THE KENIG-MERLE SCATTERING RESULT FOR THE ENERGY-CRITICAL FOCUSING NLS

TERENCE TAO

Abstract. We describe (in informal terms) how the recent result of Kenig and Merle [1] establishing scattering for radial solutions to the energy-critical focusing NLS for solutions with energy and Ḣ^{1} norm less than that of the ground state can be formulated in the framework of [2].

1. Energy trapping

Consider the focusing energy-critical NLS

i∂_{t}u + Δu = −|u|^{4}u  (1)

for u(0) ∈ Ḣ^{1}(R^{3}). This equation has a conserved energy

E(u) = E(u(t)) := ∫_{R^{3}} \frac{1}{2}|∇u(t, x)|^{2} − \frac{1}{6}|u(t, x)|^{6} dx.

We also have the non-conserved quantity

E_{0}(u(t)) := ∫_{R^{3}} |∇u(t, x)|^{2} dx.

The two quantities are of course related. We have the sharp Sobolev embedding

∥u(t)∥_{L^{6}(R^{3})} ≤ C_{0}∥∇u(t)∥_{L^{2}(R^{3})}  (2)

for some optimal constant 0 < C_{0} < ∞, which leads to the inequalities

f(E_{0}(u(t))) ≤ E(u(t)) ≤ \frac{1}{2}E_{0}(u(t))

where f(x) := \frac{1}{2}x − \frac{1}{6}C_{0}^{6}x^{3}. This function has a local maximum at x = C_{0}^{−3}. It turns out that the ground state

W (x) := (1 + \frac{|x|^{2}}{3} )^{−1/2}

attains this maximum, thus E_{0}(W) = C_{0}^{−3} and

E(W) = f(E_{0}(W)) = \frac{1}{3}C_{0}^{−3}.

W is also a nonnegative solution to the PDE

ΔW + W^{5} = 0.

Now suppose that the energy is less than that of the ground state:

E(u) < E(W).

Then we have

f(E_{0}(u(t))) < f(E_{0}(W))

which means that E_{0}(u(t)) ≠ E_{0}(W) for all time. In particular if E_{0}(u(0)) < E_{0}(W) at time zero, then by continuity E_{0}(u(t)) < E_{0}(W) for all time for which the solution exists. In fact, a refinement of the argument shows that there exists δ > 0 depending on E(u) such that

δ < E_{0}(u(t)) < E_{0}(W) − δ = C_{0}^{3} − δ  (3)

for all time t.

Next we study the virial quantity

V (t) := ∫_{R^{3}} 8|∇u(t, x)|^{2} − 8|u(t, x)|^{6} dx.

This quantity naturally arises from the (formal) virial identity

∂_{tt} ∫_{R^{3}} |x|^{2}|u(t, x)|^{2} dx = V (t).

From the sharp Sobolev embedding (2) we have

V (t) ≥ 8E_{0}(u(t)) − 8C_{0}^{6}E_{0}(u(t))^{3}

and in particular from (3) we have

V (t) ≥ c(δ) > 0

for some C(δ) > 0. Thus the quantity ∫_{R^{3}} |x|^{2}|u(t, x)|^{2} dx is (formally) strictly convex in the regime E(u) < E(W), E_{0}(u(0)) < E_{0}(W). This replusion effect suggests one has global existence and scattering in this case, and this is indeed the first main result of [1].

Henceforth we allow implicit constants to depend on E(u) and on δ.

Remark 1.1. If instead one is in the regime E(u) < E(W), E_{0}(u_{0}) > E_{0}(W) then we ahve E_{0}(u(t)) > E_{0}(W)+δ for some δ > 0 and all time t. Writing

V (t) = 48E(u) − 16E_{0}(u)

one then sees that V (t) < −c(δ) for some c(δ) > 0, thus ∫_{R^{3}} |x|^{2}|u(t, x)|^{2} is (formally) strictly concave. This suggests blowup in this case, and this is indeed the second main result of [1].

RADIAL FOCUSING H^{1}-CRITICAL NLS 3

2. Local virial identity

The above computations are not directly useful because for Ḣ^{1}(R^{3}) solutions the integral ∫_{R^{3}} |x|^{2}|u(t, x)|^{2} dx need not be convergent. However, We can localise this identity at |x| ≤ R to obtain

∂_{tt} ∫_{R^{3}} φ(|x|/R)|x|^{2}|u(t, x)|^{2} dx = ∫_{|x|≤R} 8|∇u(t, x)|^{2}−8|u(t, x)|^{6} dx+O(∫_{|x|∼R} |∇u(t, x)|^{2}+|u(t, x)|^{6})

where φ(x) is a cutoff function which equals one when |x| ≤ 1 and zero for |x| ≥ 2. Suppose we know that the Ḣ^{1} norm of u is uniformly bounded,

∫_{R^{3}} |∇u(t, x)|^{2} dx = O(1)  (4)

then by Sobolev

∫_{R^{3}} |∇u(t, x)|^{2} + |u(t, x)|^{6} dx = O(1)  (5)

and hence by Hölder

∫_{R^{3}} φ(|x|/R)|x|^{2}|u(t, x)|^{2} dx = O(R^{4})

and hence on any time interval I we have

∫ χ_{I}(t) ∫_{|x|≤R} 8|∇u(t, x)|^{2}−8|u(t, x)|^{6} dxdt = O(R^{4}/|I|)+O(∫_{I} ∫_{|x|∼R} |∇u(t, x)|^{2}+|u(t, x)|^{6} dxdt)

where χ_{I} is a smooth cutoff to I. Using (3) and suitable truncated forms of the sharp Sobolev inequality (2) (where we apply a smooth spatial cutoff to u) we obtain that

∫_{|x|≤R} 8|∇u(t, x)|^{2}−8|u(t, x)|^{6} dx ≥ c ∫_{|x|≤R} |∇u(t, x)|^{2}+O(∫_{|x|∼R} |∇u(t, x)|^{2}+|u(t, x)|^{6} dx)

for some constant c > 0 depending only on δ, and thus

∫ χ_{I}(t) ∫_{|x|≤R} |∇u(t, x)|^{2} dxdt = O(R^{4}/|I|)+O(∫_{I} ∫_{|x|∼R} |∇u(t, x)|^{2}+|u(t, x)|^{6} dxdt).

Now we replace R by 2^{j}R for j = 1, …, J and sum in J to obtain

J ∫ χ_{I}(t) ∫_{|x|≤R} |∇u(t, x)|^{2} dxdt = O(2^{4J}R^{4}/|I|)+O(∫_{I} ∑_{j=1}^{J} ∫_{|x|∼2^{j}R} |∇u(t, x)|^{2}+|u(t, x)|^{6} dxdt).

Applying (2) we conclude

J ∫ χ_{I}(t) ∫_{|x|≤R} |∇u(t, x)|^{2} dxdt = O(2^{4J}R^{4}/|I|) + O(|I|)

where C_{0} is a large constant to be chosen later. Setting J = 1/ε, R = ε^{1/4}2^{−1/ε}|I|^{1/2}, and applying the pigeonhole principle, we conclude Lemma 2.1 (Energy decay). Given any time interval I and ε > 0, there exists t ∈ I such that

∫_{|x|≤c(ε)|I|^{1/2}} |∇u(t, x)|^{2} dx ≲ ε.

where c(ε) > 0 only on ε.

This can be used to deduce an analogue of [2, Corollary 3.7] for the focusing equation assuming spherical symmetry, thus removing the need to deal with Morawetz inequalities (which do not appear to be directly useful when E_{0}(u) gets close to E(W)). One can then repeat the remainder of the arguments from [2] without much additional change to obtain the main result of [1]. Interestingly the bounds still appear to be exponential in E(W) − E(u). Also the arguments give a new proof of the main result of [2].

References

[1] C. Kenig, F. Merle, Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, preprint.

[2] T. Tao, Global well-posedness and scattering for the higher-dimensional energycritical non-linear Schrodinger equation for radial data, New York Journal of Mathematics 11 (2005), 57–80.

Department of Mathematics, UCLA, Los Angeles, CA 90024

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