You are currently browsing the tag archive for the ‘nonlinear wave equations’ tag.

I have just uploaded to the arXiv the third installment of my “heatwave” project, entitled “Global regularity of wave maps V.  Large data local well-posedness in the energy class“. This (rather technical) paper establishes another of the key ingredients necessary to establish the global existence of smooth wave maps from 2+1-dimensional spacetime ${\Bbb R}^{1+2}$ to hyperbolic space $\mathbf{H} = \mathbf{H}^m$.  Specifically, a large data local well-posedness result is established, constructing a local solution from any initial data with finite (but possibly quite large) energy, and furthermore that the solution depends continuously on the initial data in the energy topology.  (This topology was constructed in my previous paper.)  Once one has this result, the only remaining task is to show a “Palais-Smale property” for wave maps, in that if singularities form in the wave maps equation, then there exists a non-trivial minimal-energy blowup solution, whose orbit is almost periodic modulo the symmetries of the equation.  I anticipate this to the most difficult component of the whole project, and is the subject of the fourth (and hopefully final) installment of this series.

This local result is closely related to the small energy global regularity theory developed in recent years by myself, by Krieger, and by Tataru.  In particular, the complicated function spaces used in that paper (which ultimately originate from a precursor paper of Tataru).  The main new difficulties here are to extend the small energy theory to large energy (by localising time suitably), and to establish continuous dependence on the data (i.e. two solutions which are initially close in the energy topology, need to stay close in that topology).  The former difficulty is in principle manageable by exploiting finite speed of propagation (exploiting the fact (arising from the monotone convergence theorem) that large energy data becomes small energy data at sufficiently small spatial scales), but for technical reasons (having to do with my choice of gauge) I was not able to do this and had to deal with the large energy case directly (and in any case, a genuinely large energy theory is going to be needed to construct the minimal energy blowup solution in the next paper).  The latter difficulty is in principle resolvable by adapting the existence theory to differences of solutions, rather than to individual solutions, but the nonlinear choice of gauge adds a rather tedious amount of complexity to the task of making this rigorous.  (It may be that simpler gauges, such as the Coulomb gauge, may be usable here, at least in the case $m=2$ of the hyperbolic plane (cf. the work of Krieger), but such gauges cause additional analytic problems as they do not renormalise the nonlinearity as strongly as the caloric gauge.  The paper of Tataru establishes these goals, but assumes an isometric embedding of the target manifold into a Euclidean space, which is unfortunately not available for hyperbolic space targets.)

The main technical difficulty that had to be overcome in the paper was that there were two different time variables t, s (one for the wave maps equation and one for the heat flow), and three types of PDE (hyperbolic, parabolic, and ODE) that one has to solve forward in t, forward in s, and backwards in s respectively.  In order to close the argument in the large energy case, this necessitated a rather complicated iteration-type scheme, in which one solved for the caloric gauge, established parabolic regularity estimates for that gauge, propagated a “wave-tension field” by the heat flow, and then solved a wave maps type equation using that field as a forcing term.  The argument can eventually be closed using mostly “off-the-shelf” function space estimates from previous papers, but is remarkably lengthy, especially when analysing differences of two solutions.  (One drawback of using off-the-shelf estimates, though, is that one does not get particularly good control of the solution over extended periods of time; in particular, the spaces used here cannot detect the decay of the solution over extended periods of time (unlike, say, Strichartz spaces $L^q_t L^r_x$ for $q < \infty$) and so will not be able to supply the long-time perturbation theory that will be needed in the next paper in this series.  I believe I know how to re-engineer these spaces to achieve this, though, and the details should follow in the forthcoming paper.)

Today I’d like to discuss (part of) a cute and surprising theorem of Fritz John in the area of non-linear wave equations, and specifically for the equation

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

where $u: {\Bbb R} \times {\Bbb R}^3 \to {\Bbb R}$ is a scalar function of one time and three spatial dimensions.

The evolution of this type of non-linear wave equation can be viewed as a “race” between the dispersive tendency of the linear wave equation

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

and the positive feedback tendencies of the nonlinear ODE

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

More precisely, solutions to (2) tend to decay in time as $t \to +\infty$, as can be seen from the presence of the $\frac{1}{t}$ term in the explicit formula

$u(t,x) = \frac{1}{4\pi t} \int_{|y-x|=t} \partial_t u(0,y)\ dS(y) + \partial_t[\frac{1}{4\pi t} \int_{|y-x|=t} u(0,y)\ dS(y)],$ (4)

for such solutions in terms of the initial position $u(0,y)$ and initial velocity $\partial_t u(0,y)$, where $t > 0$, $x \in {\Bbb R}^3$, and dS is the area element of the sphere $\{ y \in {\Bbb R}^3: |y-x|=t \}$. (For this post I will ignore the technical issues regarding how smooth the solution has to be in order for the above formula to be valid.) On the other hand, solutions to (3) tend to blow up in finite time from data with positive initial position and initial velocity, even if this data is very small, as can be seen by the family of solutions

$u_T(t,x) := c (T-t)^{-2/(p-1)}$

for $T > 0$, $0 < t < T$, and $x \in {\Bbb R}^3$, where c is the positive constant $c := (\frac{2(p+1)}{(p-1)^2})^{1/(p-1)}$. For T large, this gives a family of solutions which starts out very small at time zero, but still manages to go to infinity in finite time.

The equation (1) can be viewed as a combination of equations (2) and (3) and should thus inherit a mix of the behaviours of both its “parents”. As a general rule, when the initial data $u(0,\cdot), \partial_t u(0,\cdot)$ of solution is small, one expects the dispersion to “win” and send the solution to zero as $t \to \infty$, because the nonlinear effects are weak; conversely, when the initial data is large, one expects the nonlinear effects to “win” and cause blowup, or at least large amounts of instability. This division is particularly pronounced when p is large (since then the nonlinearity is very strong for large data and very weak for small data), but not so much for p small (for instance, when p=1, the equation becomes essentially linear, and one can easily show that blowup does not occur from reasonable data.)

The theorem of John formalises this intuition, with a remarkable threshold value for p:

Theorem. Let $1 < p < \infty$.

1. If $p < 1+\sqrt{2}$, then there exist solutions which are arbitrarily small (both in size and in support) and smooth at time zero, but which blow up in finite time.
2. If $p > 1+\sqrt{2}$, then for every initial data which is sufficiently small in size and support, and sufficiently smooth, one has a global solution (which goes to zero uniformly as $t \to \infty$).

[At the critical threshold $p = 1 + \sqrt{2}$ one also has blowup from arbitrarily small data, as was shown subsequently by Schaeffer.]

The ostensible purpose of this post is to try to explain why the curious exponent $1+\sqrt{2}$ should make an appearance here, by sketching out the proof of part 1 of John’s theorem (I will not discuss part 2 here); but another reason I am writing this post is to illustrate how to make quick “back-of-the-envelope” calculations in harmonic analysis and PDE which can obtain the correct numerology for such a problem much faster than a fully rigorous approach. These calculations can be a little tricky to handle properly at first, but with practice they can be done very swiftly.