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Semilinear dispersive and wave equations, of which the defocusing nonlinear wave equation

$\displaystyle -\partial_{tt} u + \Delta u = |u|^{p-1} u \ \ \ \ \ (1)$

is a typical example (where ${p>1}$ is a fixed exponent, and ${u: {\bf R}^{1+n} \rightarrow {\bf R}}$ is a scalar field), can be viewed as a “tug of war” between a linear dispersive equation, in this case the linear wave equation

$\displaystyle -\partial_{tt} u + \Delta u = 0 \ \ \ \ \ (2)$

and a nonlinear ODE, in this case the equation

$\displaystyle -\partial_{tt} u = |u|^{p-1} u. \ \ \ \ \ (3)$

If the nonlinear term was not present, leaving only the dispersive equation (2), then as the term “dispersive” suggests, in the asymptotic limit ${t \rightarrow \infty}$, the solution ${u(t,x)}$ would spread out in space and decay in amplitude. For instance, in the model case when ${d=3}$ and the initial position ${u(0,x)}$ vanishes (leaving only the initial velocity ${u_t(0,x)}$ as non-trivial initial data), the solution ${u(t,x)}$ for ${t>0}$ is given by the formula

$\displaystyle u(t,x) = \frac{1}{4\pi t} \int_{|y-x|=t} u_t(0,y)\ d\sigma$

where ${d\sigma}$ is surface measure on the sphere ${\{ y \in {\bf R}^3: |y-x| = t \}}$. (To avoid technical issues, let us restrict attention to classical (smooth) solutions.) Thus, if the initial velocity was bounded and compactly supported, then the solution ${u(t,x)}$ would be bounded by ${O(1/t)}$ and would thus would decay uniformly to zero as ${t \rightarrow \infty}$. Similar phenomena occur for all dimensions greater than ${1}$.

Conversely, if the dispersive term was not present, leaving only the ODE (3), then one no longer expects decay; indeed, given the conserved energy ${\frac{1}{2} u_t^2 + \frac{1}{p+1} |u|^{p+1}}$ for the ODE (3), we do not expect any decay at all (and indeed, solutions are instead periodic in time for each fixed ${x}$, as can easily be seen by viewing the ODE (and the energy curves) in phase space).

Depending on the relative “size” of the dispersive term ${\Delta u}$ and the nonlinear term ${|u|^{p-1} u}$, one can heuristically describe the behaviour of a solution ${u}$ at various positions at times as either being dispersion dominated (in which ${|\Delta u| \gg |u|^p}$), nonlinearity dominated (in which ${|u|^p \gg |\Delta u|}$), or contested (in which ${|\Delta u|}$, ${|u|^p}$ are comparable in size). Very roughly speaking, when one is in the dispersion dominated regime, then perturbation theory becomes effective, and one can often show that the solution to the nonlinear equation indeed behaves like the solution to the linear counterpart, in particular exhibiting decay as ${t \rightarrow \infty}$. In principle, perturbation theory is also available in the nonlinearity dominated regime (in which the dispersion is now viewed as the perturbation, and the nonlinearity as the main term), but in practice this is often difficult to apply (due to the nonlinearity of the approximating equation and the large number of derivatives present in the perturbative term), and so one has to fall back on non-perturbative tools, such as conservation laws and monotonicity formulae. The contested regime is the most interesting, and gives rise to intermediate types of behaviour that are not present in the purely dispersive or purely nonlinear equations, such as solitary wave solutions (solitons) or solutions that blow up in finite time.

In order to analyse how solutions behave in each of these regimes rigorously, one usually works with a variety of function spaces (such as Lebesgue spaces ${L^p}$ and Sobolev spaces ${H^s}$). As such, one generally needs to first establish a number of function space estimates (e.g. Sobolev inequalities, Hölder-type inequalities, Strichartz estimates, etc.) in order to study these equations at the formal level.

Unfortunately, this emphasis on function spaces and their estimates can obscure the underlying physical intuition behind the dynamics of these equations, and the field of analysis of PDE sometimes acquires a reputation for being unduly technical as a consequence. However, as noted in a previous blog post, one can view function space norms as a way to formalise the intuitive notions of the “height” (amplitude) and “width” (wavelength) of a function (wave).

It turns out that one can similarly analyse the behaviour of nonlinear dispersive equations on a similar heuristic level, as that of understanding the dynamics as the amplitude ${A(t)}$ and wavelength ${1/N(t)}$ (or frequency ${N(t)}$) of a wave. Below the fold I give some examples of this heuristic; for sake of concreteness I restrict attention to the nonlinear wave equation (1), though one can of course extend this heuristic to many other models also. Rigorous analogues of the arguments here can be found in several places, such as the book of Shatah and Struwe, or my own book on the subject.

We now set aside our discussion of the finite time extinction results for Ricci flow with surgery (Theorem 4 from Lecture 2), and turn instead to the main portion of Perelman’s argument, which is to establish the global existence result for Ricci flow with surgery (Theorem 2 from Lecture 2), as well as the discreteness of the surgery times (Theorem 3 from Lecture 2).

As mentioned in Lecture 1, local existence of the Ricci flow is a fairly standard application of nonlinear parabolic theory, once one uses de Turck’s trick to transform Ricci flow into an explicitly parabolic equation. The trouble is, of course, that Ricci flow can and does develop singularities (indeed, we have just spent several lectures showing that singularities must inevitably develop when certain topological hypotheses (e.g. simple connectedness) or geometric hypotheses (e.g. positive scalar curvature) occur). In principle, one can use surgery to remove the most singular parts of the manifold at every singularity time and then restart the Ricci flow, but in order to do this one needs some rather precise control on the geometry and topology of these singular regions. (In particular, there are some hypothetical bad singularity scenarios which cannot be easily removed by surgery, due to topological obstructions; a major difficulty in the Perelman program is to show that such scenarios in fact cannot occur in a Ricci flow.)

In order to analyse these singularities, Hamilton and then Perelman employed the standard nonlinear PDE technique of “blowing up” the singularity using the scaling symmetry, and then exploiting as much “compactness” as is available in order to extract an “asymptotic profile” of that singularity from a sequence of such blowups, which had better properties than the original Ricci flow. [The PDE notion of a blowing up a solution around a singularity, by the way, is vaguely analogous to the algebraic geometry notion of blowing up a variety around a singularity, though the two notions are certainly not identical.] A sufficiently good classification of all the possible asymptotic profiles will, in principle, lead to enough structural properties on general singularities to Ricci flow that one can see how to perform surgery in a manner which controls both the geometry and the topology.

However, in order to carry out this program it is necessary to obtain geometric control on the Ricci flow which does not deteriorate when one blows up the solution; in the jargon of nonlinear PDE, we need to obtain bounds on some quantity which is both coercive (it bounds the geometry) and either critical (it is essentially invariant under rescaling) or subcritical (it becomes more powerful when one blows up the solution) with respect to the scaling symmetry. The discovery of controlled quantities for Ricci flow which were simultaneously coercive and critical was Perelman’s first major breakthrough in the subject (previously known controlled quantities were either supercritical or only partially coercive); it made it possible, at least in principle, to analyse general singularities of Ricci flow and thus to begin the surgery program discussed above. (In contrast, the main reason why questions such as Navier-Stokes global regularity are so difficult is that no controlled quantity which is both coercive and critical or subcritical is known.) The mere existence of such a quantity does not by any means establish global existence of Ricci flow with surgery immediately, but it does give one a non-trivial starting point from which one can hope to make progress.