(Apologies for typos or misquotations…; please use comments to help me improve the post. –Jim Colliander)

 

Princeton Math Colloquium 2012-03-14

Presenter: Carlos Kenig, University of Chicago

Title: A case study for critical non-linear dispersive equations: the energy critical wave equation

Abstract: We will discuss recent work on the energy critical wave equation. The issues studied are global existence, scattering, finite time blow-up, universal profiles at blow-up and soliton resolution. This is viewed not as an isolated series of results, but as a way of approaching many similar critical non-linear dispersive equations.


We hope this is a model case for how nonlinear dispersive wave equations should proceed. This is a model case. Techniques of harmonic analysis have been introduced into the subject. The focus of those studies concentrated on local well-posedness. In the last fifteen years, there has been a new concentration on global-in-time, blowup aspects.

Linear wave equation

We start with a review of the linear wave equation.

$$ (LW) \partial^2_t w – \Delta w = h, data (w_0, w_1).$$

We write the soluton as
$$
w(t) = S(t)(w_0, w_1) (t) + D(h)(t).
$$

Finite speed of propagation. Picture of a cone.

An important estimate in these studies is the Strichartz estimate.

Critical focusing NLW

$$
\partial^2_t u – \Delta u = u^5, data (w_0, w_1).$$

Defocusing has $- u^5$. Reviews scaling and explains criticality.

Small data theory for NLW

Small (in $\dot{H}^1 \times L^2$) initial data evolves uniquely into a global-in-time solution which asymptotoically approaches (in same topology) into a linear solution as $ t \rightarrow \pm \infty$. Moreover, for any data, we have short time existence and a maximal time interval of existence.

This problem has an energy which is constant throughout the solution lifetime. Focusing vs. defocusing. There is competition, in the focusing case, between these two terms.

In the defocusing case, work of Struwe, Grillakis, Shatah-Struwe, Bahouri-Shatah proves that for any finite energy data, the solution exists globally and scatters. Another factor, Grillakis showed persistence of higher regularity.

In the focusing case, this fails. Levine 1974 showed that if $E(u_0, u_1) \leq 0$ then the solution can not exist globally in time (in either direction). (This is done by obstruction.) Recently, Kreiger-Schlag-Tataru 2009 constructed solutions for which $T_+ < \infty$ but for which the energy norm remains bounded. Also, in the focusing case, the elliptic equation admits a nonnegative solution $W$. This is a solution which is independent of time. This solution satisfies $\Delta u + u^5 = 0$. This solution is called the ground state. The reason for this terminology is that it arises as the optimizer in a Sobolev inequality and is charaterized variationally. This elliptic equation has been studied in connection with the Yanage problem in differential geometry. The formula for $W$ is explicit
$$ W(x) = \frac{1}{(1 + |x|^2/3)^{1/2}}.$$
$W$ is the unique nonnegative solution of the elliptic equation (Gidas-Ni-Nirenberg 1979) and the only $\dot{H}^1$ solution (Pohozaev 1965). It does not scatter to a linear solution. It is “non-dispersive.” Donninger-Krieger 2012 have constructed global-ini-time solutions which are bounded in the energy class, are radial, and don’t scatter to either a linear solution or to $W$. There are other objects.

Recall a collection of recent results

Theorem (Kenig-Merle 2008):
If $E(u) < E(w)$ then:
1. If the gradient is smaller, we have global existence and scattering.
2. If the gradient is bigger, we have breakdown in (both directions of) finite time.
The case of equality is impossible under the hypothesis.

The proof of this result is an application of a general method. We call this a concentrated compactness rigidity method. Provides a brief summary of these ideas: small data, variational aspects, critical element extraction, rigidity.

Theorem (Duyckaerts-Merle 2008):
There exists $W_-, ~ W_+$ radial which have the same energy as $W$. Then, there are three statements which provide a characterization of the dynamics.

(original statement of DM had an extra hypothesis which is now removed with inputs from
Duyckaerts-Kenig-Merle 11, Krieger-Nakanishi-Schlag 11)

Now, we need to go beyond the energy threshold of $W$.

Existence of Type II blowup solutions

These solutions exist for a finite time but their critical norm remains bounded.

Krieger-Schlag-Tataru 2009

Describes a singularity formation process along a rescaled $W$ profile plus a continuous remnant.

We next show this is a “universal” phenomena.

DKM 2009, 2010:

(Radial) Any Type II blowup (critical norm stays bounded) solution with just a little bit bigger size than $W$. Then the soution looks like a rescaled $W$ plus errors. All type II blowups look like rescaled $W$.

Corresponding nonradial result: Slightly larger (by $\eta_0$) in the $\sup$ sense as above. After a rotation and a translation, we find that the solution looks like a rescaled and translated $W_l$ plus a small error. The rescaling rate is understood. And we have
$$\lim_{t \uparrow 1} \frac{x(t)}{1-t} = l e^1 .$$

So, the possible profiles in the nonradial case are $W$ or its Lorentz transformations.

There also exist explicit solutions which blow up in finite time. These are built with ODE techniques and are independent of $x$. You can then chop them up with finite speed of propagation. These solutions have exploding critical norm. These are called Type I solutions. Conceivably, there are solutions which are Type I along one sequence of times and Type II along a different sequence of times.

DKM 2011: $W_+$ turns out to be Type I.

Soliton resolution for radial solutions of NLW

For a long time, there has been a widespread belief in the math physics community that large, global in tie, solutions of dispersive equations, asymptotically in time, they decouple into a sum of finitely many modulated solitons, a free radiation term and a term that goes to zero at infinity. This is a kind of philosophy that guides the research. So far, this has only been proved for the integrable KdV equations. Eckhaus-Schurr carried this out using the completely integrable structure. Corresponding results for mKdV can be obtained via the Miura transform. Heuristic arguements for this conjecture, in the case of cubic NLS in 1D, were given by Ablowitz-Segur 1976 and Zakharov-Shabat 1971. These are all subcrticial equations, for which one expects that these decompositions are stable, unlike in critical equations.

For more general equatiosn, so far, results have been found for data close to the soliton, in subcritical nonlinearities due to several authors. Buslaev-Perelman 1992 NLS, Soffer-Weisntein 1990, Marterl-Merle 2001 for gKdV) For corresponding results with blowup there are works by Merle-Raphael, Martel-Merle. There have also been large solution results for critical equivariant wave maps onto the sphere due to Christodoulou-Tahvildar-Zadeh, Shatah-T-Z, Struwe. They sow convergence along some sequence of times converging to the blwup time locally in space to a soliton (harmonic map).

In the finite time blowup case, for the 1d nonlinear wave equation, Merle-Zaag have obtained reusults of this kind using a Lyapunov functional tool.

In critical elliptic problems, such as the ones mentioned earlier, in domains excluding a small ball, considering radial solutions, there have been obtained results on decompositions inot “towering bubbles” (the analog of a finite sum of modulated solitons), as the size of the ball goes to zero. Musso-Pistoia 2006

The first general results for radial solutions of NLW were for type II solutions and held only for a sequence of times. We now have the full soliton resolution for radial solutions of NLW.

Theorem: Let $u$ be radial solution of NLW. Then, one of the following holds true:

  1. Type I blowup.
  2. Type II blowup with a full decomposition as a finite sum rescaled $W$, with ordered scaling speeds.
  3. Radiation plus a sum of modulated solitons.

Consequence: Any solution which exists globally in time is bounded.

Remark 1: When the existence time is finite, the limit of the norm exists. It is either divergent or it is bounded. There are no mixed asymptotics. We do not yet have solutions which require more than one bump but we expect that they exist. For 1d NLW similar constructions were made by Cote-Zaag 2011. See also Musso-Pistoia 2006.

Remark 2: Quantifications. Each bump absorbs some energy.

Discussion….no claim of stability. This is not generic.

Ideas of the proof (global case)

The fundamental new ingredient of the proof is the following dispersive property that all radial solutions to NLW (other than 0 and $\pm W$ ) must have: $ \exists ~ R >0,, ~ \eta > 0$ such that for all $ t \geq 0$ or all $ t \leq 0$
$$
\int_{|x| > R + |t|} |\nabla_{x,t} u (x,t)|^2 dx \geq \eta.
$$

 

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