Theorem 4.2 in Section 6, but we first warm up with some finite dimensional ODE toy problems in the next section.

Quadratic circuits

Our objective is to solve an infinite-dimensional system of ODE, roughly of the form (4.17). In order to build up some intuition for doing so, we will first study a finite-dimensional “toy” model, namely ODEs of the form

We first discuss in turn the three quadratic logic gates we will be using, which we call the “pump”, the “amplifier”, and the “rotor”, and then show how these gates can be combined to build a circuit with the desired properties. It looks likely that the set of quadratic gates is sufficiently “Turing complete” in that they can perform extremely general computational tasksOf course, this is bearing in mind that, being globally well-posed ODE, circuits of the form (5.1) are necessarily limited to perform continuous (i.e. analog) operations rather than perfectly digital operations. Also, as the equation (5.1) is time reversible, only reversible computing tasks may be performed by quadratic circuits, at least in the absence of dissipation., but we will not pursueSee for a treatment of continuous computation in PDE, and for continuous computation in ODE. this matter further here.

Strictly speaking, the discussion here is not actually needed for the proof of our main results, but we believe that the model problems studied here will assist the reader in understanding what may otherwise be a highly unmotivated construction and set of arguments in the next section.

where $\alpha>0$ is a fixed coupling constant (representing the strength of the pump). We will be applying this pump in the regime where $x$ is initially positive and $y\geq 0$; by Gronwall’s inequality (or by integrating factors), we see that $x$ remains positive for all subsequent time, while $y$ is increasing. As the total energy $x^{2}+y^{2}$ is conserved, we thus see that energy is being pumped from $x$ to $y$. For instance, we have the explicit solution

for any amplitude $A>0$, which at time $t=0$ is at the initial state $(x(0),y(0))=(A,0)$. For times $0\leq t\leq\frac{1}{\alpha A}$, the $y$ component increases more or less linearly at rate comparable to $\alpha A^{2}$, with a corresponding drain of energy from $x$; after this time, $x$ decays exponentially fast (at rate $\alpha A$), with the energy in $x$ being transferred more or less completely to $y$ after time $t\geq\frac{C}{\alpha A}$ for a large constant $C$. Thus, the pump can be used to execute a delayed, but gradual, transition of energy from one mode (the $x$ mode) to another (the $y$ mode). We will schematically depict the pump by a thick arrow: see Figure 2.

If one ignores the dissipation term, the dyadic model equation (1.17) can be viewed as a sequence of pumps chained together, with the coupling constant $\lambda^{n}$ of the pump from one mode $X_{n}$ to the next $X_{n+1}$ increasing exponentially with $n$.

One useful feature of the pump which we will exploit is that it can “integrate” an alternating input $x$ into a monotone output $y$, somewhat analogously to how a rectifier in electrical engineering converts AC current to DC current. Indeed, if one couples the $x$ input of the pump to an external forcing term, thus

with $F$ highly oscillatory, then $x$ may oscillate in sign also (if the $F$ term dominates the energy drain term $-\alpha xy$), but the $y$ output continues to increase at a more or less steady rate. If for instance $F(t)=A\omega\cos(\omega t)$ with some quantities $A,\omega$ which are large compared to the coupling constant $\alpha$, and we set initial conditions $x(0)=y(0)=0$ for simplicity, then we expect $x$ to behave like $A\sin(\omega t)$, and $y$ to increase at rate about $\frac{1}{2}\alpha A^{2}$ on average.

If instead we couple the pump to an oscillatory forcing term on the output, thus

then it is possible that $y$ can turn negative, which causes the pump to reverse in energy flow to become an amplifier (see below). This behaviour will be undesirable for us, so we will take some care to design our circuit so that the output of a pump does not experience significant negative forcing at key epochs in the dynamics, unless this forcing is counterbalanced by an almost equivalent amount of positive forcing.

2. Application: finite time blowup for an exogenously truncated dyadic model

As a quick application of the pump gate, we establish blowup for the truncated version (1.21) of the dyadic model system (1.17), whenever one has supercritical dissipation:

Let $\lambda>1$ and $0<\alpha<1/2$, and let $0<\delta<1-2\alpha$. Then there exists a natural number $n_{0}$, a sequence of times

for all $t\in[0,T_{*})$ other than the times $t_{n_{0}},t_{n_{1}},\ldots$, and all $n\geq n_{0}$, with the convention that $t_{n_{0}-1}=0$ and $X_{n_{0}-1}=0$. Furthermore, we have

for every $k\geq 0$. In particular, for any $\delta^{\prime}>\delta$, we have the blowup

This proposition is not needed for the blowup results in the rest of the paper, but is easier to prove than those results, and already illustrates the basic features of the blowup solutions being constructed. Note that the blowup here is available for all values of the dissipation parameter up to the critical value of $1/2$, in contrast to the results in and for the untruncated equation (1.17) which cover the ranges $\alpha<1/4$ and $\alpha<1/3$ respectively, as well as the results in establishing global solutions when $\lambda=2$ and $2/5\leq\alpha\leq 1/2$.

We let $n_{0}$ be a sufficiently large natural number (depending on $\lambda,\alpha,\delta$) to be chosen later. We then construct $t_{n_{0}},t_{n_{0}+1},\ldots$ and $X_{n}(t)$ iteratively as follows:

Initialise $k=0$ and $t_{n_{0}}=0$. We also initialise

Now suppose that $t_{n_{0}+k}$ has been constructed, and the solution $X_{n}(t)$ constructed for all times $0\leq t\leq t_{n_{0}+k}$ and $n\geq n_{0}$. We then solve the pump system with dissipation

within the time interval $t\in[t_{n_{0}+k},t_{n_{0}+k+1}]$, where $t_{n_{0}+k+1}$ is the first time for which $X_{n_{0}+k+1}(t_{n_{0}+k+1})=\lambda^{-\delta(k+1)}$; we justify the existence of such a time below.

For each $n\neq n_{0}+k,n_{0}+k+1$, we evolve $X_{n}$ on $[t_{n_{0}+k},t_{n_{0}+k+1}]$ by the linear ODE

Increment $k$ to $k+1$ and return to Step 2.

Let us now establish that the time $t_{n_{0}+k+1}$ introduced in Step 2 is well defined for any given $k\geq 0$. If we make the change of variables

then we see from construction that we have the initial conditions

and our task is to show that $y(t)=\lambda^{-\delta}$ for some finite $t>0$. However, from the explicit solution (5.4) to the pump gate (5.3), we see that in the case $\varepsilon=0$, this occurs at time $t=\operatorname{tanh}^{-1}(\lambda^{-\delta})$; standard perturbation arguments then show that if $n_{0}$ is sufficiently large (which forces $\varepsilon$ to be sufficiently small), the claim occurs at some time $t\leq 2\operatorname{tanh}^{-1}(\lambda^{-\delta})$ (say). Undoing the scaling, we see that

so $t_{n}$ converges to a finite limit $T_{*}$ as $n\to\infty$, and the claim follows. ∎

One cannot take $\delta=0$ in the above argument, because the pump gate never quite transfers all of its energy from the $x$ mode to the $y$ mode. If however we worked with the modified equation

for some function $g:[0,+\infty)\to[0,+\infty)$ increasing to infinity, and defines $t_{n_{0}+k+1}$ to be the first time for which $X_{n_{0}+k}(t_{n_{0}+k+1})=0$ (so that $X_{n_{0}+k+1}$ is the only non-zero mode at this time), then a modification of the above argument establishes finite time blowup whenever $n_{0}$ is sufficiently large and

basically because one can show inductively that $X_{n_{0}+k}(t_{n_{0}+k})$ is comparable to $1$, $t_{n_{0}+k+1}-t_{n_{0}+k}$ is comparable to $\lambda^{-n}$, and the energy dissipation on each time interval $[t_{n_{0}+k},t_{n_{0}+k+1}]$ is comparable to $\frac{1}{g(\lambda^{n_{0}+k})^{2}}$; we omit the details. This is compatible with the heuristic calculation in [39, Remark 1.2]. In the converse direction, the arguments in or should ensure global regularity for the above equation (or for the analogous hyperdissipative version of (1.17)) under the condition

This leaves an intermediate regime (e.g. $g(s)=\log(1+s)^{\beta}$ for $1/4<\beta\leq 1/2$) in which it is unclear whether one can force blowupSince the initial release of this manuscript, it has been shown in (see also ) that blowup in fact does not occur in this intermediate regime. Roughly speaking, the basic point is that as the energy moves from low frequency modes to high frequency modes, it must transition through all intermediate frequency scales, and the cumulative energy dissipation from such transitions is enough to prevent the solution from escaping to frequency infinity in this intermediate regime. with any of these ODE models. The analysis in or suggests that this may be possible, but one would have to work with models in which many different modes are activated at once (in contrast to the situation in Proposition 5.1, in which only two modes have interesting dynamics at any given time).

3. The amplifier gate

The amplifier gate is a reversed version of the pump gate:

Here again $\alpha>0$ is a coupling constant, indicating the strength of the amplifier. We will use this gate in the regime in which $x$ is positive and large, and $y$ is positive but small. In this case, we can explicitly solve the second equation to obtain

for any $t\geq t_{0}$, which suggests that $y$ grows exponentially at rate comparable to $\alpha x(0)$, until such time that the $y$ mode begins to drain a significant fraction of energy from the $x$ mode. Thus, the $x$ mode can be viewed as causing exponential amplification in the $y$ mode. Of course, in the presence of forcing terms, we no longer have the exact formula (5.11), but we may take advantage of Gronwall’s inequality to obtain analogous control on $y$.

As with the pump gate, the amplifier gate preserves the total energy $x^{2}+y^{2}$. An explicit solution to (5.10) is given by

for any $A>0$ and $T>0$. For $0<t<T$, the quantity $y$ increases exponentially at rate about $\alpha A$, while $x$ stays roughly steady at $A$.

By using the amplifier with a large coupling constant $\alpha$, $x$ large and positive, and $y$ small and positive, we can cause $y$ to grow at a rapid exponential rate, and in particular to transition abruptly from being small (e.g. $y\leq\varepsilon$ for some threshold $\varepsilon$) to being large (e.g. $y>2\varepsilon$), if the threshold $\varepsilon$ is set low enough that $y$ does not yet begin to drain significant amounts of energy from $x$. This ability to generate abrupt transitions is of course needed in our quest to engineer an abrupt delayed transition of energy from one mode to another. This behaviour can be disrupted if $x$ becomes negative at some point, but we will avoid this in practice by making $x$ the output of a pump (which, as discussed previously, can serve to “rectify” an alternating input into a steadily increasing output). We will represent the amplifier schematically by a triangle-headed arrow (Figure 3).

4. The rotor gate

where again $\alpha>0$ is a parameter. This of course preserves the total energy $x^{2}+y^{2}+z^{2}$ and has the explicit solution

in which $(x(t),y(t))$ rotates around the origin at a contant angular rate $\alpha z(t_{0})$, while $z$ remains fixed. Thus the $z$ mode can be viewed as driving the oscillating interchange of energy between the $x$ and $y$ modes.

Because we will be coupling the rotor to various forcing terms in $x$, $y$, and $z$, we cannot rely directly on the above explicit solution, although this solution is of course very useful for supplying intuition as to how the rotor behaves. Instead, we will use energy-based analyses of the rotor, which are much more robust with respect to forcing terms. Firstly we observe that for the rotor with no forcing, the combined energy of the $x$ and $y$ modes is conserved:

In a related spirit, we have the equipartition of energy identity

Using the conserved energy $x^{2}+y^{2}=E$ and the constant nature of $z$, this becomes

and similarly with $x(t)$ replaced by $y(t)$. Thus we see that over any time interval significantly longer than the period $\frac{2\pi}{\alpha|z(t_{0})|}$, the $x$ mode absorbs about half the energy $E$ of the combined pair $x,y$, and similarly for $y$.

In our application, we will use the rotor with the driving mode $z$ being the output of an amplifier. As noted previously, amplifier outputs can transition rapidly from being small to being large, so the pair $(x,y)$ will initially be almost stationary, and then suddenly transition to a highly oscillatory state. This creates a “jolt” of “alternating current”, which we will then quickly transform to “direct current” via a pump gate.

We describe the rotor gate schematically by a loop connecting the $x$ and $y$ modes that is driven by the $z$ mode: see Figure 4.

5. A delayed and abrupt energy transition

This system looks complicated and artificial, with a rather arbitrary looking set of coupling constants of wildly differing magnitudes, but it should be viewed as a superposition of five quadratic gates:

A pump of coupling constant $\varepsilon$ that transfers a small amount of energy from $a$ to $b$;

A pump of coupling constant $\varepsilon^{2}\exp(-K^{10})$ that transfers a minute amount of energy from $a$ to $c$;

An amplifier of coupling constant $\varepsilon^{-1}K^{10}$ that uses $b$ to rapidly amplify $c$;

A rotor of coupling constant $\varepsilon^{-2}$ that uses $c$ to (eventually) rotate energy very rapidly between $a$ and $d$; and

As a caricature, the evolution of this system can be described as follows, involving a critical time $t_{c}\approx\sqrt{2}$:

At a critical time $t_{c}\approx\sqrt{2}$, there is an abrupt transition when the exponentially growing $c$ suddenly (within a time of $O(K^{-10})$ or so) transitions from being much smaller than $\varepsilon^{2}$ to being much larger than $\varepsilon^{2}$. This ignites the rotor gate, which then begins to rapidly transfer energy between $a$ and $d$. By equipartition of energy, $d^{2}$ will approximately be equal to $1/2$ on the average.

We depict these dynamics schematically in Figure 6.

If $K$ is sufficiently large, and $\varepsilon$ sufficiently small depending on $K$, then there exists a time

We shall use the usual bootstrap procedure of starting with crude estimates and steadily refining them to stronger estimates on this interval, using continuity arguments if necessary in case the crude estimates are initially only available at $t=0$ rather than for all $t\in$.

From conservation of energy and (5.18) we have

throughout this interval. In particular, we have

We can improve this bound on $b$ and $c$ as follows. From (5.14), (5.15) we have the local energy identity

(in a weak derivativeTo justify this step (a very simple example of the diamagnetic inequality (see e.g. [30, §7.19-7.22]) in action), one can first work instead with $(b^{2}+c^{2}+\delta)^{1/2}$ for some small $\delta>0$, in order to avoid any singularity, and then take distributional limits as $\delta\to 0$. sense), and so from (5.18) and the fundamental theorem of calculus we see in particular that

for $t\in$. Inserting this into (5.15), we see that

for $t\in$ (note from the initial condition $c(0)=0$ and a comparison argument that $c(t)\geq 0$ for all $t\geq 0$). By Gronwall’s inequality we thus have

for all $t\in$ and some absolute constant $C>0$. Finally, from (5.16), (5.17) we have the local energy identity

and thus by (5.18) and the fundamental theorem of calculus

which is a good bound for short times $t\leq 1/C$.

for $t\in[0,t_{c}]$. Comparing this with (5.26) we conclude that $t_{c}\gtrsim 1$. From (5.27) we have

for $t\in[0,t_{c}]$. Inserting these bounds and (5.24) back into (5.13), we have

on $[0,t_{c}]$, so from (5.18) (and assuming $\varepsilon$ sufficiently small depending on $K$) we have

for $t\in[0,t_{c}]$. This already gives all the bounds (5.20). Inserting the $a$ bound into (5.14) and using (5.29), we have

and so from (5.18) (again assuming $\varepsilon$ sufficiently small depending on $K$) we have

for $t\in[0,t_{c}]$. Inserting these bounds into (5.15), we have

and hence by (5.18) and Gronwall’s inequality

for $t\in[0,t_{c}]$. In particular (since $t_{c}\gtrsim 1$), standard asymptotics on the error function give

which, when compared against the definition of $t_{c}$, shows (5.19). In particular, $t_{c}<2$ (for $K$ large enough), and so

Having described the evolution up to time $t_{c}$, we now move to the future of $t_{c}$. From (5.30) we have

Meanwhile, from (5.26), (5.14) (discarding the non-negative $\varepsilon a^{2}$ term) we have

for $t\in[t_{c},2]$, so (for $\varepsilon$ small enough) we also have

for $t\in[t_{c},2]$. Inserting this bound into (5.15), and discarding the non-negative $\varepsilon^{2}\exp(-K^{10})a^{2}$ term, we arrive at the exponential growth

for $t\in[t_{c},2]$. From this, (5.31), and Gronwall’s inequality, we see in particular that

for $t$ in the time interval $I:=[t_{c}+K^{-9},2]$. In other words, the rotor gate will be continuously and strongly activated from time $t_{c}+K^{-9}$ onwards. On the other hand, from (5.25), (5.32) we also have

for $t\in I$, so the exponential growth rate of $c$ remains under control in this region.

Similarly, from (5.13), (5.16), and (5.23) one has

We conclude using (5.32), (5.33), and the product rule that

for $t\in I$, so if we define the modified energy

Starting with the crude bound $E_{*}(t_{c})=O(1)$ from (5.35), we thus see from Gronwall’s inequality that

whenever $t_{c}\leq t^{\prime}\leq t\leq 2$. We will use this bound with $t^{\prime}:=t_{c}+1/K$. We claim that

Suppose this is not the case; then by (5.17) we have

However, by repeating the derivation of (5.34) we have

and hence by the fundamental theorem of calculus and (5.32) we have

combining this with the previous estimate, we conclude that

in this interval, giving the required contradiction.

Inserting the bound (5.37) into (5.36), we conclude in particular that

for $t_{c}+\frac{1}{\sqrt{K}}\leq t\leq 2$, and (5.21) follows from (5.35) and (5.22). ∎

Blowup for the cascade ODE

We can now prove Theorem 4.2 (and hence Theorem 3.3 and Theorem 1.5). The idea is to chain together an infinite sequence of circuits of the form (5.13)-(5.17), so that (a more complicated version of) the analysis from Theorem 5.3 may be applied.

Let $\epsilon_{0}>0$ be fixed; we allow all implied constants in the $O()$ notation to depend on $\epsilon_{0}$. As in the previous section, we need a large constant $K\geq 1$, which we assume to be sufficiently large depending on $\epsilon_{0}$, and then a small constant $0<\varepsilon<1$, which we assume to be sufficiently small depending on both $K$ and $\epsilon_{0}$. Finally, we take $n_{0}$ sufficiently large depending on $\epsilon_{0},K,\varepsilon$.

The reader may wish to keep in mind the hierarchy of parameters

as a heuristic for comparing the magnitude of various quantities appearing in the sequel. Thus, for instance, a quantity of the form $O\left(\exp\left(O\left(K^{10}\right)\right)\left(1+\epsilon_{0}\right)^{-n_{0}/2}\right)$ will be smaller than $\exp\left(-K^{10}\right)\varepsilon^{2}$; a quantity of the form $O\left(\exp\left(O\left(K^{10}\right)\right)\varepsilon\right)$ will be smaller than $K^{-100}$; and so forth.

The dimension parameter $m$ for the system we will use to prove Theorem 4.2 will be taken to be $m=4$. We set the coefficients $\alpha_{i_{1},i_{2},i_{3},\mu_{1},\mu_{2},\mu_{3}}$ by using Table 1, with $\alpha_{i_{1},i_{2},i_{3},\mu_{1},\mu_{2},\mu_{3}}$ set equal to zero if it does not appear in the above table. It is clear that the required symmetry property (4.2) and the cancellation property (4.3) hold. Also, the hypotheses of Lemma 4.1(v) are satisfied.

It will be convenient to work with the combined energy

By Lemma 4.1(iii), we have the equations of motion

(compare with (5.13)-(5.17)) and the local energy inequality

As mentioned in the previous section, if one ignores the dissipation terms, the system (6.3)-(6.6) describes an infinite number of (rescaled) copies of the quadratic circuit analysed in Theorem 5.3, with the output of each such circuit chained to the input of a slightly faster-running version of the same circuit; see Figure 7.

By Lemma 4.1(iii), we have the initial conditions

To prove Theorem 4.2, it thus suffices to show

2. Second step: describing the blowup dynamics

We will establish the following description of the dynamics of $X_{i,n}$ and $E_{i,n}$:

Let $N\geq n_{0}$ be an integer. Then there exist times

(Scale evolution) For all $n_{0}<n\leq N$, one has the amplitude stability

(Transition state) For all $n_{0}\leq n\leq N$, we have the bounds

If $n_{0}<n\leq N$, we have the additional bounds

(Energy estimates) For all $n_{0}<n\leq N$ and $t_{n-1}\leq t\leq t_{n}$, we have the bounds

These bounds may appear somewhat complicated, but roughly speaking they assert that at each time $t_{n}$, the solution concentrates an important part of its energy at scale $n$ (and significantly less energy at adjacent scales); see Table 2 and Figure 8. The precise bounds here do have to be chosen carefully, because of a rather intricate induction argument in which the estimates for a given value of $N$ are used to prove the estimates for $N+1$. For this reason, no use of the asymptotic notation $O()$ appears in the above proposition. Of the four modes $X_{1,n},X_{2,n},X_{3,n},X_{4,n}$, it is the first mode $X_{1,n}$ that carries most of the energy at the checkpoint time $t_{n}$; the secondary modes $X_{2,n},X_{3,n}$ play an important role in driving the dynamics (and so many of the more technical bounds in (viii) are devoted to controlling these modes) but carryAs a crude first approximation (ignoring factors depending on $K$), one should think of $X_{2,n}$ as being about $\varepsilon$ the size of $X_{1,n}$ or $X_{4,n}$, and $X_{3,n}$ being about $\varepsilon^{2}$ the size of $X_{1,n}$ or $X_{4,n}$. very little energy, while the $X_{4,n}$ mode is only used as a conduit to transfer energy from the $X_{1,n}$ mode to the $X_{1,n+1}$ mode. The bounds (6.21)-(6.24) are technical; they are needed to ensure that the rotor at scale $n-1$ is rotating so quickly that the modes at scale $n-1$ do not cause any “constructive interference” with the modes at scale $n$ at time $t_{n}$ (or at slightly later times).

Let us now see how the above proposition implies Theorem 6.2 (and hence Theorem 4.2, Theorem 3.3 and Theorem 1.5). Let $N\geq n_{0}$ be arbitrary. From (6.12), (6.13), (6.14) we have

and hence by (6.11) and summing the geometric series we have

for some finite $T=T_{\epsilon_{0}}$ independent of $N$. On the other hand, from (6.15), (6.13), (6.12) we have

for any $N$. Sending $N$ to infinity, we contradict (6.1).

3. Third step: setting up the induction

It remains to prove Proposition 6.3. We do so by an induction on $N$. The base case $N=n_{0}$ is easy: one sets $t_{n_{0}}=0$ and $e_{n_{0}}=1$, and all the required claims are either vacuously true or follow immediately from the initial conditions (6.8). It remains to establish the inductive case of this proposition. For the convenience of the reader, we state this inductive case as an explicit proposition.

Assume that Proposition 6.3 has already been established for some $N\geq n_{0}$, giving times

with the properties (6.11)-(6.27) stated in that proposition. Then there exists a time

and an amplitude $e_{N+1}>0$ obeying the following properties: