LOW-REGULARITY BEHAVIOR OF THE KORTEWEG-DE VRIES AND MODIFIED KORTEWEG-DE VRIES EQUATIONS: LOCAL AND GLOBAL WELL-POSEDNESS, ASYMPTOTICS,

ILL-POSEDNESS, AND SYMPLECTIC NON-SQUEEZING

TERENCE TAO

Abstract. In these notes we discuss some recent developments in the study of the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) solutions, especially in the case of low regularity initial data. Among the results: global well-posedness of solutions for KdV in spaces rough enough to contain Dirac masses, but ill-posedness in even rougher spaces; construction and asymptotics of decaying (i.e. soliton-free) solutions to KdV and mKdV; invertibility results for the Miura transform (and its generalizations); and a symplectic non-squeezing result for KdV in the natural phase space of Ḣ^{−1/2}(T). A recurring theme in these results is the analysis of the interactions between low frequencies and high frequencies.

1. Overview

These notes are concerned with the analysis of the Korteweg-de Vries (KdV) equation

u_{t} + u_{xxx} = 6uu_{x}

and the (defocusing) modified Korteweg-de Vries equation

v_{t} + v_{xxx} = 6v^{2}v_{x}

where u is a real-valued function of one spatial variable x (which either lives on the line R or circle T = R/2πZ), and one time variable t. An old result of Kato shows that these equations are globally well-posed for sufficiently smooth data, thus allowing us to define flow maps S_{KdV} (t) and S_{mKdV} (t) defined (for Schwartz functions, at least) by

S_{KdV} (t)(u(0)) := u(t)

and

S_{mKdV} (t)(v(0)) := v(t)

respectively.

For sake of introduction we shall focus on the KdV flow S_{KdV} (t). Our interest is in analyzing this flow on very rough spaces, such as the Sobolev spaces H^{s}(R), H^{s}(T) for various low values of s. Two very basic questions are:

• (Local well-posedness) On any given ball in H^{s}, can the flow map S_{KdV} (t) be continuously defined for some non-zero interval of time t ∈ [−T, T]?

• (Global well-posedness) On all of H^{s}, can the flow map S_{KdV} (t) be continuously defined for all t ∈ R?

1991 Mathematics Subject Classification. 42B15, 35L05.

These basic questions are now close to being completely answered; we shall survey the results in the first lecture, and in particular discuss the “I-method” for obtaining global well-posedness, which is based around the manipulation of conservation laws using appropriately chosen Fourier multipliers.

Once local and global well-posedness has been established, more interesting questions can then be asked. One such question is the asymptotic behavior of solutions to KdV and mKdV as t → ∞. In the KdV case there are the famous soliton solutions, but the defocusing mKdV equation does not have these solutions, and we shall be interested in the “dispersive” portion of KdV which does not have a soliton component. Here, it turns out by selecting an appropriate ansatz (inspired by the work of Ozawa on the cubic non-linear Schrödinger equation) we can describe quite accurately a large class of soliton-free solutions to KdV and mKdV, which evolve asymptotically in a manner similar to, but not quite the same as, the Airy flow S_{Airy}(t) associated to the free evolution

u_{t} + u_{xxx} = 0.

In particular there is a “logarithmic phase shift” which enters into the ansatz, similar to the phase shift discovered by Ozawa [32]. Another interesting phenomenon is that the mKdV solutions can be localized to two frequencies (+N and −N for some large N), but KdV solutions need to be localized to five frequencies (−2N, −N, 0, N, 2N) in order to create a sufficiently accurate ansatz.

These asymptotic solutions can be used to demonstrate ill-posedness for KdV and mKdV in sufficiently low regularities, much as Kenig, Ponce and Vega [26] used breather and soliton solutions to demonstrate ill-posedness in the complex KdV and focusing mKdV cases. We remark that for the real KdV and defocusing mKdV equations, the soliton solutions which are available are not suitable for demonstrating ill-posedness, due to the inability to manipulate a modulation parameter. We shall discuss asymptotics, modulation, and ill-posedness in the second lecture.

Finally, in the last lecture we investigate how to approximate the high frequency evolution and low frequency evolution of S_{KdV} and S_{mKdV} at low regularities. In particular, we seek to approximate S_{KdV} at the symplectic regularity H^{−1/2}(T) by a finite-dimensional Hamiltonian flow. This is important because H^{−1/2}(T) is the natural symplectic phase space in order to make S_{KdV} a formally Hamiltonian flow, and if we can obtain an accurate approximation of this infinite dimensional flow by finite-dimensional Hamiltonian flows, we can transport all the finite-dimensional symplectic theory (in particular, the non-squeezing theorem of Gromov) to the infinite dimensional case. Interestingly, the naive approach to truncating the flow by simply restricting to a finite set of frequencies does not work. Instead, one must truncate the Hamiltonian in a smooth manner.

2. A brief history of KdV local well-posedness on R

Here we give a sketchy review of the local well-posedness theory for KdV on the line R.

Suppose we are solving the Cauchy problem

u_{t} + u_{xxx} = 6uu_{x} = 3(uu)_{x}; u(0) = u_{0} ∈ H^{s}

for some short range of times |t| ≤ T. The first thing we observe is that there are a number of conservation laws available. For instance, the L^{2} norm^{1}

E_{2}^{0}(t) := ∫ u^{2}(t, x) dx

is conserved:

∂_{t}E_{2}^{0}(t) = ∫ 2uu_{t}

= −2 ∫ uu_{xxx} + 12 ∫ u^{2}u_{x}

= 2 ∫ u_{x}u_{xx} + 4 ∫ (u^{3})_{x}

= 0 + 0

Thus, if s ≥ 0, we have an a priori L^{2} bound on u. One may try to obtain a similar bound on the Ḣ^{1} norm

E_{2}^{1}(t) := ∫ u_{x}^{2}(t, x) dx

but the conservation property is much less favorable:

∂_{t}E_{2}^{1}(t) = ∫ 2u_{x}u_{xt}
= −2 ∫ u_{x}u_{xxxx} + 12 ∫ u_{x}(uu_{x})_{x}
= 2 ∫ u_{xx}u_{xxx} − 6 ∫ (u_{x}^{2})_{x}u
= 0 + 6 ∫ u_{x}^{3}.

Unless one somehow has an L^{∞} bound on u_{x}, this estimate does not seem to be directly useful. However, if one adjusts E_{2}^{1} by the appropriate trilinear quantity, namely

E_{3}^{1}(t) := ∫ u_{x}^{2}(t, x) + 2u^{3} dx

then we again have a conservation law:

∂_{t}E_{3}^{1}(t) = ∂_{t}E_{2}^{1}(t) + ∫ 6u^{2}u_{t}
= 6 ∫ u_{x}^{3} − ∫ 6u^{2}u_{xxx} + ∫ 36u^{3}u_{x}
= 6 ∫ u_{x}^{3} + ∫ 12uu_{x}u_{xx} + ∫ (9u^{4})_{x}
= 6 ∫ u_{x}^{3} + ∫ 6u(u_{x}^{2})_{x} + 0
= 0.

1We call this quantity E_{2} because it plays the role that energy does in other equations such as non-linear Schrödinger, for the purposes of global well-posedness. The 2 denotes the fact that E_{2} is bilinear in u, while the 0 denotes that it requires 0 degrees of regularity on u.

Thus if s ≥ 1, we have an a priori bound on the H^{1} norm of u (using the Gagliardo-Nirenberg inequality). It turns out that because of the completely integrable structure of KdV, one can repeat this procedure for any positive integer s. (Later we will see that this procedure can also be adapted to non-integer s). In particular once s ≥ 2 one can get L^{∞} bounds on u_{x}.

Using this and some standard compactness arguments, Kato [19] (and also Bona and Smith [3]) was able to obtain global well-posedness of KdV in H^{s} for s ≥ 2, and local well-posedness for s > 3/2 (which implies L^{∞} control on u_{x} by Sobolev embedding). A key additional ingredient (in order to obtain the compactness) was the Kato smoothing effect: if u_{0} is in H_{x}^{s}, then u is in fact locally in H_{x}^{s+1} for most times t. This is most easily demonstrated when s = 0. We introduce a multiplier η(x), which equals 1 for x > 1, 0 for x < −1, and is smoothly increasing in between. Then if u_{0} is in L^{2} we have an a priori bound on the quantity

E_{2}^{η}(t) := ∫ η(x)u^{2}(x) dx.

Now we repeat the above calculations, trying to push as many derivatives onto the smooth function η as we can:

∂_{t}E_{2}^{η}(t) = ∫ 2ηuu_{t}

= −2 ∫ ηuu_{xxx} + 12 ∫ ηu^{2}u_{x}

= 2 ∫ ηu_{x}u_{xx} + ∫ η_{x}uu_{xx} + 4 ∫ η(u^{3})_{x}

= − ∫ η_{x}u_{x}^{2} − ∫ η_{xx}uu_{x} − ∫ η_{x}u_{x}^{2} − 4 ∫ η_{x}u^{3}

= −2 ∫ η_{x}u_{x}^{2} − \frac{1}{2} ∫ η_{xxx}u^{2} − 4 ∫ η_{x}u^{3}.

The latter two terms can be controlled by the L^{2} bound and Gagliardo-Nirenberg. The first term is the main term, and it is always negative, so by the fundamental theorem of calculus we obtain a bound of the form

∫_{t=O(1)} ∫ η_{x}u_{x}^{2} dxdt ≲ 1,

which shows that u(t) is locally in H_{x}^{1} for most times t.

This computation shows that one can insert a (spatial) multiplier in front of the energy and recover some useful information. Later on we will see that one can also insert a Fourier multiplier into the energy to obtain similar results.

These smoothing estimates were refined by Vega. In combination with a certain maximal function estimate (inspired by work of Carleson, Dahlberg and Kenig), Kenig, Ponce and Vega [23] were able to improve the local well-posedness to s > 3/4 (and the global well-posedness to s ≥ 1, thanks to the a priori global H^{1} bounds). Unlike the compactness methods of Kato, the methods used here were iterative, writing the equation in integral form as

u = S_{Airy}u_{0} + N(u, u)

where

N(u, v)(t) = ∫_{0}^{t} S_{Airy}(t − s)(3(uv)_{x})(s) ds

is the Duhamel term arising from the non-linearity. The idea was then to find a spacetime Banach space X^{s} with the same regularity as H^{s}, such that the bilinear operator N mapped X^{s} × X^{s} to X^{s} (with the norm of this operator being small if the time variable t was restricted to a sufficiently small interval). One then applied the Picard iteration method. As one consequence of this method one could show that the solution map S_{KdV} (t) was not only continuous, but in fact analytic on H^{s}.

The next step was taken by Bourgain [4], who introduced a new type of spacetime Banach space X^{s} based on the spacetime Fourier transform (essentially, a weighted Sobolev space adapted to the Airy flow S_{Airy}; similar spaces had been introduced earlier for the wave equation by Rauch-Reed and Beals-Bezard). In this work the local well-posedness theory was pushed down to s ≥ 0, and thus the global wellposedness theory as well thanks to a priori L^{2} bounds. (The a priori L^{2} bound, combined with the smoothing effect, are already enough to conclude the existence of global weak solutions by compactness methods, but there is not enough regularity to conclude uniqueness or continuity by classical techniques.)

Further refinements of this idea by Kenig, Ponce and Vega [24], [25] then took the local well-posedness theory down to s > −5/8, and finally to s > −3/4; at this point they observed that the nonlinear operator N began to behave badly for s < −3/4. A little later Bourgain [7] showed that in fact the solution map S_{KdV} (t) ceased to be analytic (or even C^{3}) when s < −3/4; this was refined to C^{2} by Tzvetkov [39]. At s = −3/4 the key non-linear estimate also failed in the X^{s,b} spaces of Bourgain, as was observed by Nakanishi, Takaoka, and Tsutsumi [31], although this did not preclude the map S_{KdV} (t) still being analytic in H^{−3/4}.

In the second lecture we will explain why the solution map S_{KdV} (t) still turns out to be analytic in H^{−3/4}, but is not uniformly continuous in H^{s} for any s < −3/4. Thus −3/4 is indeed the boundary for which the KdV flow is uniformly continuous (although there is still the possibility of non-uniform continuity below this regularity). In particular it is possible to construct local solutions for initial data which is a finite measure (in particular, a Dirac delta function).

The iterative methods give a time of existence T which varies inversely with the H^{s} norm of the data. At the end of this existence interval, the H^{s} norm might conceivably increase by a factor of 2 or so; this can lead to blowup in finite time unless one somehow obtains a priori H^{s} control on the solution u(t) for long times t. This can be done for non-negative integer s as indicated earlier; we explain below how one can achieve it for non-integer s.

To conclude this section, we explain informally why low regularities are harder than high regularities. Let us pretend that our solution u consists of a low frequency portion u_{low}, supported on frequencies ξ = O(1) (say), combined with a high frequency portion u_{high}, supported at frequencies |ξ| ∼ N, for some large N ≫ 1. The non-linearity N(u, u) then splits into three types of interactions: lowlow interactions N(u_{low}, u_{low}), low-high interactions 2N(u_{low}, u_{high}), and high-high interactions N(u_{high}, u_{high}).

As s gets lower, the high frequency portion u_{high} can get larger, but the low frequency portion stays about the same. Thus there is no difference to the low-low interactions, and also no significant difference to the low-high interactions since they are linear in u_{high}. (Note that N(u_{low}, u_{high}) is also high frequency, so it will not cause a “high to low cascade” in which a lot of mass at high modes moves down to low modes). However, the high-high interactions become increasingly severe as s gets lower, and eventually causes either a high-to-low cascade, or increasingly unstable high frequency self-interaction, or perhaps a cascade from high frequencies to even higher frequencies, leading to concentration and blowup. (Presumably, even weirder scenarios are possible, but these are the three basic ones that we know how to analyze).

3. The I-method

In the previous section we saw that the KdV flow S_{KdV} was locally well-posed in H^{s} for s > −3/4. Because of the a priori L^{2} bounds, we can also extend this to global well-posedness for s ≥ 0. (Once one has a priori control on some H^{s}, it is relatively easy from the iterative theory to gain control on higher H^{s} norms. This is the reverse of the informal argument at the end of the previous section; as s gets higher, u_{high} gets weaker, and one has less of a movement to high frequencies).

So to extend the global well-posedness theory to match the local one, we need to get a priori H^{s} control on solutions for −3/4 < s < 0. Actually it will suffice to do so for smooth solutions, since the local well-posedness theory is strong enough to allow us to then take a limiting argument. Also, it suffices to pick a target time T and just try to get H^{s} bounds up to this time T; the bounds are allowed to depend on T, as long as T is allowed to be arbitrarily large. Observe that the L^{2} conservation law does not appear to be directly useful since the L^{2} norm of arbitrary H^{s} data is infinite.

The first step in this direction was taken by Bourgain (but for a slightly different set of equations, including the two-dimensional nonlinear Schrodinger equation). The idea was to pick a large frequency cutoff N depending on T, and split u = u_{low} + u_{high} accordingly. The function u_{low}(0) is in H^{s}, and thus by the frequency localization is also in L^{2} (but with a large norm, namely O(N^{−s})). Thus if u_{low} simply evolved by itself, with no interference from u_{high}, it would exist globally in time, with global L^{2} bounds (and thus global H^{s} bounds). Meanwhile, u_{high} is much rougher; it is not in L^{2}, but if one goes to a norm H^{s′} for some s′ < s we see that u_{high} is quite small (it has an H^{s′} norm of O(N^{−(s−s′)})). Thus one can hope to neglect the contribution of u_{high} (at least for times up to T, if N is large enough depending on T).

The idea is then to show that the low-high and high-high interactions N(u_{low}, u_{high}), N(u_{high}, u_{high}) are small in L^{2} in the sense that they have norm O(N^{−σ}) for some σ > 0, at least for some short time t; this is feasible if s is not too negative. At the end of this time, all the effect of these interactions are added by fiat to the low frequency term u_{low}; this disturbs the L^{2} conservation law, but not by much, and so one can still continue to evolve u_{low} for long times. Meanwhile, u_{high} simply evolves by the free evolution S_{Airy}. If s is not too negative, one can hope to hold this scheme together all the way to time T.

In order for this strategy to work, it is imperative that all the low-high and high-high interactions are in L^{2}. In particular one needs a smoothing estimate of the form

∥N(u, v)∥_{L_{t}^{∞}L_{x}^{2}} ≲ ∥u∥_{X^{s}}∥v∥_{X^{s}}

where X^{s} is the appropriate spacetime norm with the regularity of H^{s} that one is iterating in. (In comparison, the estimate that one needs for local well-posedness only places N(u, v) in L_{t}^{∞}H_{x}^{s}). Colliander, Staffilani, and Takaoka carried out this computation and observed that this estimate actually failed unless some vanishing of u and v was assumed near the frequency origin (i.e. u and v needed to obey some moment conditions). Assuming such conditions, they were able to carry out the above scheme and obtain global well-posedness in a space which was basically H^{s} for s > −1/24, with some additional assumptions of vanishing near the frequency origin.

As one consequence of the argument, we see that the high frequency evolution is very close to the free Airy evolution, in particular there is a bound of the form

∥S_{KdV} (t)u_{0} − S_{Airy}(t)u_{0}∥_{L^{2}} ≤ C(t, ∥u_{0}∥_{H^{s}}).

(Actually one has to modify the spaces H^{s} as indicated above, but we ignore this technicality). Thus, as measured in the H^{s} norm, the divergence between the KdV and Airy flows are mostly concentrated in the low frequencies.

However, it seems difficult to carry this scheme much further, mainly because the low-high interactions are not negligible if no vanishing is assumed near the frequency origin, and even if vanishing is assumed, it is increasingly difficult to demand a smoothing effect from H^{s} to L^{2} for this low-high interaction. However, the key observation (which motivates the “I-method” of Keel, Colliander, Staffilani, Takaoka, and myself) is that this low-high interaction, while it does cause a significant effect on the high-frequency evolution u_{high}, it does not significantly affect the energy of u_{high}. (This is similar to how the KdV nonlinearity 6uu_{x} is not negligible, but nevertheless it does not affect L^{2} conservation). To explain this, let us ignore everything except the high-low interaction, and look at the equation

(∂_{t} + ∂_{xxx})u_{high} = 3(u_{low}u_{high})_{x}

where u_{low} is some low frequency function we will not specify. If we define

E_{2}^{high}(t) := ∫ u_{high}^{2}

then by repeating the previous types of calculations we have

∂_{t}E_{2}^{high}(t) = ∫ 2u_{high}∂_{t}u_{high}
= −2 ∫ u_{high}∂_{xxx}u_{high} + 6 ∫ u_{high}(u_{low}u_{high})_{x}
= 0 − 6 ∫ u_{low}u_{high}(u_{high})_{x}
= 3 ∫ (u_{low})_{x}u_{high}u_{high}.

This is quite favorable because the derivative is only on the low frequency term u_{low}. In conjunction with the Kato smoothing effect (and the more refined smoothing effects of Kenig-Ponce-Vega and Bourgain), one thus has quite a bit of hope that one can hope to keep E_{2}^{high} under control).

As long as the high-high interaction is negligible, the energy of the low modes u_{low} should also be controlled, since the only significant nonlinearity is the low-low interaction, in which case the evolution behaves like

(∂_{t} + ∂_{xxx})u_{low} = 3(u_{low}u_{low})_{x}

which is just the KdV flow again, which conserves L^{2}. Thus we should be able to control the energies of u_{low} and u_{high} separately.

Of course, there is a problem in that u_{high} is merely in H^{s} for some s < 0, so the energy of u_{high} can be infinite. To resolve this we use a damping Fourier multiplier (cf. the spatial multiplier used to prove the Kato smoothing effect). This operator will conveniently handle u_{low} and u_{high} together as a single unit, avoiding any need to actually decompose the solution (at least until one gets to proving the required bilinear estimates, see below). Nevertheless, the above low-high picture is very useful to keep in mind when following the method.

We introduce the operator I defined by

Îu(ξ) := m(ξ)û(ξ)

where m(ξ) equals 1 for |ξ| ≤ N, equals (|ξ|/N)^{s} for |ξ| ≥ 2N, and is a smooth interpolant in between. Thus I is the Identity for the low modes u_{low}, and an Integration operator for frequencies u_{high}. From this we see that ∥Iu∥_{L^{2}} is somewhat comparable with ∥u∥_{H^{s}}:

∥Iu∥_{L^{2}} ≲ ∥u∥_{H^{s}} ≲ N^{−s}∥Iu∥_{L^{2}};

one can see this by testing on low frequencies and high frequencies separately. We then introduce the modified energy E_{2}^{I} by

E_{2}^{I}(t) := ∫ (Iu(t))^{2},

thus E_{2}^{I}(t) is comparable to the H^{s} norm of u(t) squared (up to factors of N^{−s}. So to get a priori bounds on ∥u(t)∥_{H^{s}} it will suffice to get a priori bounds on E_{2}^{I}.

We compute once again

∂_{t}E_{2}^{I}(t) = ∫ IuIu_{t}
= − ∫ IuIu_{xxx} + ∫ Iu6I(uu_{x})
= 0 + 3 ∫ Iu∂_{x}(I(u^{2}) − (Iu)^{2}) + 3 ∫ Iu∂_{x}((Iu)^{2})
= 3 ∫ Iu∂_{x}(I(u^{2}) − (Iu)^{2}) + 0.

Thus we are left with estimating a “commutator” expression between I and the quadratic expression u^{2}. Of course in the limit N → +∞, I becomes 1, and the above expression vanishes. When N is finite but large, the hope is then to get a bound such as

∂_{t}E_{2}^{I}(t) = O(N^{−σ}E_{2}^{I}(t)^{3/2})

for some σ > 0. Since we have an initial bound of E_{0}^{I}(t) ⪅ N^{−2s}∥u_{0}∥_{H^{s}}^{2} ⪅ N^{−2s}, one can thus get control of E_{2}^{I}(t) up to time T if σ is large enough compared to s, and N is large enough compared to T.

Actually, in practice one does not quite obtain such a bound (because it is difficult to control the derivative ∂_{x}, among other things, just by using the energy E_{2}^{I}(t). Instead one obtains a discretized analogue of the form

E_{2}^{I}(t + δ) − E_{2}^{I}(t) = O(N^{−σ}δE_{2}^{I}(t)^{3/2})

for some small time increment δ, which turns out to basically be E_{2}^{I}(t)^{−1/2}. To prove this it is first convenient to rescale using the KdV rescaling to make E_{2}^{I}(t) equal to 1, which makes δ essentially equal to 1 (this is basically possible thanks to the sub-criticality of our problem here). One can then apply (a variant of) the local well-posedness theory to give some spacetime bounds on the solution u in the interval between t and t + δ (basically, because we have bounded Iu in L^{2} at time t, we get to bound Iu in X^{0} on the time interval [t, t + δ]. Note that estimating Iu in L^{2} is stronger than estimating u in H^{s}, and so by the informal argument at the end of the previous section, we expect the local well-posedness theory at H^{s} to imply a local well-posedness theory in the norm ∥Iu∥_{2}. This can be made rigorous by using, for instance, the fractional Leibnitz rule.

It remains to see why we would expect the commutator expression I(u^{2}) − (Iu)^{2} would be “small” in the sense that there is a decay of N^{−σ} in some appropriate norm. To see this it is convenient to once again split u = u_{low} + u_{high}, so that the commutator splits into the “low-low” part

I(u_{low}u_{low}) − (Iu_{low})(Iu_{low})

the “low-high” part

2(I(u_{low}u_{high}) − (Iu_{low})(Iu_{high}))

and the “high-high” part

2(I(u_{high}u_{high}) − (Iu_{high})(Iu_{high})).

The low-low part vanishes because I is the identity on low frequencies (perhaps after tweaking the frequency cutoff for low frequencies by a factor of 2 or so); this is essentially the same observation that Bourgain’s method uses to ignore the low-low interactions. The high-high part does not vanish, but fortunately in the context of KdV it is fairly small (at least for s not too close to −3/4). The main improvement in the I-method over Bourgain’s method lies in the low-high part. This is because Iu_{low} = u_{low}, and so we can write the above expression as a genuine commutator

2(I(u_{low}u_{high}) − u_{low}(Iu_{high})) = 2[I, u_{low}]u_{high}.

If u_{low} was constant, this would vanish. While u_{low} is not quite constant, it is low frequency, and this allows one to extract a certain gain out of the commutator. (On the Fourier multiplier side, this represents the fact that m(ξ + η) − m(ξ) is smaller than m(ξ) if |η| is much lower than |ξ|). Thus the low-high interactions become quite small, and one is just left with the high-high interactions.

In terms of the frequency distribution, the I-method asserts that for times up to some time T = N^{α}, the high frequencies do not move much in energy (so that one can damp them with the I without affecting conservation of energy much), while the low frequencies stay low but can slosh around between low modes. As time increases, more and more of the frequencies are free to move around, causing a possible polynomial growth in the energy. Note that we do not get to show that the high frequencies evolve like the linear flow, unlike Bourgain’s method, because we do not show that the low-high interaction is small (instead using commutator estimates to finesse this issue).

By pursuing these techniques rigorously one can eventually obtain a global wellposedness result at s > −3/10, which unfortunately is still short of the local result at s > −3/4. The problem is that the decay N^{−σ} obtained by this method is only N^{−3/4+}; the commutator yields some cancellation, but not an infinite amount. One can do a little better by refining the estimates a bit more, but this improvement only goes so far. To get a significant gain one needs to replace the modified energy E_{2}^{I}(t) by something with a smaller derivative. In analogy with our experience with E_{2}^{1} and E_{3}^{1} it seems natural to do so by adding a cubic or higher “correction term” to the energy E_{2}^{I}, and this is exactly what we will do.

To explain the argument it is convenient to move to Fourier space and use some notation for multilinear multipliers. If n ≥ 2 is an integer, we define a (spatial) n-multiplier to be any function M_{n}(ξ_{1}, …, ξ_{n}) on the hyperplane

Γ_{n} := {(ξ_{1}, …, ξ_{n}) ∈ R^{n} : ξ_{1} + … + ξ_{n} = 0}.

If M_{n} is a n-multiplier and u_{1}, …, u_{n} are functions on R, we define the n-linear functional Λ_{n}(M_{n}; u_{1}, …, u_{n}) by

Λ_{n}(M_{n}; f_{1}, …, f_{n}) := ∫_{Γ_{n}} M_{n}(ξ_{1}, …, ξ_{n}) ∏_{j=1}^{n} û_{j}(ξ_{j}).

We adopt the notation

Λ_{n}(M_{n}; u) := Λ_{n}(M_{n}; u, …, u).

Observe that Λ_{n}(M_{n}; f) is invariant under permutations of the ξ_{j} indices. In particular we have

Λ_{n}(M_{n}; u) = Λ_{n}([M_{n}]_{sym}; u)

where

[M_{n}]_{sym}(ξ) := \frac{1}{n!} ∑_{σ∈S_{n}} M_{n}(σ(ξ))  (1)

is the symmetrization of M_{n}.

Thus, for instance, we have ∫ u^{2} = Λ_{2}(1; u). Now suppose that u obeys the KdV evolution, and M_{n} is a symmetric multiplier. Then a direct computation yields the differentiation law

\frac{d}{dt}Λ_{n}(M_{n}; u(t)) = Λ_{n}(M_{n}α_{n}; u(t)) − 3inΛ_{n+1}(M_{n}(ξ_{1}, …, ξ_{n−1}, ξ_{n} + ξ_{n+1})(ξ_{n} + ξ_{n+1}); u(t)) (2)

where

α_{n} := ξ_{1}^{3} + … + ξ_{n}^{3}.

Thus for instance we have

\frac{d}{dt}Λ_{2}(1; u(t)) = Λ_{2}(α_{2}; u(t)) − 6iΛ_{3}(ξ_{2} + ξ_{3}; u(t))
= Λ_{2}(ξ_{1}^{3} + ξ_{2}^{3}; u(t)) − 4iΛ_{3}(ξ_{1} + ξ_{2} + ξ_{3}; u(t))
= 0 − 0,

demonstrating the conservation of the L^{2} norm.

Henceforth we shall omit the u(t) from the Λ_{n} notation for brevity. We also adopt the convenient notation that ξ_{ij} := ξ_{i} +ξ_{j}, etc., thus for instance ξ_{145} = ξ_{1} +ξ_{4} +ξ_{5}. Also we write m_{i} := m(ξ_{i}), m_{ij} := m(ξ_{ij}), etc, and N_{i} for |ξ_{i}|, N_{ij} for |ξ_{ij}|, etc.

By construction we have

E_{2}^{I}(t) := Λ_{2}(m_{1}m_{2}).

From (2), (1) and the fact that α_{2} = 0 we have

\frac{d}{dt}E_{2}(t) = −6iΛ_{3}(m_{1}m_{23}ξ_{23})
= 6iΛ_{3}(m_{1}^{2}ξ_{1})
= Λ_{3}(M_{3})

where M_{3} is the 3-multiplier

M_{3} := 2i(m_{1}^{2}ξ_{1} + m_{2}^{2}ξ_{2} + m_{3}^{2}ξ_{3}).

Now define the modified energy E_{3}^{I}(t) by

E_{3}^{I}(t) := E_{2}^{I}(t) + Λ_{3}(σ_{3})

where σ_{3}(ξ_{1}, ξ_{2}, ξ_{3}) is the 3-multiplier

σ_{3} := −M_{3}/α_{3}.

This multiplier may appear to be singular at first glance, but we observe that

α_{3} = ξ_{1}^{3} + ξ_{2}^{3} + ξ_{3}^{3} = 3ξ_{1}ξ_{2}ξ_{3}  (3)

and that M_{3} vanishes whenever ξ_{1}ξ_{2}ξ_{3} = 0. Then by (2), (1) we have

\frac{d}{dt}E_{3}^{I}(t) = Λ_{3}(M_{3}) + Λ_{3}(σ_{3}α_{3}) − −9iΛ_{4}(σ_{3}(ξ_{1}, ξ_{2}, ξ_{34})ξ_{34})
= Λ_{4}(M_{4})

where M_{4} is the 4-multiplier

M_{4} := −9i[σ_{3}(ξ_{1}, ξ_{2}, ξ_{34})ξ_{34}]_{sym}.

Now define the modified energy E_{4}^{I}(t) by

E_{4}^{I}(t) := E_{3}^{I}(t) + Λ_{4}(σ_{4})

where σ_{4}ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}) is the 4-multiplier

σ_{4} := −M_{4}/α_{4}

This multiplier may appear to be singular at first glance, but we observe that

α_{4} = ξ_{1}^{3} + ξ_{2}^{3} + ξ_{3}^{3} + ξ_{4}^{3} = 3ξ_{12}ξ_{13}ξ_{14}  (4)

and one can check that M_{4} vanishes when ξ_{12}ξ_{13}ξ_{14} = 0. Then as before we have that

\frac{d}{dt}E_{4}^{I}(t) = Λ_{5}(M_{5})  (5)

where

M_{5} := −12i[σ_{4}(ξ_{1}, ξ_{2}, ξ_{3}, ξ_{45})ξ_{45}]_{sym}.

We could continue this procedure indefinitely, but E_{4}^{I} will turn out to be a suitable almost conserved quantity for our purposes, giving the more than adequate decay of N^{−5/2+}. (If one just uses E_{3} then one gets decay of N^{−3/2+}, which ultimately translates to s > −1/2; the above decay of N^{−5/2+} will get us all the way down to s > −3/4 (and in principle a bit beyond, but the local well-posedness theory gives out after −3/4, at which point a priori bounds on the H^{s} norm don’t help much)). The point is that the denominators α_{2}, α_{3}, α_{4} are giving additional damping on the multiplier and giving more and more decay in N. The main difficulty is when the α_{j} degenerate to zero, but here there is some cancellation in the numerator (similar to the commutator cancellation observed earlier) which comes to the rescue. It seems that this procedure is aided by the completely integrable structure of KdV; the same argument can be adapted to non-integrable situations, such as the nonintegrable versions of the derivative non-linear Schrodinger equation, but it is not clear whether one can get arbitrary decay by this method (which would eventually translate to, not just global well-posedness, but a polynomial growth bound of the form ∥u(t)∥_{H^{s}} ≲ (1 + |t|)^{α} with arbitrarily small α).

We remark that this procedure can also be used to discover the infinite number of conservation laws for KdV without explicitly using integrable systems techniques.

The same strategy works on the circle T. Here the relevant endpoint is s = −1/2 rather than s = −3/4, in that local well-posedness has already been established up to and including the endpoint s = −1/2 by Kenig, Ponce and Vega [25], with ill-posedness below −1/2. The spaces X^{s} are a little different, and so the proof of the key bilinear estimates (actually we need some higher order estimates too, because of our use of higher order correction terms, eventually ending up with using a quintilinear estimate), but otherwise the basic strategy is the same. (One technicality is that the scaling argument becomes a little more subtle, since when one scales the circle one gets a new circle with larger period. This causes an unpleasant amount of technicality which I don’t want to discuss here).

This method is actually rather general. In addition to KdV (and thence to mKdV by the Miura transform, which we discuss later), the method has also been applied to wave maps, the Maxwell-Klein-Gordon system, nonlinear Schrodinger equations in one, two, and three dimensions, and generalized KdV. It is not just restricted to proving global well-posedness for low regularities; it can also prove related results such as scattering or orbital stability for low regularities.

4. The Miura transform

The Miura transform M is defined by

M(v) := v_{x} + v^{2}.

Observe that if v is a smooth real-valued solution to the mKdV equation, then u = M(v) is a smooth real-valued solution to KdV equation. Indeed:

(∂_{t} + ∂_{xxx})u = (∂_{t} + ∂_{xxx})(v_{x} + v^{2})
= ∂_{x}(∂_{t} + ∂_{xxx})v + 2v(∂_{t} + ∂_{xxx})v + 6v_{x}v_{xx}
= ∂_{x}(6v^{2}v_{x}) + 12v^{3}v_{x} + 6v_{x}v_{xx}
= ∂_{x}(6v^{2}v_{x} + 3v^{4} + 3v_{x}^{2})
= ∂_{x}(3u^{2}).

The Miura transform acts roughly like a derivative, and in particular maps H_{x}^{s} to H_{x}^{s−1}. From this fact (and a little extra work, for instance using the L^{2} conservation of mKdV), we can show that if KdV is locally or globally well-posed in H^{s−1}, then mKdV is similarly well-posed in H^{s}. In particular, one has local and global wellposedness of mKdV^{2} in R for H^{s}, s > 1/4, and in T for H^{s}, s ≥ 1/2.

However, the local theory for mKdV on R is slightly better than this; Kenig, Ponce, and Vega were able to use the Kato smoothing effect and another maximal function estimate to obtain local well-posedness of mKdV at s = 1/4. This suggests that we should be able to resolve the endpoint well-posedness of KdV at s = −3/4 in the affirmative, by inverting the Miura transform.

Unfortunately the Miura transform is not surjective, for the same reason that the derivative map v ↦ v_{x} is not surjective from H^{s} to H^{s−1}. Namely, the map vanishes at low frequencies. Fortunately, this can be rectified by adding a low frequency (i.e. smooth) error term to the Miura transform.

2Similar results hold for the focussing mKdV equation, but it gets a little trickier because one needs to use the complex Miura transform v → v_{x} + iv^{2}. See [13].

Motivated by this, we define a generalized Miura transform M : H_{x}^{1/4} × H_{x}^{1} → H_{x}^{−3/4} by

M(v, w) := v_{x} + v^{2} + w,

where v and w are complex valued. (The H^{1} regularity of the error is not particularly special - anything between H^{1/4} and H^{5/4} will suffice).

It is possible to show (basically by iterative arguments) that the generalized Miura transform is surjective, and has an inverse which is locally Lipschitz; see [10]. Of course, the modified Miura transform M(v, w) no longer maps mKdV to KdV. However, if (v, w) are smooth solutions to the mKdV-like system

v_{t} + v_{xxx} = 6(v^{2} + w)v_{x}
w_{t} + w_{xxx} = 6(v^{2} + w)w_{x}
v(x, 0) = v_{0}(x)
w(x, 0) = w_{0}(x).

Then the function u(t) := M(v(t), w(t)) will solve KdV with u_{0} := M(v_{0}, w_{0}). Indeed, we have

(∂_{t} + ∂_{xxx})u = (∂_{t} + ∂_{xxx})(v_{x} + v^{2} + w)
= ∂_{x}(∂_{t} + ∂_{xxx})v + 2v(∂_{t} + ∂_{xxx})v + 6v_{x}v_{xx} + (∂_{t} + ∂_{xxx})w
= ∂_{x}(6(v^{2} + w)v_{x}) + 12v(v^{2} + w)v_{x} + 6v_{x}v_{xx} + 6(v^{2} + w)w_{x}
= ∂_{x}(6v^{2}v_{x} + 3v^{4} + 3v_{x}^{2} + 3w^{2} + 6wv_{x} + 6wv^{2})
= ∂_{x}(3u^{2}).

(This remarkable identity presumably has some explanation in the general AKNS framework [1]).

The generalized Miura transform is an extension of Gardner’s generalization [30] of the Miura transform: If a, b are constants and y solves

y_{t} + y_{xxx} = 6(a^{2}y^{2} + by)y_{x}

then u := ay_{x} + ay^{2} + by solves KdV. (Indeed, one applies the generalized Miura transform with v := ay and w := by).

One can show that the mKdV system described above is locally well-posed in H^{1/4}, and hence KdV is locally well-posed in H^{−3/4}. It seems likely that by combining this transform with the arguments in [13] one can also get global well-posedness for mKdV and KdV at these respective regularities.

This Miura transform can similarly be generalized on the circle. In fact, on the circle it suffices to take w to be a constant, and specifically one can take w to be the lowest eigenvalue of the operator −\frac{d^{2}}{dx^{2}} + u. (See Appendix).

5. Asymptotics, modulation and ill-posedness

In the previous sections we discussed global solutions to KdV and mKdV, with particular emphasis on low regularity solutions. Now we change tack slightly, and now return to smooth global solutions, but instead focus on understanding the asymptotic behavior of such solutions, especially for small data.

The KdV picture is complicated by the presence of soliton solutions, so we instead look at mKdV (after all, we can convert our arguments here back to KdV via the Miura transform). Actually, it will be more convenient to first look at the simpler NLS equation

−iu_{t} + u_{xx} = |u|^{2}u

where u is now complex-valued. It is well known (see e.g. [37]) that solutions to the free Schrödinger equation with Schwarz initial data behave asymptotically like

u(x, t) ≈ t^{−1/2} exp(−ix^{2}/4t))w(x/t)

for some Schwarz w (which is essentially the Fourier transform of the initial data).

One may conjecture that for small data, the NLS equation also has similar asymptotics (i.e. there is scattering). But surprisingly, Ozawa [32] showed that for certain small data the asymptotics instead behave like

u(x, t) ≈ t^{−1/2} exp(−ix^{2}/4t)w(x/t) exp(i|w(x/t)|^{2} log(t));

in other words, there is a logarithmic phase shift. (The precise statement of Ozawa was that for any sufficiently small Schwartz function w, there existed a solution u with these asymptotics; in other words, he constructed modified wave operators. This fundamental result has since been refined and extended by many authors, notably Hayashi, Ginibre and Velo, and Nakanishi).

How can one explain the presence of this phase shift? One clue comes from the observation that

t^{−1/2} exp(−ix^{2}/4t) exp(i log(t))

is an explicit solution of NLS (this seems to have first been observed in [1]), but this does not completely explain the matter. The argument in [32] proceeds via the pseudo-conformal transformation

(y, s) := (\frac{x}{t}, \frac{1}{t}); (x, t) = (\frac{y}{s} , \frac{1}{s})

u(t, x) := t^{−1/2} exp(−ix^{2}/4t)v(s, y)

to obtain

iv_{s} + v_{yy} = s^{−1}|v|^{2}v;

ignoring the dispersive term v_{yy} we obtain an ODE which can be explicitly solved as

v = w(y) exp(−i|w(y)|^{2} log s).

Inverting the pseudo-conformal transformation we obtain (heuristically at least) the asymptotics.

However, once one has the right guess for the asymptotics, one can verify them directly using energy methods. If one sets

u^{[w]} := t^{−1/2} exp(−ix^{2}/4t)w(x/t) exp(i|w(x/t)|^{2} log(t))

one can verify that u^{[w]} is an approximate solution to NLS in the sense that the error

e := −iu_{t}^{[w]} + u_{xx}^{[w]} − |u^{[w]}|^{2}u^{[w]}

has^{3} an H^{2} norm of t^{−2} log^{2} t as t → ∞ (in fact, it looks quite a lot like the original function u^{[w]}, but with a decay of t^{−5/2} log^{2} t or better instead of t^{−1/2}). Since this is integrable in time, one can thus use energy methods (and many applications of Gronwall’s inequality) to construct an exact solution u to NLS with the asymptotics of u^{[w]}, at least for times t ≥ 1 and assuming w is sufficiently small.

3Actually, this is a lie; the H^{2} norm is only like t^{−1} log t. One has to add a correction term to the ansatz to make the above claim rigorous, just as with mKdV; see the discussion below.

This logarithmic phase shift shows that solutions to NLS eventually decohere in phase: if one varies w even by an epsilon, then eventually as t → ∞, the phase of u^{[w]} will change substantially (though the magnitude is still quite stable).

One can take advantage of the Gallilean invariance

u(x, t) ↦ e^{iNx}e^{iN^{2}t}u(x + 2tN, t)

to thus create a family of highly oscillating solutions to NLS, with the asymptotics

u^{[w]} := t^{−1/2} exp(−ix^{2}/4t)w(x/t + 2N) exp(i|w(x/t + 2N)|^{2} log(t));

the phase exp(−ix^{2}/4t) has spatial frequency ∼ N on the support of w(x/t + 2N).

By rescaling these oscillating solutions it is possible to construct, for any s < 0, a pair of solutions which are arbitrarily close in H^{s} at time zero, but (due to the decoherence property) are quite far apart in H^{s} at some later time which can be rescaled to be arbitrarily small. This then implies that the solution map for NLS is ill-posed (in the sense that the solution map was not uniformly continuous) in H^{s} for any s < 0. (This was first achieved for the focusing NLS in [26] using soliton solutions instead of Ozawa’s asymptotics; the argument above is in [10] and is otherwise similar).

In [10], a similar family of highly oscillatory global solutions were constructed for mKdV, and the corresponding scaling argument used to demonstrate ill-posedness in H^{s} for s < 1/4, exactly matching the well-posedness theory^{4}. The ansatz however is a little more complicated. The known counterexamples [25] to the trilinear estimate needed for well-posedness to mKdV suggests that the solution should have frequency ±N + O(N^{−1/2}). An oscillating solution to the Airy equation with this Fourier support and normalized in H^{1/4} has the asymptotics

u(x, t) ≈ 2N^{−1/2}t^{−1/2} cos ((−4x^{3}/27t)^{1/2})w(z)

where w is a bump function and we use the shorthand

z = N^{−1/2}t^{−1}(x + 3N^{2}t).

Note that x is extremely negative (x ∼ −3N^{2}t) on the support of w, so the phase in the cosine is real.

Following Ozawa, we thus seek a solution to mKdV with a similar form but with an additional logarithmic phase correction. One can determine this phase correction by inserting it into the above ansatz, substituting the ansatz into the mKdV equation, and determining what is needed to make the two top order terms in the resulting equation cancel each other. After some laborious computation, the correct guess arises as

u(x, t) ≈ 2N^{−1/2}t^{−1/2} cos ((−4x^{3}/27t)^{1/2} + 6 log(t)N^{−1}(−x/3t)^{1/2}w^{2}(z))w(z).

If one inserts this ansatz into the mKdV equation, it solves mKdV up to an error which looks vaguely like

N^{−1/2}t^{−3/2} log^{C} t cos (3[(−4x^{3}/27t)^{1/2} + 6 log(t)N^{−1}(−x/3t)^{1/2}w^{2}(z)])a(z) for some bump function a, plus terms of lower order. This is not suitable for our purposes; for instance, the H^{1/4} norm only decays like t^{−1} log^{C} t, which is not integrable. It turns out that we have to add a correction term

4Alternatively, one can proceed by exploiting the well-known connection between KdV or mKdV and NLS via the long wave approximation; see [10].

N^{−7/2}t^{−3/2} cos (3[(−4x^{3}/27t)^{1/2} + 6 log(t)N^{−1}(−x/3t)^{1/2}w^{2}(z)])w̃(z)

to u to cancel the biggest term in the error (of course, at the cost of introducing many other terms of lower order). Once one adds this term to the ansatz, the error consists of terms which look like

N^{−1/2}t^{−5/2} log^{C} t cos (k[(−4x^{3}/27t)^{1/2} + 6 log(t)N^{−1}(−x/3t)^{1/2}w^{2}(z)])a_{k}(z)

for −9 ≤ k ≤ 9, plus terms of lower order. This expression has an H^{1/4} norm of t^{−2} log^{C} t, which is integrable, so we now have a fighting chance of using energy estimates to remove the error and get a genuine global solution.

There is however one problem: even though the H^{1/4} norm is suitably small, the H^{2} norm (for instance) of these functions are rather large, because the phase (−4x^{3}/27t)^{1/2} behaves like Nx (recall that x ∼ −3N^{2}t). As a caricature, the functions we are considering are of the form

cos(Nx + z)a(z)

times various factors of N and t which are not important for this discussion. The high oscillation in Nx defeats a naive attempt to use energy methods.

However, the well-known method of slow and fast variables can resolve this. If we define y ∈ T by y := Nx(mod1), we can rewrite the above function as a function

cos(y + z)a(z)

on the cylinder (y, z) ∈ T × R. This function has bounded energy in y and z. One can then lift the mKdV equation up onto this cylinder and construct solutions up on this level. Then at the very end one can descend back to the real line, which is embedded in the cylinder via the tight spiral x ↦ (Nx mod 1, z), to recover the global solution to mKdV. See [10] for details.

It is interesting that we need to add a small correction term at frequencies ±3N in order to make the ansatz at frequencies ±N sufficiently accurate, although from the point of view of the H^{1/4} norm (for instance) such a correction term is negligible. If one were to repeat the same analysis for KdV, one would also need to add correction terms at frequencies ±N and 0 (basically because the ansatz would be the Miura transform of our current ansatz).

A somewhat similar argument works on the circle; there is no decay any more as t → ∞, but by the same token the phase correction grows linearly t instead of logarithmically. This is easiest to explain in the case of the periodic NLS

−iu_{t} + u_{xx} = |u|^{2}u

where we have explicit solutions of the form

u(x, t) = a exp(iNx − iN^{2}t) exp(i|a|^{2}t)

(compare this with the free solution a exp(iNx − N^{2}t)). For mKdV, if one starts with initial data of the form aN^{−1/2} cos(Nx) (which is normalized at the endpoint regularity of H^{1/2}, and ignores all modes other than ±N, one would similarly expect a solution of the form

u(x, t) ≈ aN^{−1/2} cos(Nx + N^{3}t + 6|a|^{2}t).

In practice one would have to add a correction term at frequencies ±3N in order to make the error term in this ansatz small enough to make the ansatz rigorous.

6. Symplectic non-squeezing

The KdV equation can be written in Hamiltonian form. On the real line R there are some technical difficulties in making this rigorous, but on the circle one can do so using the natural phase space H_{0}^{−1/2}(T) of H^{−1/2} functions of mean zero. (Recall that the mean is preserved by the KdV flow).

Observe that on this space we can define a formal inverse ∂_{x}^{−1} : H_{0}^{−1/2}(T) → H_{0}^{1/2}(T) to the differential operator ∂_{x} via the Fourier transform by

{ \atop ∂_{x}^{−1}f}(k) := \frac{1}{ik} f(k).

The anti-symmetric form

{u, v} := ∫T u(x)∂_{x}^{−1}v(x) dx  (6)

thus turns H_{0}^{−1/2}(T) into an infinite-dimensional symplectic space. The KdV flow is then formally the Hamiltonian vector flow corresponding to the (densely defined) Hamiltonian

H(u) := ∫T −\frac{1}{2}u_{x}^{2} − u^{3}.  (7)

Indeed one easily verifies the formal identity

\{u_{t}, v\} = \frac{d}{dε}H(u + εv)|_{ε=0}

whenever v ∈ H_{0}^{−1/2}(T) and u solves KdV. In particular, the flow maps S_{KdV} (t) are formally symplectomorphisms on H_{0}^{−1/2}(T).

For any u_{∗} ∈ H_{0}^{−1/2}(T), r > 0, k_{0} ∈ Z^{∗}, and z ∈ C, we consider the infinitedimensional ball

B^{∞}(u_{∗}; r) := {u ∈ H_{0}^{−1/2}(T) : ∥u − u_{∗}∥_{H_{0}^{−1/2}} ≤ r}

and the infinite-dimensional cylinder

C_{k_{0}}^{∞}(z; r) := {u ∈ H_{0}^{−1/2}(T) : |k_{0}|^{−1/2}|û(k_{0}) − z| ≤ r}.

We have the infinite-dimensional symplectic non-squeezing theorem:

Theorem 6.1. [15] Let 0 < r < R, u_{∗} ∈ H_{0}^{−1/2}(T), k_{0} ∈ Z^{∗}, z ∈ C, and T > 0. Then

S(T)(B^{∞}(u_{∗}; R)) ⊈ C_{k_{0}}^{∞}(z; r).

In other words, there exists a global H_{0}^{−1/2}(T) solution u to KdV such that

∥u(0) − u_{∗}∥_{H_{0}^{−1/2}} ≤ R

and

|k_{0}|^{−1/2}|{ \atop u(T)}(k_{0}) − z| > r.

Informally, this Theorem asserts that the KdV flow cannot squash a large ball into a thin cylinder. Notice that the balls and cylinders can be arbitrarily far away from the origin, and the time T can also be arbitrary. Formally, this result follows from the finite-dimensional non-squeezing theorem of Gromov [17], [18], but the extension to the infinite-dimensional case is not completely trivial. Similar results have been obtained in [29], [8] for other infinite-dimensional flows (for instance the cubic non-linear Schrödinger flow on L^{2}(T) was treated in [8]), however the treatment in the KdV case is more delicate. To indicate this difficulty let us first attempt to repeat the argument in [8]. Let N ≥ 1 be an integer, and let P_{≤N} denote projection to frequencies |k| ≤ N. The space P_{≤N}H_{0}^{−1/2}(T) is thus a finitedimensional real vector space (specifically, it is 2N-dimensional) and can easily be checked to be a symplectic space under the form (6) restricted to this space.

The Hamiltonian H(u) is well-defined on the finite-dimensional symplectic space P_{≤N}H_{0}^{−1/2}(T), and the corresponding Hamiltonian flow on this space can easily be computed to be given by the equation

u_{t} + u_{xxx} = P_{≤N}(6uu_{x})  (8)

which can be viewed either as a PDE or as a system of 2N ODE. The associated flow maps S_{P_{≤N} KdV} (t) are then symplectomorphisms on the finite-dimensional space P_{≤N}H_{0}^{−1/2}(T). One can also define, for any u_{∗}^{(N)} ∈ P_{≤N}H_{0}^{−1/2}(T), r > 0, 0 < |k_{0}| ≤ N, and z ∈ C, the finite-dimensional ball

B^{N}(u_{∗}^{(N)}; r) := {u^{(N)} ∈ P_{≤N}H_{0}^{−1/2}(T) : ∥u^{(N)} − u_{∗}^{(N)}∥_{H_{0}^{−1/2}} ≤ r}

and the finite-dimensional cylinder

C_{k_{0}}^{N}(z; r) := {u^{(N)} ∈ P_{≤N}H_{0}^{−1/2}(T) : |k_{0}|^{−1/2}|û^{(N)}(k_{0}) − z| ≤ r}.

From the finite-dimensional theory of symplectic capacity [17], [18] we thus have the finite-dimensional analogue of Theorem 6.1:

Theorem 6.2. Let N ≥ 1, 0 < r < R, u_{∗}^{(N)} ∈ P_{≤N}H_{0}^{−1/2}(T), 0 < |k_{0}| ≤ N, z ∈ C, and T > 0. Then

S_{P_{≤N} KdV} (T)(B^{N}(u_{∗}^{(N)}; R)) ⊈ C_{k_{0}}^{N}(z; r).

In other words, there exists a global P_{≤N}H_{0}^{−1/2}(T) solution u = u^{(N)} to (8) such that

∥u^{(N)}(0) − u_{∗}^{(N)}∥_{H_{0}^{−1/2}} ≤ R

and

|k_{0}|^{−1/2}|û^{(N)}(T)(k_{0}) − z| > r.

To deduce Theorem 6.1 from Theorem 6.2, one would like to let N → ∞ and show that the flow S_{N}(T) converged to S(T) in some weak sense. More precisely, one would need

Conjecture 6.3. Let k_{0} ∈ Z^{∗}, T > 0, A > 0, 0 < ε ≪ 1. Then there exists an N_{0} = N_{0}(k_{0}, T, ε, A) > |k_{0}| such that

|k_{0}|^{−1/2}|{ \atop S_{KdV} (T)u_{0}}(k_{0}) − { \atop S_{P_{≤N} KdV} (T)u_{0}}(k_{0})| ≪ ε

for all N ≥ N_{0} and all u_{0} ∈ B^{N}(0, A).

If Conjecture 6.3 were true then it would be an easy matter to conclude Theorem 6.1. Indeed, let r, R, u_{∗}, k_{0}, z, T be as in that Theorem, and choose 0 < ε < (R − r)/2. The ball B^{∞}(u_{∗}; R) is contained in some ball B^{∞}(0; A) centered at the origin. We choose N ≥ N_{0}(k_{0}, T, ε, A) so large that ∥u_{∗} − P_{≤N}u_{∗}∥_{H_{0}^{−1/2}} ≤ ε. From

Theorem 6.2 we can find a solution u^{(N)} to (8) such that

∥u^{(N)}(0) − P_{≤N}u_{∗}∥_{H_{0}^{−1/2}} ≤ R − ε

and

|k_{0}|^{−1/2}|Ŝ(T)u^{(N)}(0)(k_{0}) − z| > r + ε.

If we then apply Conjecture 6.3 and the triangle inequality we obtain Theorem 6.1 with u_{0} := u_{0}^{(N)}.

The analogue of Conjecture 6.3 has been established for several Hamiltonian PDE, notably the cubic NLS flow on L^{2}(T) (see [6], [8]). Unfortunately, in [15] it is shown that Conjecture 6.3 fails. We can sketch the problem as follows. Take initial data u_{0} which is something like cos(x) + N^{1/2} cos(Nx); the Airy flow of this is cos(x + t) + N^{1/2} cos(Nx + N^{3}t). The KdV flow will then generate a secondary term which looks something like N^{−1/2} cos((N + 1)x + (N^{3} + 1)t), but this term is missing from the truncated flow S_{P_{≤N} KdV}. This secondary term ought to have created a tertiary term of the form cos(x), but this term will again be missing from S_{P ≤N}KdV. Since this term has large H^{−1/2} norm, one is now in trouble in the sense that Conjecture 6.3 fails.

Despite the failure of the above argument, it is still possible to conclude symplectic non-squeezing by using an alternative finite-dimensional Hamiltonian flow to approximate S_{KdV}.

Before doing so, though, one will have to go through a series of other results concerning the approximation of the low and high frequency evolutions of KdV and mKdV.; it turns out that at the endpoint regularity of H^{−1/2} (for KdV) or H^{1/2} (for mKdV) the evolution is surprisingly delicate. I’ll have to leave you in suspense as to how to do the finite dimensional approximation of KdV properly for the time being, and turn instead to mKdV.

7. The high and low evolutions of mKdV

To analyze the mKdV flow at H_{0}^{1/2} it is convenient to make a Gallilean transformation and instead study the renormalized mKdV flow

v_{t} + v_{xxx} = F (v); v(0, x) = v_{0}(x)  (9)

where the non-linearity F (v) is given by

F (v) := 6(v^{2} − P_{0}(v^{2}))v_{x}  (10)

and P_{0} is the mean operator

P_{0}u := \frac{1}{2π} ∫_{0}^{2π} u.

Since the L^{2} norm P_{0}(v^{2}) is conserved by both flows, it is easy to see that the two flows are linked by a Gallilean transformation (see e.g. [36] for a detailed discussion). More precisely, if we use \overline{S}_{mKdV} to denote the nonlinear flow associated with (9), then

\overline{S}_{mKdV} (t)v_{0}(x) = S_{mKdV} (t)v_{0}(x − 6P_{0}(v_{0}^{2})t).

Note that the flow \overline{S}_{mKdV} (t) is mean-preserving and is thus densely defined as a map from H_{0}^{s} to itself.

We now focus on the renormalized mKdV flow (9) in H_{0}^{1/2}, and especially on the evolution of the high frequency modes. We shall need to understand the evolution of these modes in order to truncate the high frequencies, and approximate the mKdV flow by a finite-dimensional low-frequency model.

A naive guess for the high-frequency evolution would be that it should resemble the linear Airy flow S_{Airy}, which has the explicit solution

{ \atop S_{Airy}(t)v_{0}}(k) = exp(ik^{3}t)v̂_{0}(k).  (11)

One might then conjecture that S_{Airy}(t) stays close to \overline{S}_{mKdV} at high frequencies. A precise version of this statement would be that

sup_{|t|≤T} ∥\overline{S}_{mKdV} (t)v_{0} − S_{Airy}(t)v_{0}∥_{H_{0}^{s+σ}} ≤ C(s, T, ∥v_{0}∥_{H_{0}^{s}})  (12)

for all t and v_{0} ∈ H_{0}^{s}, where s ≥ 1/2 and σ > 0 was an absolute constant depending only on s. This would mean that at high frequencies |k| ∼ N, for N ≫ 1, the error between the linear and non-linear flows would only be O(N^{−σ}) in the H_{0}^{s} topology.

Following [8], [11], it would seem that a natural approach to proving (12) would be to prove an extra smoothing estimate of the form

∥(uv − P_{0}(uv))w_{x}∥_{Z^{s+σ}} ≲ ∥u∥_{Y^{s}}∥v∥_{Y^{s}}∥w∥_{Y^{s}}.  (13)

Estimates of this form are available on the real line R (see [11]) but one can easily show^{5} that the estimate (13) fails for any s ∈ R and any σ > 0. Nevertheless, as we shall see, the approximation property (12) still holds for s > 1/2 but fails for s = 1/2.

To understand the mKdV flow better, we follow [6] and split the non-linearity F (v) defined in (10) into “resonant” and “non-resonant” components. In the following analysis we shall always assume that v has mean zero.

We start with the Fourier inversion formula

v(x) = ∑_{k∈Z^{∗}} v̂(k) exp(ikx)

for v ∈ H_{0}^{s}, where Z^{∗} := Z\{0} is the set of the non-zero integers. A direct computation gives that the Fourier transform of F (v) is

{ \atop F (v)}(k) = 6 ∑_{k_{1},k_{2},k_{3}∈Z^{∗}:k_{1}+k_{2}+k_{3}=k;k_{1}+k_{2}≠0} v̂(k_{1})v̂(k_{2})ik_{3}v̂(k_{3})  (14)

for all k ∈ Z^{∗}; the constraint k_{1} + k_{2} ≠ 0 arises since we have subtracted the mean P_{0}(v^{2}) from v^{2} in the definition of F (v). Observe that F (v) has mean zero and thus has no Fourier component at 0.

Lemma 7.1. We have

F (v) = F_{0}(v, v, v) + F_{≠0}(v, v, v)

where the “resonant” trilinear operator F_{0} is given by

{ \atop F_{0}(u, v, w)}(k) := 6ikû(k)v̂(−k)ŵ(k)

5This is a simple modification of the counterexamples in [25]. Informally, one lets u, v, w be concentrated near the points (k, τ) = (N, N^{3}), (1, 1), (1, 1) in spacetime frequency space respectively.

for k ∈ Z^{∗}, and the “non-resonant” trilinear operator F_{≠0} is defined by

{ \atop F_{≠0}(u, v, w)}(k) := ∑_{k_{1},k_{2},k_{3}∈Z^{∗}:k_{1}+k_{2}+k_{3}=k;(k_{1}+k_{2})(k_{1}+k_{3})(k_{2}+k_{3})≠0} 2i(k_{1} + k_{2} + k_{3})û(k_{1})v̂(k_{2})ŵ(k_{3}) (15)

for k ∈ Z^{∗}.

Proof Consider the right-hand side of (14). It is not possible for k_{1} + k_{2}, k_{1} + k_{3}, and k_{2} + k_{3} to all simultaneously be zero, since k_{1}, k_{2}, k_{3} are individually non-zero. If two of k_{1} + k_{2}, k_{2} + k_{3}, k_{1} + k_{3} are zero, then the summation is ±F_{0}(v, v, v), with the total contribution of the three cases being +F_{0}(v, v, v). If just k_{2} + k_{3} is zero, then the total contribution of this case vanishes since the summand is antisymmetric with respect to swapping k_{2} and k_{3}. Similarly if just k_{1} +k_{3} is zero. The remaining portion of the summation can be seen to be F_{≠0}(v, v, v) by a symmetrization in k_{1}, k_{2}, k_{3}.

If k = k_{1} + k_{2} + k_{3}, then we have the fundamental resonance identity

k^{3} − (k_{1}^{3} + k_{2}^{3} + k_{3}^{3}) = 3(k_{1} + k_{2})(k_{1} + k_{3})(k_{2} + k_{3})  (16)

(see e.g. [4]). This justifies the terminology that F_{0} is “resonant” but F_{≠0} is “non-resonant”.

Despite the extra smoothing estimate failing for the nonlinearity F, (a weaker form of^{6}) this estimate does hold for F_{≠0}. Because of this, one is eventually able to approximate \overline{S}_{mKdV} by the effective (renormalized) mKdV flow

v_{t} + v_{xxx} = F_{0}(v, v, v); v(0) = v_{0}  (17)

and denote the associated flow map by \overline{S}_{eff−mKdV}; in other words, we discard the non-resonant portions of (9) and retain only the resonant portion. On the Fourier side, we can rewrite (17) as

∂_{t}v̂(t, k) − ik^{3}v̂(t, k) = 6ik|v̂(t, k)|^{2}v̂(t, k); v̂(0, k) = v̂_{0}(k)

which has the explicit solution

(\overline{S}_{eff−mKdV} (t)v_{0}ˆ)(k) = exp(i(k^{3} + 6k|v̂_{0}(k)|^{2})t)v̂_{0}(k)  (18)

(compare with (11) and the construction of global solutions from previous sections). Note that \overline{S}_{eff−mKdV} (t) preserves the H_{0}^{s} norm (indeed, it preserves the magnitude of each individual Fourier coefficient), but it is not linear, and it is only continuous on H_{0}^{s} when s ≥ 1/2.

If v is in H_{0}^{s}, then v̂_{0}(k) = O(⟨k⟩^{−s}). Thus when s > 1/2, the phase correction 6k|v̂_{0}(k)|^{2} becomes small for large k, and one can easily verify that \overline{S}_{eff−mKdV} (t) is close to S_{Airy}(t) for large frequencies in the sense that

sup_{|t|≤T} ∥\overline{S}_{eff−mKdV} (t)v_{0} − S_{Airy}(t)v_{0}∥_{H_{0}^{s+2(s−1/2)}} ≤ C(s, T, ∥v_{0}∥_{H_{0}^{s}}). (19)

However at the endpoint s = 1/2 one can easily check that there is no extra smoothing property, and the two flows can be genuinely different at high frequencies.

6More precisely, one does not gain the regularity σ in the output, but instead one can put less derivatives on two of the three inputs. It is not entirely automatic that this weaker estimate suffices to prove the approximation property; in fact, at the endpoint s = 1/2 one has to modify the approximating Airy flow with an additional correction term (similar to the one used to construct the global solutions in previous sections). See [15].

By using some trilinear estimates of extra smoothing type to handle the nonresonant nonlinearity F_{≠0}, one can obtain

Theorem 7.2. [15] For any s ≥ 1/2, T > 0, and v_{0} ∈ H_{0}^{s}, we have

sup_{|t|≤T} ∥\overline{S}_{mKdV} (t)v_{0} − \overline{S}_{eff−mKdV} (t)v_{0}∥_{H_{0}^{s+σ}} ≤ C(s, T, ∥v_{0}∥_{H_{0}^{s}})  (20)

for some σ = σ(s) > 0.

As a consequence we see that the high frequencies of \overline{S}_{mKdV} only behave like the Airy flow for s > 1/2, and there is an additional phase correction at s = 1/2.

To summarize so far, the high frequencies evolve as if the (complicated) nonlinearity F (v) was replaced with the (simpler) nonlinearity F_{0}(v, v, v). From this and a little more work it is possible to create a smooth transition between F and F_{0}, and show that this smoothed out flow is a good approximation at both high and low frequencies:

Theorem 7.3. [15] Fix s ≥ 1/2, T > 0, and N ≫ 1. Let v_{0} ∈ H_{0}^{s} have Fourier transform supported in the range |k| ≤ N. Let b(k) be a bump function adapted to [−N, N] which equals one on [−N/2, N/2], and let B the associated Fourier multiplier. Let v be given by the finite dimensional evolution

v_{t} + v_{xxx} = BF (v) + (P_{≤N} − B)F_{0}(v, v, v); v(0) = v_{0}.  (21)

Then v(t) approximates the renormalized mKdV flow (9) in the sense that

sup_{|t|≤T} ∥v(t) − \overline{S}_{mKdV} (t)v_{0}∥_{H_{0}^{s}} ≤ N^{−σ}C(s, T, ∥v_{0}∥_{H_{0}^{s}})

for some σ = σ(s) > 0.

This gives us an effective finite-dimensional approximation to mKdV. Pushing this forward by the Miura transform it is possible to obtain a finite-dimensional approximation to KdV. For instance, let N and B be as in the previous Theorem, and consider the evolution

u_{t} + u_{xxx} = B(6uu_{x}); u(0) = u_{0}.  (22)

We will later show that this map is (after some simple transformations) essentially a symplectomorphism on the finite dimensional space P_{≤N}H_{0}^{−1/2}.

Let S_{B KdV} denote the flow map associated to (22). Then we have

Theorem 7.4. [15] Fix s ≥ −1/2, T > 0, and N ≫ 1. Let u_{0} ∈ H_{0}^{s} have Fourier transform supported in the range |k| ≤ N. Then

sup_{|t|≤T} ∥P_{≤N^{1/2}}(S_{B KdV} v(t) − S_{KdV} (t)u_{0})∥_{H_{0}^{s}} ≤ N^{−σ}C(s, T, ∥u_{0}∥_{H_{0}^{s}})

for some σ = σ(s) > 0.

Thus, if one truncates the KdV non-linearity smoothly at frequencies ∼ N, one does not really affect the flow at frequencies ≲ \sqrt{N}. This is in contrast to the sharp truncation in (8), which creates large low-frequency errors.

This theorem is surprisingly delicate to prove, especially at the endpoint s = −1/2, because there is very little slack in the bilinear estimates for KdV wellposedness at this endpoint. Instead, one must use a (modified) Miura transform to pull back to mKdV, and use the mKdV truncation theorems there.

Once one has this theorem, one can (finally!) approximate KdV by a finitedimensional Hamiltonian flow, and conclude the proof of symplectic non-squeezing.

We begin by considering the modified Hamiltonian H_{N} on P_{N}H_{0}^{−1/2}(T), defined by

H_{N}(u) := ∫T −\frac{1}{2}u_{x}^{2} − (Bu)^{3} dx.

We compute the Hamiltonian flow on P_{N}H_{0}^{−1/2} corresponding to H_{N}. Fix u, v ∈ H_{0}^{−1/2}. We see that

\frac{d}{dε}H_{N}(u + εv)|_{ε=0} = ∫T −u_{x}v_{x} − 3(Bu)^{2}Bv dx
={−u_{xxx} + 6B((Bu)(Bu_{x})), v}.

Since −u_{xxx} + 6B((Bu)(Bu_{x})) is in P_{N}H_{0}^{−1/2}, we thus see that the Hamiltonian flow of H_{N} on P_{N}H_{0}^{−1/2} is given by

u_{t} + u_{x}xx = 6B((Bu)(Bu_{x})); u(0) = u_{0} ∈ P_{N}H_{0}^{1/2}(T).  (23)

Making the change of variables ũ := Bu we see that this flow is just (22) in disguise (with the trivial change that B is replaced with B^{2}). From this and a little more work we obtain the desired approximation property.

8. Appendix: Inverting Miura on the circle

Let C_{0}^{∞}(T) be the smooth functions of mean zero on the circle. Consider the linearized Miura transform f → f′ on the circle. This maps C^{∞}(T) to C^{∞}(T), but the map is neither surjective nor injective because derivatives always have mean zero, and constants have vanishing derivative. However, by adding a constant and restricting the domain to C_{0}^{∞} we can recover a bijection:

Proposition 8.1. For any real V ∈ C^{∞}(T), there exists a unique potential F ∈ C_{0}^{∞}(T) and a unique real number c such that V = F′ + c.

Indeed, c is just the mean of V, and F is the primitive of V − c, normalized to have mean zero.

We now give the nonlinear analogue of this for the Miura transform V = F′+F^{2}.

Proposition 8.2. For any real V ∈ C^{∞}(T), there exists a unique potential F ∈ C_{0}^{∞}(T) and a unique real number c such that V = F′ + F^{2} + c.

For F and V small one can verify this by solving iteratively the integral equation F = D^{−1}(V − F^{2}), where D^{−1} is the inverse of the operator f ↦ f′, thought of as a map from C_{0}^{∞}(T) to C_{0}^{∞}(T). (In other words, the Fourier multiplier with symbol 1/(2ik) for k ≠ 0 and 0 for k = 0). However, to obtain existence and uniqueness for large V and F one needs a more “elliptic” approach. Proof Fix V. Consider the self-adjoint elliptic operator L := −\frac{d^{2}}{dx^{2}} + V on the circle. From standard theory this operator has discrete spectrum, with eigenvalues λ_{1} ≤ λ_{2} ≤ … tending to infinity. (To see this, observe that for large enough C, L + C is positive definite and invertible, and the inverse (L + C)^{−1} is compact self-adjoint, hence has discrete spectrum tending to 0). Let λ_{1} be the smallest eigenvalue of L; it can be defined variationally as

λ_{1} := inf{E(u) : u ∈ C^{∞}(T); ∫ u^{2} = 1}

where the energy E(u) is defined by

E(u) := ⟨Lu, u⟩ = ∫ u_{x}^{2} + V u dx.

We first observe that if V = F′ + F^{2} + c for some F ∈ C_{0}^{∞}(T), then we can factorize

L − c = (\frac{d}{dx} + F)(−\frac{d}{dx} + F)

and hence

E(u) = c ∫ u^{2} dx + ∫ (−u_{x} + F u)^{2} dx.

In particular we have λ_{1} ≥ c. Indeed we have λ_{1} = c, as can be seen by setting u := e^{G}, where G is the primitive of F chosen so that u is L^{2}-normalized.

This shows that c is uniquely determined by V (being equal to λ_{1} of L). It also shows that F is uniquely determined by V, since if we had two choices F, F̃ for F then by the above argument we see that a primitive of F must equal a primitive of F̃, a contradiction.

Now to prove existence. Of course we set c = λ_{1}; in fact by subtracting λ_{1} from V if necessary we may assume that c = λ_{1} = 0. Let u be a L^{2}-normalized zero eigenfunction for L; thus E(u) = 0, and E(v) ≥ 0 for all other potentials v (not necessarily L^{2}-normalized). Also we have u_{xx} = V u. Without loss of generality we may assume that u is positive somewhere. We in fact claim that u cannot change sign; for otherwise max(u, 0) would have equal or lower energy than u, and then by smoothing out max(u, 0) slightly at the zeroes we contradict the minimality of u. Also u cannot have a repeated zero anywhere since the local well-posedness of the second-order ODE u_{xx} = V u would then force u to be zero everywhere. Thus u is always positive. If we then set e^{G} := u and F := G′ (i.e. F is the logarithmic derivative F = u_{x}/u of u) then we see from the ODE u_{xx} = V u that F has the desired properties.

One can in fact show that the map V → (F, c) is a locally bilipschitz homeomorphism between H^{s} and H^{s+1} × R for any s ≥ −1/2 [15].

This proposition allows one to connect mKdV and KdV on the torus. Suppose that V is a real potential on the torus obeying the KdV equation

V_{t} = V_{xxx} + 6V V_{x}.

The quantity λ_{1} = λ_{1}(\frac{−d^{2}}{dx^{2}} + V (t)) is an invariant of the KdV flow, so we write c = λ_{1}. Setting V (t) = F_{x}(t) + F (t)^{2} + c as per the above proposition, we see that F evolves by the drifting mKdV equation

F_{t} = F_{xxx} + 6F^{2}F_{x} + 6cF_{x}.

If we make the Gallilean change of variables F (x, t) := F̃(x + 6ct, t) then we can erase the drift term:

F̃_{t} = F̃_{xxx} + 6F̃^{2}F̃_{x}.

note that all the F̃ functions have mean zero, which is consistent with the fact that the mean is preserved by the mKdV flow.

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Department of Mathematics, UCLA, Los Angeles, CA 90024

   E-mail address: tao@math.ucla.edu