MORAWETZ AND THE 3D QUINTIC NLS

J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO

Abstract. This might be the Morawetz estimate needed to complete Project gopher.

1. Review of Project Gopher

Consider an Ḣ^{1} solution to the quintic defocusing NLS

iu_{t} + Δu = |u|^{4}u  (1)

in R^{3}. Henceforth we assume that energy is bounded, but not small:

∥u∥_{L_{t}^{∞}Ḣ_{x}^{1}} + ∥u∥_{L_{t}^{∞}L_{x}^{6}} ≲ 1.  (2)

We wish to obtain scale-invariant global control on u (e.g. global L_{t,x}^{10} bounds) and thus obtain GWP and scattering for this equation in Ḣ^{1} (and above). The interaction Morawetz inequality gives us

∫_{0}^{T} ∫_{R_{3}} |u(t, x)|^{4} dxdt ≲ sup_{0≤t≤T} ∥u(t)∥_{L^{2}}^{2}∥u(t)∥2 ; Ḣ^{1/2}

but this is not good enough because both sides of this equation are supercritical, which means that this inequality is useless for preventing concentration.

In the previous notes on Project Gopher, I wished for a variant of the above estimate, namely

∫_{0}^{T} ∫_{R_{3}} |P_{>N}u(t, x)|^{4} dxdt ≲ N^{−3}  (3)

for any number N > 0 for which we have the low frequencies estimate

sup_{0≤t≤T} ∥P_{≤N/ε}u(t)∥Ḣ^{1} ≲ ε  (4)

for some small absolute constant 0 < ε ≪ 1 (depending only on the energy). Here we use P_{>N} to denote Littlewood-Paley projection to frequencies > N, etc. Thus the above estimate is an interaction Morawetz estimate on high frequencies under the assumption that low frequencies have small energy. Morally speaking, the reason we gain N^{−3} as N increases is that the L^{2} and Ḣ^{1/2} norm of u will decay at high frequencies thanks to the bounded energy assumption.

J.E.C. is supported in part by an N.S.F. Postdoctoral Research Fellowship.

M.K. is supported in part by N.S.F. Grant DMS 9801558.

G.S. is supported in part by N.S.F. Grant DMS 9800879 and by a grant from Hewlett and Packard and the Sloan Foundation.

H.T. is supported in part by J.S.P.S. Grant No. 13740087.

T.T. is a Clay Prize Fellow and is supported in part by grants from the Packard Foundation.

Morally speaking, the estimate (3) has the scaling of

∫_{0}^{T} ∫_{R_{3}} ||∇_{x}|^{3/4}u(t, x)|^{4} dxdt ≲ 1

which is a scale-invariant estimate. (It is also consistent with the Strichartz estimates for the linear equation). In the Project Gopher notes I indicate how Bourgain’s induction on energy argument will go through if we have this estimate (3). I still believe that argument is correct.

In these notes I believe I have found a way to prove (3) which is basically following the path we took in Minnesota. The main new ingredient here is that we are exploiting not only the positivity of the L^{4} term ∫_{0}^{T} ∫R^{3} |u(t, x)|^{4} dxdt, but also the non-linear positive term ∫_{0}^{T} ∫R^{3} ∫R^{3} |u(t, x)|^{2}|u(t, y)|^{6}/|x − y| dxdydt.

So now let’s prove (3). The first thing is that we may use scale invariance to take N = 1, so we’ll do so consistently throughout. Note that this breaks the scale invariance, so we now have the freedom to use non-scale-invariant norms. Fix the time interval [0, T], which we assume to be compact (the bounds will not depend on T, so we can remove this assumption eventually). The local well-posedness theory will allow us to take u to be smooth (again, the bounds will not depend on the smoothness of u).

We now write P_{hi} for P_{>1} and P_{lo} = 1 − P_{hi} for P_{≤1}, and decompose u = u_{hi} + u_{lo} where u_{hi} = P_{hi}u and u_{lo} = P_{lo}u. Let Q denote the quantity

Q := ∫_{0}^{T} ∫_{R_{3}} |u_{hi}|^{4} dxdt;  (5)

our objective is to prove that Q is bounded. We may thus assume that Q ≫ 1 since the claim is trivial otherwise.

The idea is to run the interaction Morawetz argument for u_{hi}. If it were not for the interaction with u_{lo} (i.e. if u_{hi} just solved the equation (i∂_{t} + Δ)u_{hi} = |u_{hi}|^{4}u_{hi}), then the standard interaction Morawetz argument will give

Q ≲ sup_{0≤t≤T} ∥u_{hi}(t)∥_{L^{2}}^{2}∥u_{hi}(t)∥2 ; Ḣ^{1/2}.

But the right-hand side is bounded because of the high-frequency localization and (2). So we would be done.

Of course, we pick up error terms because of the interaction between high and low frequencies. However, the strategy is to use the L^{4} bound (5) on the high frequencies and the small energy assumption on the low frequencies (4) in order to control those error terms by Cε^{1/2}Q. This gives us a bound of the form Q ≲ 1 + Cε^{1/2}Q and then we will be done.

That’s the general idea. The first obstacle is that we have too few bounds on u_{hi} and u_{lo} to realistically have a chance of estimating all the error terms; for instance, we do not have any global Strichartz bounds on [0, T] (other than the trivial energy bound (2)). So the first step is to use local Strichartz theory to bootstrap our meager collection (5), (2), (4) of bounds to a much wider class of estimates in which we control a number of Strichartz-type norms.

2. Control of Strichartz-type norms

In this section all spacetime norms are on the slab [0, T] × R^{3} unless otherwise indicated. For ease of reference we place here all the estimates we currently know on u_{lo} and u_{hi}, adding a few more that can be easily obtained from Sobolev or Bernstein and the frequency localization:

∥u_{lo}∥_{L_{t}^{∞}Ḣ_{x}^{1}} + ∥u_{lo}∥_{L_{t}^{∞}L_{x}^{6}} + ∥u_{lo}∥_{L_{t}^{∞}L_{x}^{∞}} ≲ ε  (6)
∥P_{<1/ε}u_{hi}∥_{L_{t}^{∞}H_{x}^{1}} + ∥P_{<1/ε}u_{hi}∥_{L_{t}^{∞}L_{x}^{6}} ≲ ε  (7)
∥u_{hi}∥_{L_{t}^{∞}H_{x}^{1}} + ∥u_{hi}∥_{L_{t}^{∞}L_{x}^{6}} ≲ 1  (8)
∥u_{hi}∥_{L_{t}^{4}L_{x}^{4}} ≲ Q^{1/4}.  (9)

From (7), (8) we see in particular that

∥u_{hi}∥_{L_{t}^{∞}L_{x}^{2}} ≲ ε^{1/2},  (10)

as can be seen by splitting into frequencies < 1/ε^{1/2} and > 1/ε^{1/2}.

In what follows it may help to think of Q as having the units of time; more precisely, every time an L_{t}^{q} estimate will appear on the left-hand side, a Q^{1/q} will inevitably appear on the right-hand side. The quantity Q has an interpretation as the number of time intervals for which the L_{t,x}^{4} norm is under control; this is not quite as good as L_{t,x}^{10} control as Bourgain uses, but as we shall see we can still get some reasonably satisfactory estimates using Q.

We can see from this rather sparse list of estimates that we do not have any time integrability control on u_{lo}, and the only time integrability estimate on u_{hi} is (9). This compares poorly with the linear equation, where we have a large range of Strichartz estimates available (and we can get integrability in time all the way up to L_{t}^{2}, not just L_{t}^{4}).

We will now address this problem. Let δ > 0 be a small absolute constant (depending only on the energy). Using (9), we subdivide the time interval [0, T] into ∼ Q/δ^{4} intervals I such that

∥u_{hi}∥_{L_{t}^{4}L_{x}^{4}(I×R^{3})} ≲ δ  (11)

on each interval. Note that we have no control on the length of these intervals, so we should not use any Hölder in time (we don’t want any constants depending on T).

Pick any one of these intervals I, let’s say it has initial time t_{I}. We run the local well-posedness theory for the low frequencies u_{lo} on this interval; for the next few paragraphs we restrict spacetime norms to I × R^{3}. The low frequencies obey the equation

(i∂_{t} + Δ)u_{lo} = P_{lo}(|u|^{4}u).

We apply endpoint Strichartz and (6) to obtain

∥∇u_{lo}∥_{L_{t}^{2}L_{x}^{6}} + ∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}} ≲ ε + ∥∇P_{lo}(|u|^{4}u)∥_{L_{t}^{2}L_{x}^{6/5}};  (12)

The L_{t}^{8}L_{x}^{12} can be obtained by Sobolev from the L_{t}^{8}L_{x}^{12/5} Strichartz estimate. We now split u = u_{hi} + u_{lo}. Ignoring irrelevant constants and conjugations, we essentially have

|u|^{4}u = u_{hi}^{5} + u_{hi}^{4}u_{lo} + u_{hi}^{3}u_{lo}^{2} + u_{hi}^{2}u_{lo}^{3} + u_{hi}u_{lo}^{4} + u_{lo}^{5}.  (13)

The first four terms are schematically of the form u_{hi}^{2}u^{3} can be estimated using (11) twice and three applications of (2) to give

∥u_{hi}^{2}u^{3}∥_{L_{t}^{2}L_{x}^{1}} ≲ δ^{2}.  (14)

Applying Bernstein we thus have

∥∇P_{lo}u_{hi}^{2}u^{3}∥_{L_{t}^{2}L_{x}^{6/5}} ≲ δ^{2}.

Let us skip the fifth term for the moment and consider the last term u_{lo}^{5}. Applying the derivative, we see that the contribution of this term to (12) is basically

∥P_{lo}(u_{lo}^{4}∇u_{lo})∥_{L_{t}^{2}L_{x}^{6/5}}.

We discard P_{lo} (which is pretty useless anyway since everything was low frequency already) and apply Hölder and (6) to bound this term by

ε^{4}∥∇u_{lo}∥_{L_{t}^{2}L_{x}^{6}}

which can then be absorbed into the left-hand side of (12). Finally we consider the fifth term u_{hi}u_{lo}^{4}. We again apply the derivative. We pick up some terms like

∥P_{lo}(u_{hi}u_{lo}^{3}∇u_{lo})∥_{L_{t}^{2}L_{x}^{6/5}}.

which can be treated as the preceding term (we only gain ε^{3} instead of ε^{4}, but this is irrelevant). There is also a term of the form

∥P_{lo}(∇u_{hi}u_{lo}^{4})∥_{L_{t}^{2}L_{x}^{6/5}}

but if we once again discard P_{lo} and use (8) and Hölder, we can bound this by ∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{4}.

Putting all of this together, we see that

∥∇P_{lo}(|u|^{4}u)∥_{L_{t}^{2}L_{x}^{6/5}} ≲ δ^{2} + ∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{4}

and hence

∥∇u_{lo}∥_{L_{t}^{2}L_{x}^{6}} + ∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}} ≲ ε + δ^{2} + ∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{4}.

Thus if we choose δ to be sufficiently small (depending only on the energy), and assume ε is much smaller than δ, then we obtain (by the usual continuity arguments) the bound

∥∇u_{lo}∥_{L_{t}^{2}L_{x}^{6}} + ∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}} ≲ δ^{2}.

and

∥∇P_{lo}(|u|^{4}u)∥_{L_{t}^{2}L_{x}^{6/5}} ≲ δ^{2}.

Now let us go back to the global interval [0, T]. Adding up all the bounds just obtained, we have the spacetime bounds

∥∇P_{lo}(|u|^{4}u)∥_{L_{t}^{2}L_{x}^{6/5}} ≲ Q^{1/2}.  (15)

and

∥∇u_{lo}∥_{L_{t}^{2}L_{x}^{6}} ≲ Q^{1/2}.  (16)

Interpolating this with (6) we obtain

∥∇u_{lo}∥_{L_{t}^{8}L_{x}^{12/5}} ≲ ε^{3/4}Q^{1/8}

and hence by Sobolev

∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}} ≲ ε^{3/4}Q^{1/8}.  (17)

Now we go back to u_{hi} and see if there are any spacetime estimates available on [0, T] other than what can be obtained from (9) and interpolation. There are indeed such estimates, but they will cost us quite a bit of regularity. Starting with the evolution equation

(i∂_{t} + Δ)u_{hi} = P_{hi}(|u|^{4}u)

and using endpoint Strichartz and (8), we obtain

∥|∇|^{−1}u_{hi}∥_{L_{t}^{2}L_{x}^{6}} ≲ 1 + ∥|∇|^{−1}P_{hi}(|u|^{4}u)∥_{L_{t}^{2}L_{x}^{6/5}}.

As before, we expand the non-linearity using (13). The first four terms are of the form u_{hi}^{2}u^{3} and by (2), (9) we have

∥u_{hi}^{2}u^{3}∥_{L_{t}^{2}L_{x}^{1}} ≲ Q^{1/2}

so by Bernstein we have

∥|∇|^{−1}P_{hi}(|u|^{4}u)∥_{L_{t}^{2}L_{x}^{6/5}} ≲ Q^{1/2}.

Now consider the fifth term u_{hi}u_{lo}^{4}. We discard the bounded |∇|^{−1}P_{hi} multiplier, and use Hölder to bound

∥|∇|^{−1}P_{hi}(u_{hi}u_{lo}^{4})∥_{L_{t}^{2}L_{x}^{6/5}} ≲ ∥u_{hi}∥_{L_{t}^{∞}L_{x}^{2}}∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{4}.

But this is O(ε^{3}Q) by (17), (8). Finally, we consider u_{lo}^{5}. Multiplying and dividing by a derivative, we can estimate this contribution by

∥|∇|^{−2}P_{hi}(u_{lo}^{4}∇u_{lo})∥_{L_{t}^{2}L_{x}^{6/5}}.

We discard the bounded multiplier |∇|^{−2}P_{hi} and use (16), (6) to bound this by O(ε^{4}Q^{1/2}). Putting this all together, we see that we have obtained the bound

∥|∇|^{−1}u_{hi}∥_{L_{t}^{2}L_{x}^{6}} ≲ Q^{1/2}.  (18)

This is a full two derivatives worse than what we really expect (compare with (16), for instance), but it is still a useful estimate. For instance, we may interpolate this with (8) to obtain

∥u_{hi}∥_{L_{t}^{4}L_{x}^{3}} ≲ Q^{1/2},  (19)

which is a variant of (9). (From a scaling point of view, it is a worse estimate than (9), however it will be convenient to be able to have as much flexibility in the space integrability exponents as possible).

The estimates (15), (16), (17), (18), (19) greatly supplement the list of estimates already available in (6)-(10), and will give us enough room to handle all the error terms which arise.

3. Running the interaction Morawetz argument

Suppose that u is any solution to an inhomogeneous Schrödinger equation

iu_{t} + Δu = F

on R^{3}. Then the interaction Morawetz argument will eventually give us the bound

c ∫_{0}^{T} ∫_{R_{3}} |u|^{4} dxdt + C ∫_{0}^{T} ∫_{R_{3}} ∫R^{3} |u(t, y)|^{2} \frac{x − y}{|x − y|} · \{F, u\}(t, x) dxdydt ≲ sup_{0≤t≤T} ∥u(t)∥_{L^{2}}^{2}∥u(t)∥2 ; Ḣ^{1/2}, (20)

where c, C are positive constants, and {F, u} denotes the Poisson bracket-like quantity

\{F, u\} := Re(F \overline{∇u}(t, x) − ∇F \overline{u}).

When F = 0 this is just our standard Morawetz inequality; it is then just a matter of seeing how the forcing term F affects the Morawetz potentials M_{y}(t).

For instance, if F = |u|^{4}u, then a computation shows that

\{|u|^{4}u, u\} = −\frac{2}{3}∇|u|^{6}  (21)

and so after integrating by parts, the second term in (20) becomes a positive multiple of

∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u(t, y)|^{2}|u(t, x)|^{6}}{|x − y|} dxdydt.

Now we run the same argument with the high frequencies u_{hi}. Writing F (u) := |u|^{4}u, we see that u_{hi} obeys the equation

i∂_{t}u_{hi} + Δu_{hi} = P_{hi}F (u)

and hence (20) becomes

Q + C ∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} |u_{hi}(t, y)|^{2} \frac{x − y}{|x − y|} · \{P_{hi}F (u), u_{hi}\}(t, x) dxdydt ≲ 1 (22)

where we have estimated the right-hand side using (8). This is almost what we want - a bound on Q - but we must deal with the pesky integrand on the left-hand side of (22).

From (21) we have

\{F (u), u\} = −\frac{2}{3}∇|u|^{6}  (23)

and

\{F (u_{lo}), u_{lo}\} = −\frac{2}{3}∇|u_{lo}|^{6}.  (24)

To exploit these facts, we split

{P_{hi}F (u), u_{hi}} ={F (u), u} − {F (u_{lo}), u_{lo}}  (25)
− {P_{lo}F (u), u_{hi}}  (26)
− {F (u) − F (u_{lo}), u_{lo}}.  (27)

Contribution of (25).

Consider first the contribution of (25). After integrating by parts using (23), (24), we see that these terms give us a contribution of

C ∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, y)|^{2}(|u(t, x)|^{6} − |u_{lo}(t, x)|^{6})}{|x − y|} dxdydt.

Now observe the elementary pointwise bound

|u(t, x)|^{6}−|u_{lo}(t, x)|^{6} ≥ c|u_{hi}(t, x)|^{6}−O(|u_{hi}(t, x)|^{2}|u_{lo}(t, x)|^{4})−( terms like u_{hi}u_{lo}^{5}).

(We cannot put absolute values inside the last term yet because we will integrate by parts shortly). MK? About the other terms, (u_{lo}^{3}u_{hi}^{3}, u_{lo}^{1}u_{hi}^{5}) - my guess is that these terms have fewer u_{lo} terms hanging around so they can be handled like the second term on the right here. Is this what you’re saying, Terry? Thus we have a useful positive term

C ∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, x)|^{2}|u_{hi}(t, y)|^{6}}{|x − y|} dxdydt.  (28)

which we will save for future use, plus two errors, one of the form

O(∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, y)|^{2}|u_{hi}(t, x)|^{2}|u_{lo}(t, x)|^{4}}{|x − y|} dxdydt)  (29)

and one which schematically has the form

O(| ∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, y)|^{2}u_{hi}(t, x)u_{lo}(t, x)^{5}}{|x − y|} dxdydt|).  (30)

Let us estimate (29). Noting that convolving with 1/|x| is essentially the same as fractional integration |∇|^{−2}, we rewrite this expression as

O(∫_{0}^{T} ∫_{R_{3}}(|∇|^{−2}|u_{hi}|^{2})|u_{hi}|^{2}|u_{lo}|^{4} dxdt).

We split this using Hölder as

O(∥|∇|^{−2}|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{12}}∥u_{hi}∥_{L_{t}^{4}L_{x}^{4}}^{2}∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{2}∥u_{lo}∥_{L_{t}^{∞}L_{x}^{8}}^{2}).

From (6) and Bernstein we have ∥u_{lo}∥_{L_{t}^{∞}L_{x}^{8}}^{2} = O(ε^{2}). Also from fractional integration, Hölder, (9), (10) we have

∥|∇|^{−2}|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{12}} ≲ ∥|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{4/3}} ≲ ∥u_{hi}∥_{L_{t}^{4}L_{x}^{4}}∥u_{hi}∥_{L_{t}^{∞}L_{x}^{2}} ≲ ε^{1/2}Q^{1/4}. (31)

Applying (9), (17) we thus see that^{1}

(29) = O(ε^{5/2}Q).

Now we estimate (30). The above approach will not quite work because there are too many factors u_{lo} and they do not decay very fast in space. To rectify this we shall integrate by parts, taking derivatives away from a high frequency factor and placing them on a low frequency. We can write u_{hi}(t, x) = ∇ · ∇^{−1}u_{hi}(t, x), where ∇^{−1} := Δ^{−1}∇, and so integrating by parts we can estimate (30) by a linear combination of

O(∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, y)|^{2}|∇^{−1}u_{hi}(t, x)||∇(u_{lo}(t, x)^{5})|}{|x − y|} dxdydt)  (32)

and

O(∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, y)|^{2}|∇^{−1}u_{hi}(t, x)||u_{lo}(t, x)|^{5}}{|x − y|^{2}} dxdydt).  (33)

Consider first (32). Using fractional integrals as before, and expanding the derivative on u_{lo}, we can write this schematically as

O(| ∫_{0}^{T} ∫_{R_{3}}(|∇|^{−2}|u_{hi}|^{2})|∇^{−1}u_{hi}||u_{lo}^{4}∇u_{lo}| dxdt).

Notice how the negative derivatives are falling on high frequencies and the positive derivatives on low frequencies; this is very advantageous. We estimate this using Hölder by MK? I started fiddling with the text here.

O(∥|∇|^{−2}|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{12}}∥∇^{−1}u_{hi}∥_{L_{t}^{4}L_{x}^{12}}∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{4}∥∇u_{lo}∥_{L_{t}^{∞}L_{x}^{2}}).

We bound the factor ∥∇^{−1}u_{hi}∥_{L_{t}^{∞}L_{x}^{6}} using Sobolev, the high-frequency localization, and (9),

∥∇^{−1}u_{hi}∥_{L_{t}^{4}L_{x}^{12}} ≲ ∥∇^{− \frac{1}{2}} u_{hi}∥_{L_{t}^{4}L_{x}^{4}}
≲ ∥u_{hi}∥_{L_{t}^{4}L_{x}^{4}}
≲ Q^{\frac{1}{4}} ,  (34)

while from (6), we have ∥∇u_{lo}∥_{L_{t}^{∞}L_{x}^{2}} ≲ ε. Applying (17), (31) we thus see that

(32) = O(ε^{9/2}Q).

MK? Stopped fiddling here. Now consider (33). Again we use fractional integrals (noting that the kernel 1/|x|^{2} corresponds to |∇|^{−1} to write this as

O(∫_{0}^{T} ∫_{R_{3}}(|∇|^{−1}|u_{hi}(t, y)|^{2})|∇^{−1}u_{hi}||u_{lo}|^{5} dxdt.)

we again use Hölder to split this as

O(∥|∇|^{−1}|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{12/5}}∥∇^{−1}u_{hi}∥_{L_{t}^{4}L_{x}^{12}}∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{4}∥u_{lo}∥_{L_{t}^{∞}L_{x}^{6}}).

But by the obvious modification of (31) we see that

∥|∇|^{−1}|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{12/5}} = O(ε^{1/2}Q^{1/4})  (35)

1The reader is advised to ignore the powers of ε which appear here; any positive power of ε will be sufficient.

, while from (9), Sobolev, and the high-frequency localization we have ∥∇^{−1}u_{hi}∥_{L_{t}^{4}L_{x}^{12}} = O(Q^{1/4}). From (17) and (6) we thus see that

(33) = O(ε^{7/2}Q).

MK? I get ε^{\frac{5}{2}} here...though I know it doesn’t matter! To summarize, we see that the contribution of (25) to (22) is bounded from below by

c(28) + O(ε^{1/2}Q).

Contribution of (26).

It is of course natural to try to move the derivative over from the high frequency term to the low frequency term, so we write

\{P_{lo}F (u), u_{hi}\} = ∇Re[(P_{lo}F (u))\overline{u_{hi}}] − 2Re[(∇P_{lo}F (u))\overline{u_{hi}}].

The first term can be integrated by parts as usual. The second term is not yet in a good enough form to be estimated, and we have to move one more derivative. Writing u_{hi} = ∇ · ∇^{−1}u_{hi} as before, we can decompose

(∇P_{lo}F (u))\overline{u_{hi}} = ∇ · ((∇P_{lo}F (u))\overline{∇^{−1}u_{hi}}) − (∇^{2}P_{lo}F (u))\overline{∇^{−1}u_{hi}}.

MK? There were some ‘hi’ subscripts missing on two u’s here and I added them. Thus, we can estimate the contribution of (26) by the three terms

O(∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, y)|^{2}|∇P_{lo}F (u)(t, x)||∇^{−1}u_{hi}(t, x)|}{|x − y|} dxdydt), (36)

O(∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, y)|^{2}P_{lo}F (u)(t, x)u_{hi}(t, x)}{|x − y|} dxdydt),  (37)

and

O(∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} |u_{hi}(t, y)|^{2}|∇^{2}P_{lo}F (u)(t, x)||∇^{−1}u_{hi}(t, x)| dxdydt). (38)

Let us first consider (36). As usual, we write the kernel 1/|x − y| as a fractional integral:

O(∫_{0}^{T} ∫_{R_{3}}(|∇|^{−2}|u_{hi}|^{2})|∇P_{lo}F (u)||∇^{−1}u_{hi}| dxdt).

We estimate this by Hölder as

O(∥|∇|^{−2}|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{12}}∥∇P_{lo}F (u)∥_{L_{t}^{4}L_{x}^{4/3}}∥∇^{−1}u_{hi}∥_{L_{t}^{2}L_{x}^{6}}).

By (31), (18) we can bound this by

O(ε^{1/2}Q^{3/4}∥∇P_{lo}F (u)∥_{L_{t}^{4}L_{x}^{4/3}}).

Now we expand out ∇P_{lo}F (u). First consider any term which contains a u_{hi}, thus u_{hi}u^{4}. Using (9), (2) we have

∥u_{hi}u^{4}∥_{L_{t}^{4}L_{x}^{12/11}} ≲ Q^{1/4}  (39)

and hence by Bernstein

∥∇P_{lo}u_{hi}u^{4}∥_{L_{t}^{4}L_{x}^{4/3}} ≲ Q^{1/4}.

It remains to consider the pure low-frequency term u_{lo}^{5}. For this term we distribute the derivative and use Bernstein to estimate

∥∇P_{lo}u_{lo}^{5}∥_{L_{t}^{4}L_{x}^{4/3}} ≲ ∥u_{lo}^{4}∇u_{lo}∥_{L_{t}^{4}L_{x}^{1}} ≲ ∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{2}∥u_{lo}∥_{L_{t}^{∞}L_{x}^{6}}^{2}∥∇u_{lo}∥_{L_{t}^{∞}L_{x}^{2}}

which is equal to O(ε^{9/2}Q^{1/4}) by (17), (6). Thus we have bounded

(36) = O(ε^{1/2}Q).

MK? Another way to do the above is to use the estimate you got back in (15). That is, go back up to where it’s written ‘As usual, we write the kernel...’, and estimate now using Hölder:

O(∥|∇|^{−2}|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{12}}∥∇P_{lo}F (u)∥_{L_{t}^{2}L_{x}^{6/5}}∥∇^{−1}u_{hi}∥_{L_{t}^{4}L_{x}^{12}}).

By (15), (31), (18) we can bound this by

O(ε^{1/2}Q^{3/4}∥∇^{−1}u_{hi}∥_{L_{t}^{4}L_{x}^{12}}).

But the remaining norm here is handled exactly as before by Sobolev and (9) and is bounded by Q^{\frac{1}{4}} .

Now we consider (37). We again integrate by parts, writing u_{hi} = ∇·∇^{−1}u_{hi}. This allows us to rewrite (37) as a term looking like (36) (which has just been estimated), plus a term of the form

O(∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, y)|^{2}|P_{lo}F (u)(t, x)||∇^{−1}u_{hi}(t, x)|}{|x − y|^{2}} dxdydt).  (40)

We rewrite the 1/|x − y|^{2} kernel as a fractional integral, obtaining

O(∫_{0}^{T} ∫_{R_{3}}(|∇|^{−1}|u_{hi}|^{2})|P_{lo}F (u)||∇^{−1}u_{hi}| dxdt).

We use Hölder again to estimate this as

O(∥|∇|^{−1}|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{12/5}}∥P_{lo}F (u)∥_{L_{t}^{4}L_{x}^{12/5}}∥∇^{−1}u_{hi}∥_{L_{t}^{2}L_{x}^{6}}).

Using (35), (18) we can bound this by

O(ε^{1/2}Q^{3/4}∥P_{lo}F (u)∥_{L_{t}^{4}L_{x}^{12/5}})

and once again we must expand F (u). Again, we first consider the u_{hi}u^{4} terms. By Bernstein and (39) we have

∥P_{lo}(u_{hi}u^{4})∥_{L_{t}^{4}L_{x}^{12/5}} ≲ ∥u_{hi}u^{4}∥_{L_{t}^{4}L_{x}^{12/11}} ≲ Q^{1/4}.

Meanwhile, the u_{lo}^{5} term can be estimated by Bernstein and (17), (6):

∥P_{lo}(u_{lo}^{5})∥_{L_{t}^{4}L_{x}^{12/5}} ≲ ∥u_{lo}^{5}∥_{L_{t}^{4}L_{x}^{12/7}} ≲ ∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{2}∥u_{lo}∥_{L_{t}^{∞}L_{x}^{6}}^{3} ≲ ε^{9/2}Q^{1/4}.

Combining all these estimates we see that

(36) = O(ε^{1/2}Q).

MK? I’m being a pain, I know - but the analogue of the above without expanding F (u) would be, starting with the Holder:

O(∥|∇|^{−1}|u_{hi}|^{2}∥_{L_{t}^{4}L_{x}^{12/5}}∥P_{lo}F (u)∥_{L_{t}^{2}L_{x}^{2}}∥∇^{−1}u_{hi}∥_{L_{t}^{4}L_{x}^{12}}

MORAWETZ 11

In equation (35), we bounded the first factor by ε^{1/2}Q^{1/4}. The second factor is bounded by Q^{\frac{1}{2}} using Sobolev and (15). The third factor is bounded by Q^{1/4} by Sobolev as in (34) above. Finally, we deal with (38). The y integral can be bounded by (10), leaving us to estimate

O(ε^{1/2} ∫_{0}^{T} ∫_{R_{3}} |∇^{2}P_{lo}F (u)||∇^{−1}u_{hi}| dxdt).

By (18) and (15), this is O(ε^{1/2}Q). To summarize we have shown that the total contribution of (26) is O(ε^{1/2}Q). Contribution of (27). Now we deal with (27). Following our strategy for controlling (??), we shall aim to move the derivatives onto u_{lo}. Accordingly, we write

\{F (u) − F (u_{lo}), u_{lo}\} = 2Re[(F (u) − F (u_{lo}))\overline{∇u_{lo}}] − 2∇Re[(F (u) − F (u_{lo}))\overline{u_{lo}}]. The second term is a total derivative and so we can again integrate by parts. The contribution to (22) of these terms can thus be bounded by the sum of

O(∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} |u_{hi}(t, y)|^{2}|F (u)(t, x) − F (u_{lo})(t, x)||∇u_{lo}(t, x)| dxdy (41)

and what is schematically

O(| ∫_{0}^{T} ∫_{R_{3}} ∫_{R_{3}} \frac{|u_{hi}(t, y)|^{2}(F (u)(t, x) − F (u_{lo})(t, x))u_{lo}(t, x)}{|x − y|} dxdydt|). (42)

Let us first consider (41). The y integral can be estimated by (10), leaving us with

O(ε^{1/2} ∫_{0}^{T} ∫R^{3} |F (u) − F (u_{lo})||∇u_{lo}| dxdt.

We observe the pointwise estimate

|F (u) − F (u_{lo})| ≲ |u_{hi}|^{5} + |u_{hi}||u_{lo}|^{4}.

Consider the contribution of |u_{hi}|^{5}. By Hölder we can bound this contribution by

O(ε^{1/2}∥u_{hi}∥_{L_{t}^{4}L_{x}^{4}}^{2}∥u_{hi}∥_{L_{t}^{∞}L_{x}^{6}}^{3}∥∇u_{lo}∥_{L_{t}^{2}L_{x}^{∞}}).

The last factor is O(Q^{1/2}) by (16) and Bernstein, so by (9), (8) this contribution is O(ε^{1/2}Q). Now consider the contribution of |u_{hi}||u_{lo}|^{4}. For this term the Hölder splits as

O(ε^{1/2}∥u_{hi}∥_{L_{t}^{4}L_{x}^{3}}∥u_{lo}∥_{L_{t}^{8}L_{x}^{12}}^{2}∥u_{lo}∥_{L_{t}^{∞}L_{x}^{6}}^{2}∥∇u_{lo}∥_{L_{t}^{2}L_{x}^{6}}.

But by (19), (17), (6), (16) we can bound this by O(ε^{1/2}Q). Thus we have established

(41) = O(ε^{1/2}Q).

12 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO

Finally, we turn to (42). Here we split schematically

(F (u) − F (u_{lo}))u_{lo} = u_{hi}^{5}u_{lo} + u_{hi}^{4}u_{lo}^{2} + u_{hi}^{3}u_{lo}^{3} + u_{hi}^{2}u_{lo}^{4} + u_{hi}u_{lo}^{5}.

The very last term u_{hi}u_{lo}^{5} gives rise to an expression of the form (30), which has already been shown to be O(ε^{1/2}Q). The other three terms can be bounded pointwise by O(σ|u_{hi}|^{6}) + O(C_{σ}|u_{hi}|^{2}|u_{lo}|^{4}) for any 0 < σ ≪ 1. If σ is chosen small enough, we can absorb the O(σ|u_{hi}|^{6}) term under (28); meanwhile, the O(C_{σ}|u_{hi}|^{2}|u_{lo}|^{4}) term is of the form (29), which has already been shown to be O(ε^{1/2}Q). Putting all this together we have shown that the contribution of (27) is

O(σ(28)) + O(C_{σ}ε^{1/2}Q),

and hence (on choosing σ appropriately) we can deduce from (22) that

Q + O(ε^{1/2}Q) ≲ 1

and hence Q ≲ 1 as desired. This proves the Morawetz estimate.

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