Low regularity semi-linear

wave equations

Terence Tao Mathematics Department

UCLA

www.math.ucla.edu/∼tao/preprints

The purpose of this talk is to:

• State the local existence problem for semilinear wave equations

• Discuss some of the known techniques and results

• Describe some recent improvements for the low regularity case Let n ≥ 1, and let F (u) be a power-type non-linearity, i.e. a function such that

|F (u)| ∼ |u|^{p}, |∇F (u)| ∼ |u|^{p−1}

for some p > 1. (u can be scalar or vector valued). We consider the following Cauchy problem for the semilinear wave equation

\begin{aligned}□u & = F (u) \\ u(0, ·) & = f ∈ H^{γ}(R^{n}) \\ ∂_{t}u(0, ·) & = g ∈ H^{γ−1}(R^{n}), \end{aligned}

where u is a function on [0, T] × R^{n}, γ ≥ 0, and □ = −\frac{∂^{2}}{∂t^{2}} + Δ is the D’Alembertian.

• We consider the question of which p, γ does one have local existence of solutions u for arbitrary data f, g in H^{γ} × H^{γ−1}.

• Global solutions are unrealistic (without further assumptions on F or the data) as one can easily construct examples of blowup in finite time, based on the ODE u_{tt} = |u|^{p}.

• Smooth classical solutions are also unrealistic unless p is an integer, so we shall only consider weak solutions.

• One can also ask if the problem is locally well-posed, i.e. one has uniqueness and continuous dependence in H^{γ} of the data in addition to existence. In practice these properties are easy once one obtains existence.

Necessary conditions on (p, γ) for local existence

• By scaling the example of blowup in finite time, one can obtain blowup in arbitrarily small time for data of fixed norm, unless the scaling condition

p(\frac{n}{2} − γ) ≥ \frac{n + 4}{2} − γ  (1)

holds.

• For radial data the above scaling condition is known to be sharp; that is, one does indeed have local existence whenever (1) holds. (Lindblad, Sogge)

• However, this is false for non-radial data. By Lorentztransforming and scaling the finite blowup example, one can make the solution blowup along a light ray in arbitrarily small time, for data of fixed norm, unless the concentration condition

p(\frac{n + 1}{4} − γ) ≤ \frac{n + 5}{4} − γ  (2)

holds.

• Finally, Hans Lindblad has an example that shows that one does not have local existence for the equation □u = −u^{2} in L^{2} for n = 3. (This is an endpoint to (2)). The three-dimensional case is pathological in a number of ways.

• It is tentatively conjectured that there are no further obstructions to local existence. In other words, one has local existence whenever (1), (2) hold and (n, p, γ) ≠ (3, 2, 0).

The high regularity case: γ > \frac{n}{2}

• When the data is very smooth (γ > \frac{n}{2}) and the nonlinearity is sufficiently smooth, one can obtain local existence via the classical energy method.

• The energy method relies on two estimates. The first is a linear estimate (the “energy estimate”)

∥u∥_{L_{t}^{∞}H^{γ}} ≲ ∥f∥_{H^{γ}} + ∥g∥_{H^{γ−1}} + ∥F ∥_{L_{t}^{1}H^{γ−1}}

which holds for all f, g, F, where u is the solution to the linear problem

\begin{aligned}□u & = F \\ u(0, ·) & = f \\ ∂_{t}u(0, ·) & = g, \end{aligned}

• The second estimate is a non-linear estimate (Schauder’s lemma), which states that if u is in H^{γ} for γ > \frac{n}{2} and F is sufficiently smooth, then F (u) is also in H^{γ} (and thus in H^{γ−1}).

• With these two estimates local existence can be obtained by the contraction mapping principle (aka Picard iteration). In other words, we show that the non-linear problem is a small perturbation of the linear problem in some Banach space.

• For function v, consider the solution u to the problem

\begin{aligned}□u & = F (v) \\ u(0, ·) & = f \\ ∂_{t}u(0, ·) & = g, \end{aligned}

• To find a solution to the non-linear problem we need to find a fixed point of the map v ↦ u. It suffices to find a Banach space X such that v ↦ u is a contraction on X. In practice it’s enough to show that v ↦ u maps a ball in X to itself, since proving a contraction only requires a little more computation.

• For the energy method we choose the space X = L_{t}^{∞}H^{γ}. If v is bounded in this space, then by Schauder’s lemma F (v) is in the space L_{t}^{∞}H^{γ−1}, and so (if time T is small) F (v) will be small in L_{t}^{1}H^{γ−1}. But by the energy estimate this implies that u is bounded in L_{t}^{∞}H^{γ} = X. By taking a little more care one can show that v ↦ u is actually a contraction on X, and so we get our solution.

• From the contraction mapping theorem one can also get uniqueness and continuous dependence on the data.

The general case.

• Virtually all local existence results for hyperbolic equations are based on the above contraction mapping principle. There are variants, but basically we need three ingredients:

• A Banach space X to contain the solution, and a supplementary space Y to contain the inhomogeneity. (For the energy method, X = L_{t}^{∞}H^{γ} and Y = L_{t}^{1}H^{γ−1}).

• A linear estimate

∥u∥_{X} ≲ ∥f∥_{H^{γ}} + ∥g∥_{H^{γ−1}} + ∥F ∥_{Y};

• A non-linear estimate, which has the qualitative form

u is small in X ⇒ F (u) is small in Y.

• Once we have the above ingredients, we can set up an iteration scheme to find a solution, and the finer details (actually obtaining a contraction, showing that the solution u has the same regularity as the data, etc.) tend to be pretty routine.

Example: the conformal wave equation for n = 3

• We illustrate the above formalism with the equation

□u = |u|^{3}

in three dimensions, with data in H^{1/2} × H^{−1/2}.

• This equation is somewhat special, being invariant under conformal transformations. The choice of data regularity is also “conformal” as it is essentially independent of the choice of co-ordinates. It lies on the intersection of the two conditions (1), (2).

• We can use the above machinery to obtain local existence for this problem (indeed, one can even get global existence for small data). We choose the solution space to be X = L_{t,x}^{4} and the inhomogeneity space to be Y = L_{t,x}^{4/3}.

• The non-linear estimate is trivial:

u is small in L_{t,x}^{4} ⇒ |u|^{3} is small in L_{t,x}^{4/3}.

• The linear estimate

∥u∥_{L_{t,x}^{4}} ≲ ∥f∥_{H^{1/2}} + ∥g∥_{H^{−1/2}} + ∥F ∥_{L_{t,x}^{4/3}}

is due to Strichartz and is proven next.

Sketch of proof of Strichartz’ estimate

• By linearity we can assume that either F = 0 (homogeneous case), or f = g = 0 (inhomogeneous case). One can obtain the homogeneous estimate from the inhomogeneous estimate by the T T^{∗} method, so we concentrate on the inhomogeneous estimate ∥u∥_{4} ≲ ∥F ∥_{4/3}.

• Duhamel’s formula states that

u(t) = ∫_{0}^{t} \frac{sin(s\sqrt{−Δ})}{\sqrt{−Δ}}F (t − s) ds.

Thus it suffices to prove the fixed-time bounds

∥\frac{sin(s\sqrt{−Δ})}{\sqrt{−Δ}}f∥_{L_{x}^{4}} ≲ \frac{1}{|s|^{1/2}}∥f∥_{L_{x}^{4/3}},

since the Hardy-Littlewood fractional integration inequality states that

∥g ∗ \frac{1}{|s|^{1/2}}∥_{L_{t}^{4}} ≲ ∥g∥_{L_{t}^{4/3}}.

• To prove

∥\frac{sin(s\sqrt{−Δ})}{\sqrt{−Δ}}f∥_{L_{x}^{4}} ≲ \frac{1}{|s|^{1/2}}∥f∥_{L_{x}^{4/3}}

we interpolate between the energy estimate

∥ sin(s\sqrt{−Δ})f∥_{L_{x}^{2}} ≲ ∥f∥_{L_{x}^{2}},

proven by Plancherel’s theorem, and the decay estimate

∥\frac{sin(s\sqrt{−Δ})}{−Δ}f∥_{L_{x}^{∞}} ≲ \frac{1}{|s|}∥f∥_{L_{x}^{1}},

proven by a stationary phase computation of the kernel corresponding to the multiplier ^{\frac{sin(s\sqrt{−Δ})}{−Δ}}.

• Strictly speaking, the decay estimate is false, but it is true when you localize in frequency, and you can fix matters up using Littlewood-Paley theory.

• The decay estimate can be viewed as a statement on the decay and regularity of the fundamental solution to the linear wave equation.

What about other powers and regularities?

• For data which is smoother than conformal H^{1/2} but rougher than energy H^{n/2} one can adapt the above estimates by placing derivatives in the Banach spaces X and Y. The linear estimate commutes with differentiation, so adding the derivatives there is trivial. The non-linear estimate gets a little more complicated and you have to use Leibnitz’ rule for fractional differentiation. This method turns out to be sharp, and one can conclude that one has local wellposedness as long as the scaling condition is satisfied.

• For data which is rougher than H^{1/2} but not too rough one can modify the above argument by splitting the time and space integrations, using mixed space-time norms L_{t}^{q}L_{x}^{r} instead of pure norms L_{t,x}^{p}. The non-linear estimate remains trivial, and the linear Strichartz estimate is proven in much the same way as the original Strichartz estimate. This technique is (somewhat surprisingly) also sharp, and one can conclude that one has local well-posedness as long as the concentration condition is satisfied and the data is not too rough.

More precisely, we have

Theorem 0.1 (Kapitanski, Lindblad, Sogge, Keel, T.) The conjecture holds if n ≤ 3. For n > 3 the conjecture holds assuming the technical condition

p(\frac{n + 1}{4} − γ) ≤ \frac{n + 1}{2n}(\frac{n + 3}{2} − γ).

• The argument for (n, p, γ) = (3, 2, 0) fails because the corresponding Strichartz estimate fails:

∥u∥_{L_{t}^{∞}L_{x}^{2}} ≴ ∥f∥_{H^{1}} + ∥g∥_{L^{2}} + ∥F ∥_{L_{t}^{2}L_{x}^{1}}

• This theorem settles the conjecture stated earlier for n ≤ 3. For n > 3 there is a gap for very low regularity problems, because one begins to run out of Strichartz estimates. (More on this later).

• Our main result is that we have been able to narrow this gap, relaxing the above condition to

p(\frac{n}{4} − γ) ≤ \frac{1}{2}(\frac{n + 3}{2} − γ),

except for an endpoint.

Local well-posedness results for n = 3. Local well-posedness results for n = 4.

What is the problem for very low regularities?

• The above arguments relied on linear estimates of the form

∥u∥_{L_{t}^{q}L_{x}^{r}} ≲ ∥f∥_{H^{γ}} + ∥g∥_{H^{γ−1}} + ∥F ∥_{L_{t}^{q̃′}L_{x}^{r̃′}}

for some exponents q, r, q̃, r̃; the power p of nonlinearity that this estimate can handle is based on the gap between L_{t}^{q}L_{x}^{r} and L_{t}^{q̃′}L_{x}^{r̃′}.

• There are several limitations as to the type of estimates above that can occur. Firstly, there is a dimensional analysis condition that links the regularity γ of the data to the choice of exponents q, r, q̃, r̃:

\frac{1}{q} + \frac{n}{r} = \frac{n}{2} − γ = \frac{1}{q̃′} + \frac{n}{r̃′} − 2.

• Translation invariance considerations give

q, q̃, r, r̃ ≥ 2.

• Finally the Knapp example gives more constraints which are independent of the regularity of the data:

\frac{1}{q} + \frac{(n − 1)/2}{r}, \frac{1}{q̃} + \frac{(n − 1)/2}{r̃} ≤ \frac{(n − 1)/2}{2}.

• As γ gets small, these constraints become increasingly incompatible, and one cannot make the gap between L_{t}^{q}L_{x}^{r} and L_{t}^{q̃′}L_{x}^{r̃′} as wide as one might hope.

• In other words, the less regular the data becomes, the fewer L_{t}^{q}L_{x}^{r} spaces one can hope to place the solution in, and eventually the options become too limited to obtain optimal results.

How to get around this problem?

• For high regularity data, one placed the solution in a high-regularity space such as L_{t}^{∞}H^{s} for s > \frac{n}{2}. For medium regularity data one placed the solution in a medium regularity space L_{t}^{q}L_{x}^{r}. What do we do for low regularity?

• One obvious idea is to place negative derivatives on the solution space, e.g. trying to place u in a negativeorder Sobolev space L_{−α}^{r}. The linear estimate carries through fine, but the non-linear estimate is problematic. Basically, if u is only assumed to be in a space such as L_{−α}^{r}, there is no way to control F (u).

• For instance, take F (u) = |u|^{2} and let u(x) = e^{ix·ξ}ψ(x) for some cutoff ψ and some very high frequency ξ. As ξ → ∞, u tends to zero in L_{−α}^{r}, but F (u) remains constant.

• As the above example shows, the enemy is the fact that the high frequencies of u can corrupt the low frequencies of F (u). (The low frequencies of u can also corrupt the high frequencies of F (u), but this is an obstruction for the high-regularity problem rather than the low regularity problem). If one could somehow prevent the frequencies from interacting with each other, then one could use negative order Sobolev spaces with impunity, and get sharp results (down to concentration and scaling) for all regularities.

• So we have to limit the corruption effect of the high frequencies of u. Key observation: the high frequencies of u tend to be supported on “thin” sets. For instance, if you consider the linear problem □u = 0, u(0) = δ, then u(t) is mainly supported on the sphere of radius t (Huygens’s principle), and so the 2^{j}-frequency portion of u is concentrated in an annulus of width 2^{−j} around that sphere. So even though u is spread out in most directions, in the radial direction u is quite thin.

Why does “thin-ness” help?

• In the counterexample u = e^{ix·ξ}ψ(x), u had very high frequency but was “fat”: spread out over a set of dimension 1 in all dimensions. Thus F (u) was also fat, and low-frequency. If u was supported on a thinner set, then F (u) would also have thinner support, and F would consist mainly of high-frequency modes instead. Thus: if u is thin, then F (u) does not have very many low-frequency components. In other words, the frequency-corruption effect is not as bad as one might think (assuming we can show that u is thin).

• To make this thin-ness quantitative, we need to estimate u not in a Lebesgue spaces L_{x}^{r}, but in two-scale spaces X_{0}^{r,p} defined by

∥f∥_{X_{0}^{r,p}} = (∑_{Q} ∥f∥_{L^{p}(Q)}^{r})^{1/r},

where Q ranges over a partition of R^{n} into unit cubes. If u is fat then the X_{0}^{r,p} norm is pretty much the same as the L^{r} norm, but if u is thin and p < r then one expects to gain something in the X_{0}^{r,p} norm.

• One also has to define norms X_{k}^{r,p} for spatial scales 2^{−k} other than the unit scale.

• To formally obtain an improvement on local wellposedness results, we need an improved linear estimate and an improved non-linear estimate.

• The improved linear estimate states that we can replace the L_{x}^{r} space in the standard Strichartz estimate

∥u∥_{L_{t}^{q}L_{x}^{r}} ≲ ∥f∥_{H^{γ}} + ∥g∥_{H^{γ−1}} + ∥F ∥_{L_{t}^{q̃′}L_{x}^{r̃′}}

by the spaces X_{k}^{r,p} and gain something, especially if f, g, F are high frequency. These improved estimates are still proven by the same techniques used to prove the ordinary Strichartz estimates (i.e. T T^{∗}, energy estimate, decay estimate, interpolation). The gain ultimately stems from the fact that the fundamental solution of the wave equation is “thin”.

• To close the argument we need non-linear estimates

which state, roughly, that \sqrt{−Δ}^{−α}u is small in X_{k}^{r,p} ⇒

\sqrt{−Δ}^{−pα}|u|^{p} is small in L_{x}^{r/p}.

This is achieved by Littlewood-Paley theory; if u is in X_{k}^{r,p}, then |u|^{p} is locally integrable at the scale of 2^{−k}, and so the 2^{k} frequency piece of |u|^{p} is well controlled.