LECTURE NOTES 4 FOR 254B

TERENCE TAO

1. Non-linear dispersive equations

A major portion of modern PDE theory is concerned with evolution equations - partial differential equations which involve a time variable. There are three classic linear equations of this type: the heat equation

u_{t} − Δu = 0,

the Schrödinger equation^{1}

iu_{t} − Δu = 0,

and the wave equation

u_{tt} − Δu = 0.

There are also other important evolution equations such as the ones related to fluids and conservation laws (Burger, Euler, Navier-Stokes, etc), but we will not consider them here. (They are much more non-linear, and cannot really be studied by the methods I will be discussing).

For evolution equations, the natural problem to study is the Cauchy problem (as opposed to, say, the Dirichlet problem): specify initial data u(0) = f, and try to solve for u at a later time t. (For the wave equation, which is second order in time, one also needs to specify initial velocity u_{t}(0) = g). There are of course many questions one can ask about such equations, but perhaps the three most basic ones are:

• Existence. Does a solution u(t) exist at all? In what sense is it a solution? Does it exist for all time, or just for a finite interval?

• Uniqueness. Can there be more than one solution with the same initial data? Are there some extra conditions we need to impose (e.g. some sort of regularity condition) to force uniqueness? If there are several solutions, is there one ”good” solution which is somehow better than the others?

1For this equation, of course, u must be complex valued.

• Stability. Suppose we perturb the initial data slightly. How does this affect the solution? Specifically: does the solution depend continuously on the data (as measured in some Banach space norm, for instance)?

If an equation satisfies all three of these properties, we say that the equation is well-posed. (One can of course qualify well-posedness as being local or global in time, or being subject to some regularity condition, etc.)

For linear equations these questions are fairly easy to answer, but they become more subtle for non-linear equations.

Despite the apparent similarity of the above three equations, there is an important difference between the heat equation and the other two equations. The heat equation is dissipative - this means that any irregularities or singularities in the initial data will quickly attenuate and disappear as time goes by. Related to this is the fact that this equation is not time reversible. The wave and Schrödinger equations, on the other hand, are dispersive - irregularities do not go away at all, but instead they propogate around in space, with different frequency irregularities moving in different directions or at different speeds. As such, they do not get smoother as time goes by, but they do tend to spread out and decay.

(A slightly more formal definition of dispersiveness: if the initial data is a plane wave with frequency ξ, then the solution will travel with group velocity v(ξ), where v(ξ) is not stationary with respect to ξ.)

For this discussion we shall just focus on variants of the Cauchy problem for the free linear Schrödinger equation

iu_{t} − Δu = 0
u(0) = f

in two spatial dimensions, so that u(t, x) is a function on R × R^{2}.

For this equation we have an exact solution

u(t, x) = e^{−itΔ}f(x) = \frac{1}{4πt} ∫ e^{−i|x−y|^{2}/4t}f(y) dy,

valid for all t ≠ 0.

As one can see, for fixed f in L^{1}, this solution decays in time. Indeed, we have

∥e^{−itΔ}f∥_{∞} ≤ \frac{1}{4πt}∥f∥_{1}. (1)

On the other hand, we have the Fourier identity

{ \atop e^{−itΔ}f}(ξ) = e^{4π^{2}it|ξ|^{2}} f̂(ξ),

which comes from the standard Fourier fact

Δ̂f(ξ) = −4π^{2}|ξ|^{2}f̂(ξ).

This implies that { \atop e^{−itΔ}f} and f̂ have the same L^{2} norm. By Plancherel, we thus have

∥e^{−itΔ}f∥_{2} = ∥f∥_{2}. (2)

To physicists, this identity represents the conservation of probability. So even though the solution decays, it does not alter its L^{2} norm. This is reflecting the dispersive rather than dissipative nature of the equation.

Very roughly speaking, it is the Δ term in the Schrödinger equation which is causing the dispersion. Without this term, the equation becomes u_{t} = 0, which obviously has no dispersion.

Let us now consider variants of the free Schrödinger equation. Some obvious choices for perturbation include placing the equation on a manifold on R^{n}; adding an obstacle (and providing some sort of boundary condition); adding a potential; coupling it with some other equation; or adding a non-linearity. We will consider the last option, and just consider for illustration the so-called meson equation:

\begin{aligned}iu_{t} − Δu & = λ|u|^{2}u \\ u(0) & = f \end{aligned} (3)

where λ ∈ C is a constant.

In other words, we have a cubic non-linearity as a forcing term. One could consider other non-linearities than cubic, but this particular choice is special because it is L^{2}-critical. More precisely, if we require the L^{2} norm of u to be dimensionless (as it is physically, when u has a probabilistic interpretation), then λ must also be dimensionless. So if we want to keep the L^{2} norm of u constant, we can’t scale λ away, which we could do with other powers.

To get some idea of what this equation is doing, suppose that we remove the dispersive −Δu term, and just look at the equation

u_{t} = −iλ|u|^{2}u.

For this equation the spatial variable is irrelevant, and we may as well consider this equation as an ODE. It is easy to verify this ODE has the solution

u(t) = (|u(0)|^{−2} − 2at)^{− \frac{a+ib}{2a}} \frac{u(0)}{|u(0)|}

where −iλ = a + ib.

In particular, if a > 0, then this equation will blow up in finite time (t = 1/(2a|u(0)|^{2}), in fact). Basically the non-linearity causes a positive feedback loop which causes u to increase very rapidly.

However, we expect the dispersive Δ term to try to stop this blow up from happening, by spreading u around as soon as it gets too large. There is no guarantee that this dispersion will be enough to stop the blowup; for instance, if f were constant in space, then u would also be constant in space (by translation symmetry), and the Δ term would be completely irrelevant. On the other hand, we do have

Theorem 1.1. Suppose ∥f∥_{2} = 1. Then if λ is sufficiently small, we have a global solution u to (3) such that ∥u(t)∥_{2} ≲ 1 for all t. Furthermore we have the spacetime estimate ∥u∥_{L_{x,t}^{4}} ≲ 1. This solution u is unique subject to the above conditions, and the solution depends continously in the norms just mentioned on the initial data f in L^{2}.

Finally, we have scattering in the following sense: there exists some initial data f_{+} such that

∥u(t) − e^{−itΔ}f_{+}∥_{2} → 0 as t → ∞.

In the PDE jargon, we’ve just claimed that the meson equation is globally wellposed with scattering in L^{2} for small λ. For large lambda, see the last section of this paper.

Note that this theorem does not care about the sign of λ. Basically it says that the decay inherent in the Schrödinger equation has tamed the effect of the nonlinearity, and in fact as time goes to infinity the non-linearity becomes increasingly irrelevant.

The techniques used to prove this result are by no means restricted to this one particular equation; they can be extended to all kinds of non-linear dispersive equations which are in some sense a small perturbation of a well-understood linear equation. Unfortunately we still do not really understand how to push the well-posedness theory beyond perturbation theory, and into equations which are far more non-linear than (3), at this stage.

The treatment of these equations is connected, spiritually at least, with restriction theory. An informal link is as follows. Suppose that u is a global solution to the free Schrödinger equation

iu_{t} − Δu = 0.

Assuming that u has a space-time Fourier transform, we get (formally at least)

−2πτû(τ, ξ) + 4π^{2}|ξ|^{2}û(τ, ξ)

where

û(τ, ξ) = ∫ ∫ e^{−2πi(τt+ξ·x)}u(t, x) dtdx.

This implies that û is supported on the paraboloid

S = {(τ, ξ) : τ = 2π|ξ|^{2}},

so we may write

û = gdσ

for some g, where dσ is some surface measure on S. It turns out that the best choice of dσ is the spatial Lebesgue measure dξ, or more precisely the pullback of this measure under the projection map (τ, ξ) ↦ ξ.

If we require that the initial data of u is in L^{2}, this turns out to imply a L^{2} estimate on g, by Plancherel’s theorem. (Exercise!)

In other words, we have a representation of u as u = ĝdσ, where we have L^{2} control on g.

We’d like to say that u decays at infinity, so that the non-linear effects will also die away. It turns out that the right estimate to use is

∥ĝdσ∥_{L_{x,t}^{4}} ≲ ∥g∥_{2}.

If you take the adjoint of this, this is

∥f̂∥_{L^{2}(S,dσ)} ≲ ∥f∥_{4/3},

and this turns out to just be the Tomas-Stein restriction estimate R_{S}(4/3 → 2)!

Actually, we won’t explicitly invoke Tomas-Stein in what follows - I lied a little bit in the above exposition. (First of all, u is supposed to solve the non-linear Schrödinger equation rather than the linear Schrödinger equation). But the Tomas-Stein philosophy - particularly that of squaring an estimate, and interpolating between an (L^{1}, L^{∞}) and an (L^{2}, L^{2}) estimate - will be very much present.

2. Proof of well-posedness

Let’s now begin the proof of the main theorem. Fix f such that ∥f∥_{2} = 1; we want to solve the equation

iu_{t} − Δu = λ|u|^{2}u

with initial data u(0) = f.

We first observe that if we want to solve any equation of the form

iu_{t} − Δu = F, u(0) = f

then the only solution is

u(t) = e^{−itΔ}f + ∫_{0}^{t} e^{−i(t−s)Δ}F (s) ds,

as can be easily verified. One should think of the first term as the influence of the initial data, and the second term as the cumulative influence of the forcing term F.

Thus, we may rewrite our non-linear differential equation as a non-linear integral equation:

u(t) = e^{−itΔ}f + λ ∫_{0}^{t} e^{−i(t−s)Δ}(|u|^{2}u) ds. (4)

This equation is equivalent to the differential form, but is much easier to handle when it comes to proving questions of existence and uniqueness.

To find a solution to (4), we shall use an iterative method. We first approximate u by the linear solution

u_{0}(t) = e^{−itΔ}f. (5)

Then we make a better approximation

u_{1}(t) = e^{−itΔ}f + λ ∫_{0}^{t} e^{−i(t−s)Δ}(|u_{0}|^{2}u_{0}) ds.

More generally, we define the non-linear map u ↦ N(u) by

N(u)(t) = e^{−itΔ}f + λ ∫_{0}^{t} e^{−i(t−s)Δ}(|u|^{2}u) ds

and define u_{k+1} = N(u_{k}) for all k = 0, 1, . . .. We hope to show that this sequence of approximations converges to a limit u, so that u = N(u). This would give us a solution to (4).

(As an analogy, suppose we wished to solve the equation

x = b + λAx

where b was a vector, A was a matrix, and λ was a small number. We could try to solve this iteratively by starting with x_{0} = b, and then inductively defining x_{k+1} = b + λAx_{k}. As long as λA was a contraction, this sequence would converge to

x = (1 + λA + (λA)^{2} + (λA)^{3} + …)b,

which is the unique solution. Note that this approach only works if λ is small. This trick is at the heart of perturbation theory for non-linear PDE.)

In short, we want to find a fixed point of N, show that it is unique, and that it depends continuously on the initial data. We can accomplish all this in one stroke from the contraction mapping theorem, as soon as we show that N is a contraction on some metric space X which contains u_{0}. This sounds easy enough - and a very large number of existence results in PDE are ultimately derived from this very simple idea. What’s the catch? Well, we have to pick the right metric space to get the contraction working.

After much experimentation and looking at the behaviour of the first few iterates u_{0}, u_{1}, etc., we ultimately decide that the correct space to use is

X = {u : ∥u∥_{L_{x,t}^{4}} ≤ C},

with the metric induced from the L^{4} norm. where C is an absolute constant. To be able to apply the contraction mapping principle we need to know that u_{0} is in the space, so that

∥u_{0}∥_{L_{x,t}^{4}} ≲ 1; (6)

and that N is a contraction on X:

∥N(u) − N(v)∥_{L_{x,t}^{4}} ≤ \frac{1}{2}∥u − v∥_{L_{x,t}^{4}} for all u, v ∈ X. (7)

This will be accomplished by the following three estimates, known as Strichartz estimates:

∥e^{−itΔ}f∥_{L_{x,t}^{4}} ≲ ∥f∥_{2} (8)

∥ ∫_{0}^{∞} e^{itΔ}F (t) dt∥_{2} ≲ ∥F ∥_{L_{x,t}^{4/3}} (9)

∥ ∫_{0}^{t} e^{−i(t−s)Δ}F ds∥_{L_{x,t}^{4}} ≲ ∥F ∥_{L_{x,t}^{4/3}}. (10)

We’ll prove these estimates in the next section. For now, let’s see how these estimates give what we want.

Firstly, (6) follows straight from (8) and (5), since ∥f∥_{2} = 1. Now let’s look at (7). We can simplify N(u) − N(v) as

N(u) − N(v) = λ ∫_{0}^{t} e^{−i(t−s)Δ}(|u|^{2}u − |v|^{2}v) ds.

Thus by (10) we have

∥N(u) − N(v)∥_{4} ≲ |λ|∥|u|^{2}u − |v|^{2}v∥_{4/3}.

We now use the pointwise estimate

|u|^{2}u − |v|^{2}v = O(|u|^{2}|u − v|) + O(|v|^{2}|u − v|)

and Hölder’s inequality to estimate this as

∥N(u) − N(v)∥_{4} ≲ |λ|(∥u∥_{4}^{2}∥u − v∥_{4} + ∥v∥_{4}^{2}∥u − v∥_{4}.

But since u, v ∈ X, the L^{4} norms of u and v are bounded. Thus (7) holds if λ is sufficiently close to zero.

This shows the existence, uniqueness, and continuous dependence of u on the data. We also get a bonus, that u is in X, so that the L^{4} norm is finite. We still have to show a couple things. Firstly, we show that u stays in L^{2}. From (4) we have

∥u(t)∥_{2} ≲ ∥e^{−itΔ}f∥_{2} + ∥e^{−itΔ} ∫_{0}^{t} e^{isΔ}(|u|^{2}u(s)) ds∥_{2}.

From (2) the first term is bounded. As for the second, we use (2) and (9) to estimate it by

∥|u|^{2}u∥_{4/3} = ∥u∥_{4}^{3} ≲ 1

as desired.

Finally, we show scattering. Define f_{+} by

f_{+} = f + ∫_{0}^{∞} e^{itΔ}(|u|^{2}u)(t) dt.

In other words, f_{+} is equal to f modified by the backdated effect of the non-linearity. From (9) and the argument just given, we see that f_{+} is in L^{2}. We wish to show that

∥u(t) − e^{−itΔ}f_{+}∥_{2} → 0 as t → ∞.

From (4) we have

u(t) − e^{−itΔ}f_{+} = e^{−itΔ} ∫_{t}^{∞} e^{isΔ}(|u|^{2}u)(s) ds.

From (2) and (9) we thus have

∥u(t) − e^{−itΔ}f_{+}∥_{2} ≲ ∥χ_{[t,∞]}|u|^{2}u∥_{4/3}.

The claim then follows from the monotone convergence theorem.

3. Proof of the Strichartz estimates

We still have to prove (8), (9), and (10). We first observe that (8) will follow from (9) by duality. Indeed, suppose (9) were true. Then by Hölder’s inequality, we have

| ∫ (∫_{0}^{∞} e^{itΔ}F (t, x) dt)f(x) dx| ≲ ∥F ∥_{L_{x,t}^{4/3}}∥f∥_{2}.

Rearranging the left-hand side, this becomes

| ∫ ∫ F (t, x)e^{−itΔ}f(x) dtdx| ≲ ∥F ∥_{L_{x,t}^{4/3}}∥f∥_{2}.

Taking the sup over all F with unit L^{4/3} norm and applying the converse to Hölder, we obtain (8) as desired.

Next, we observe that (9) follows from (10). For, suppose (10) were true. To show (9), we square it as

⟨∫_{0}^{∞} e^{itΔ}F (t) dt, ∫_{0}^{∞} e^{isΔ}F (s) ds⟩ ≲ ∥F ∥_{L_{x,t}^{4/3}}^{2}.

We rewrite this as

∫_{0}^{∞} ∫_{0}^{∞}⟨e^{itΔ}F (t), e^{isΔ}F (s)⟩ dsdt ≲ ∥F ∥_{L_{x,t}^{4/3}}^{2}.

By symmetry it suffices to consider the portion of the double integral where s ≤ t:

∫_{0}^{∞} ∫_{0}^{t}⟨e^{itΔ}F (t), e^{isΔ}F (s)⟩ dsdt ≲ ∥F ∥_{L_{x,t}^{4/3}}^{2}.

We rewrite this as

∫ ∫ F (t, x)(∫_{0}^{t} e^{−i(t−s)Δ}F (s, x) ds) dtdx ≲ ∥F ∥_{L_{x,t}^{4/3}}^{2}.

By Hölder, the left-hand side is bounded by

∥F ∥_{4/3}∥ ∫_{0}^{t} e^{−i(t−s)Δ}F (s, x) ds∥_{4},

and the result follows from (10).

Now we turn to (10). Instead of delegating this to yet another estimate, we shall actually get some work done here. We rewrite the left-hand side of (10) as

(∫_{0}^{∞} ∥ ∫_{0}^{t} e^{−i(t−s)Δ}F (s) ds∥_{L_{x}^{4}}^{4})^{1/4}.

By Minkowski’s inequality, we may estimate this by

(∫_{0}^{∞}(∫_{0}^{t} ∥e^{−i(t−s)Δ}F (s)∥_{L_{x}^{4}} ds)^{4})^{1/4}.

On the other hand, by interpolating between (2) and (1) we have

∥e^{isΔ}F (s)∥_{L_{x}^{4}} ≲ |t − s|^{−1/2}∥F (s)∥_{L_{x}^{4/3}},

so we may estimate this by

(∫_{0}^{∞}(∫_{0}^{t} |t − s|^{−1/2}∥F (s)∥_{L_{x}^{4/3}} ds)^{4})^{1/4}.

Writing a(s) = ∥F (s)∥_{L_{x}^{4/3}}, this is bounded by

∥a ∗ \frac{1}{|t|^{1/2}} ∥_{L_{t}^{4}}.

By the Hardy-Littlewood-Sobolev lemma on fractional integration, this is bounded by ∥a∥_{4/3}, which is just ∥F ∥_{L_{x,t}^{4/3}} as desired.

4. Large values of λ

When λ is large, one cannot guarantee global existence; the non-linear term can destroy the wave function before the dispersive effect of the Laplacian can repair the damage. However, one can still get a local solution.

Theorem 4.1. Suppose ∥f∥_{2} = 1, and λ is arbitrary. Then there exists a time T_{0} > 0 and a local solution u to (3) such that ∥u(t)∥_{2} ≲ 1 for all 0 ≤ t ≤ T_{0}, and estimate ∥u∥_{L_{x,t}^{4}(R^{3}×[0,T_{0}])} ≲ 1. This solution u is unique subject to the above conditions, and the solution depends continously in the norms just mentioned on the initial data f in L^{2}.

The proof of this theorem is virtually identical to that of the main theorem. There are two main differences. Firstly, all our norms and estimates are restricted to the time interval [0, T_{0}] rather than the whole real line. Secondly, we iterate on a much smaller ball, namely

X = {u : ∥u∥_{L_{x,t}^{4}(R^{3}×[0,T_{0}])} ≤ ε},

where ε = ε(λ) is a small number. One can check that our map N is still a contraction if we choose ε small enough. Finally, we have to guarantee that the zeroth iterate u_{0} is in X. Well, from (8) we know that ∥u∥_{L_{x,t}^{4}} ≲ 1 globally; so by the monotone convergence theorem, if we choose T_{0} small enough, we’ll end up in the space X.

References

1. J. Ginebre, G. Velo, Smoothing Properties and Retarded Estimates for Some Dispersive Evolution Equations, Comm. Math. Phys., 123 (1989) , 535–573.

2. J. Ginebre, G. Velo, Generalized Strichartz Inequalities for the Wave Equation, Jour. Func. Anal., 133 (1995), 50–68.

3. L. Kapitanski, Some Generalizations of the Strichartz-Brenner Inequality, Leningrad Math. J., 1 (1990), 693–676.

4. L. Kapitanski, Weak and Yet Weaker Solutions of Semilinear Wave Equations, Comm. Part. Diff. Eq., 19 (1994), 1629–1676.

5. M. Keel, T. Tao, Endpoint Strichartz Estimates, Amer. Math. J. 120 (1998) , 955–980.

6. H. Lindblad, C. D. Sogge, On Existence and Scattering with Minimal Regularity for Semilinear Wave Equations, Jour. Func. Anal., 130 (1995 ), 357–426.

7. C. D. Sogge, Lectures on Nonlinear Wave Equations, International Press, 1995.

LECTURE NOTES 4 11

8. R. S. Strichartz, Restriction of Fourier Transform to Quadratic Surfaces and Decay of Solutions of Wave Equations, Duke Math. J., 44 (1977), 70 5–774.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555 E-mail address: tao@math.ucla.edu