The Nirenberg Trick and Wave Maps

The "Nirenberg trick" or "Nirenberg transformation" is the observation¹ that the semilinear wave equation \begin{equation}\label{eq:semi} - \partial^2_{tt} u + \triangle u = (\partial_t u)^2 - |\nabla u|^2 \end{equation} can be transformed into the linear wave equation by noting that \[ (- \partial^2_{tt} + \triangle) e^{u} = e^{u} \left[ - \partial_{tt}^2 u + \triangle u - (\partial_t u)^2 + |\nabla u|^2\right] .\] Therefore, if $u$ solves $\eqref{eq:semi}$ with initial data \begin{equation} u(0,x) = f(x), \qquad \partial_t u(0,x) = g(x) \end{equation} where $f, g$ are both Schwartz functions, then we can solve for $u$ by

1. Solving the linear wave equation $\Box v = 0$ with initial data $v(0,x) = e^{f(x)} - 1$ and $\partial_t v(0,x) = e^{f(x)} g(x)$ (note that the data are both in Schwartz class also).
2. Reconstructing $u = \ln(v + 1)$.

A few consequences of this procedure are:

• Immediately we see that as long as $v > -1$, the reconstructed function $u$ is smooth.
• Therefore we have automatically small data global well-posedness for the Cauchy problem for $\eqref{eq:semi}$: for all sufficiently small (in various suitable senses) initial data, the corresponding data for $v$ is small, and therefore the solution $v$ to the linear equation have uniform (small) $L^\infty$ bound.
• The singularity for $u$, if it occurs, are due to $u \searrow -\infty$. Furthermore, there seems to be a somewhat canonical way of making sense of the solution past the singularity.

The goal of this post is to make sense of the Nirenberg trick from a geometric perspective.

Wave-maps

Given a Minkowski space $\mathbb{R}^{1,d}$ and a Riemannian manifold $(M,g)$ (where we let $x^i$ with $i = 0, \ldots, d$ be the standard coordinate for $\mathbb{R}^{1,d}$ and $y^A$ for abstract local coordinates for $M$), a wave-map $u: \mathbb{R}^{1,d} \to M$ is a formal critical point of the functional \[ S[u] = \int g(u)_{AB} \partial_i u^A \partial_j u^B m^{ij} ~\mathrm{d}x. \] The symbol $m^{ij}$ is of course the Minkowski metric. Its Euler-Lagrange equation takes the form \begin{equation}\label{eq:genwm} \Box u^A = \Gamma^A_{BC} \partial_i u^B \partial_j u^C m^{ij} \end{equation} where $\Gamma$ is the Christoffel symbols for $(M,g)$ in the coordinates $y^A$. In the case where $(M,g)$ is the Euclidean space (of $N$ dimensions), the Euler-Lagrange equations can be reduced to $N$ decoupled linear wave equations.

Now let us consider the case where $(M,g)$ is a one-dimensional complete Riemannian manifold; this means that either $(M,g)$ is the real line $\mathbb{R}$ with the standard metric or its quotient $\mathbb{R} / \lambda \mathbb{Z}$ which is just a circle with circumference $\lambda$. In the latter case we can equivalently study the wave-map problem by first lifting to the universal cover (which is just $\mathbb{R}$) and then projecting the solution to the circle. Hence we see that in these cases the wave-map equation is identical to the linear wave equation.

However, what if we chose a non-standard coordinate system for $\mathbb{R}$?

Let $I$ be an interval and suppose $y: I \to \mathbb{R}$ is a diffeomorphism to its image. Parametrize $I$ by $z$. The pull-back metric on $I$ is $(y')^2~ \mathrm{d}z^2$. Let's write $\phi = y^{-1}\circ u: \mathbb{R}^{1,d} \to I$. The action functional is then equivalently \[ S[\phi] = \int (y'(\phi))^2 m^{ij} \partial_i \phi \partial_j \phi ~\mathrm{d}x. \] And the corresponding Euler-Lagrange equation is \begin{equation}\label{eq:1dwm} \Box \phi + \frac{y''(\phi)}{y'(\phi)} m^{ij} \partial_i\phi \partial_j\phi = 0. \end{equation}

Observe that $\eqref{eq:semi}$ is merely the special case of $\eqref{eq:1dwm}$ with $y'(\phi) = e^\phi$.

Semilinear equations

In general, we can reverse the derivation of $\eqref{eq:1dwm}$. Given a wave equation \[ \Box \phi + F(\phi) m^{ij} \partial_i \phi \partial_j\phi = 0 \] we have that if we set \[ y(\phi) = \int_0^\phi \exp\left( \int_0^\sigma F(s) ~\mathrm{d}s \right) ~\mathrm{d}\sigma.\] Then the equation can be written in the form of $\eqref{eq:1dwm}$. Therefore, if we rewrite $u = y\circ \phi$ the function $u$ now must solve a linear wave equation.

The singularity that we can observe for $\phi$ now really is merely due to $y(I)$ possibly not covering the entirety of $\mathbb{R}$. In other words, $I$ with the pull-back metric may be an incomplete Riemannian manifold. The singularity is merely just us running to the "end" of the target domain.

Back to geometry

Returning to the wave-maps equation, one sees that the "best" coordinate system is the one obtained from the arc-length parametrization of the one-dimensional manifold $(M,g)$. This can be also generalized to the wave-map equation when $M$ is higher dimensional. Suppose the coordinate system $y^A$ on $M$ is the geodesic normal coordinate system based at some point $p\in M$. The geodesic normal coordinate system is, in many ways, the natural generalization of arc-length parametrization in the one-dimensional case; for general manifolds, however, the coordinate system is only local, covering a neighborhood within the injectivity radius at $p$.

Returning to the general equation $\eqref{eq:genwm}$, in the small data regime we know that in geodesic normal coordinates $p$ is associated to the origin, and $\Gamma^A_{BC}(0) = 0$. Therefore the nominally quadratically nonlinear equation $\eqref{eq:genwm}$ can always be cast into "normal form" with $\Gamma^A_{BC}(u) = O(u)$, and we thus see that the nonlinearity are in fact at least cubic. This "normal form" transformation is nothing more than the statement that any Riemannian manifold is locally Euclidean.

Incidentally, noting that the Riemannian curvature tensor gives the coefficients of the quadratic Taylor expansion of the metric in geodesic normal coordinates, we see that if $p$ is such that $(M,g)$ has vanishing curvature there, then the corresponding wave-maps equation into its neighborhood will be a quartic nonlinear equation.

Now, the normal form implies that the small data Cauchy problem for $\eqref{eq:genwm}$ is automatically globally well-posed when the domain $\mathbb{R}^{1,d}$ has spatial dimension $d\geq 3$. When $d = 2$ that cubic nonlinear equations have small data global well-posedness needs the null structure; I verified this in a recent paper. In the $d = 1$ case the small data global well-posedness follows from a easy generalization² of a result of Kevin Luli, Shiwu Yang, and Pin Yu.