Energy Estimates for Quasilinear Wave Systems

Energy estimates control a quasilinear wave solution without losing derivatives. The top-order commutator contains one derivative of the metric coefficient and one top-order derivative of the unknown; Sobolev–Moser estimates close this term when the metric is controlled in $H^s$, $s>5/2$. Successive iterates contract in a lower norm, while a uniform top-order bound preserves the desired regularity.

Required background. Analysis for Mathematical Relativity supplies commutator and Grönwall estimates, The Einstein Equation supplies the nonlinear coefficients, and PDEs and Green Functions for Gravitation supplies wave energies.

Helpful background. Causality and Global Structure supplies the geometry of spacelike energy surfaces.

Hypotheses for the estimate

Assume the slices carry induced metrics uniformly equivalent to a fixed reference metric, the lapse is bounded above and away from zero, and the shift is controlled in the same coefficient class. Assume also that the wave coefficients are symmetric, their first derivatives lie in $L^1_tL^\infty_x$, and the source has the Sobolev regularity appearing on the right-hand side of the estimate. On a noncompact slice, require either compactly supported fields or decay eliminating the flux at infinity. On a bounded domain, prescribe boundary conditions for which the incoming energy flux is controlled.

These hypotheses do two separate jobs. Uniform hyperbolicity makes the energy positive and equivalent to the desired Sobolev norm. Coefficient-derivative control bounds the error created by differentiating that energy. If either fails, the displayed inequality need not hold even though every frozen coefficient matrix has real characteristic speeds.

The basic variable-coefficient energy

Consider a scalar model

$$g^{\alpha\beta}(t,x)\partial_\alpha\partial_\beta u=f,$$

where $g$ is Lorentzian and $t=\mathrm{constant}$ is uniformly spacelike. On a slice $\Sigma_t$, an energy equivalent to the first-derivative norm is

$$E_1(t) \simeq \int_{\Sigma_t} \left(|\partial_tu|^2+h^{ij}\partial_i u\partial_j u\right)d\mu_h.$$

Multiply the equation by a future timelike derivative of $u$ and integrate by parts. For compact support, a compact slice, or boundary conditions controlling the flux, one obtains

$$\frac{dE_1}{dt} \le C\|\partial g\|_{L^\infty}E_1 +C\|f\|_{L^2}E_1^{1/2}.$$

The derivative of the coefficient appears because integration by parts differentiates $g^{\alpha\beta}$. Thus uniform Lorentzian bounds on $g$ are not enough; first coefficient derivatives must be integrable in time. Grönwall’s inequality then gives

$$E_1(t)^{1/2} \le e^{C\int_0^t\|\partial g\|_\infty ds} \left(E_1(0)^{1/2} +C\int_0^t\|f(s)\|_2ds\right).$$

Ringström develops these hyperbolic estimates and their geometric use in Ringström 2009, Chs. 3–12, pp. 21–144.

Commuting derivatives

First take integer $s=m\ge3$, and let $D^I$ be a spatial derivative of order $|I|\le m-1$ in a fixed chart. Commuting it through the principal wave operator gives

$$g^{\alpha\beta}\partial_\alpha\partial_\beta D^Iu =D^If+[g^{\alpha\beta}\partial_\alpha\partial_\beta,D^I]u.$$

At the top spatial order, its terms have the form

$$(D^Jg^{\alpha\beta})\, \partial_\alpha\partial_\beta D^{I-J}u, \qquad 1\le |J|\le m-1.$$

The coefficient carries at most $m-1$ spatial derivatives, while the field carries at most $m$ total derivatives. Time derivatives must also be counted: mixed derivatives use $\partial_tu\in H^{m-1}$, and $\partial_t^2u$ is eliminated using the wave equation and its source. The relevant norm is therefore

$$\mathcal N_s(u) =\|u\|_{H^s}+\|\partial_tu\|_{H^{s-1}},$$

not a spatial norm of $u$ alone. For example, at $m=3$ the product $D^2g\,D^2u$ is bounded in $L^2$ by putting both factors in $L^4$ and using $H^1(\mathbb R^3)\hookrightarrow L^4$. It is unnecessary and generally false at this regularity to demand that a second derivative lie in $L^\infty$. Terms with one derivative of $g$ instead use its $L^\infty$ bound. Product and commutator estimates interpolate between these cases; fractional $s\gt5/2$ uses a fractional derivative operator rather than a nonexistent multi-index of fractional order.

For Einstein’s reduced system, the inverse metric is a smooth function of the unknown metric and the source is quadratic in first derivatives. Under uniform hyperbolicity, these estimates yield the schematic nonlinear inequality

$$\frac{dE_s}{dt} \le C(E_s^{1/2})E_s.$$

A continuity argument chooses $T$ so that $E_s$ remains within a bootstrap bound. Friedrich and Rendall describe this quasilinear closure in Friedrich and Rendall 2000, §§2–4.

Iteration without derivative loss

Let $u^{(m+1)}$ solve the linear equation with coefficients $g(u^{(m)})$. Uniform top-order estimates bound every iterate in $H^s$. For the difference $w^{(m)}=u^{(m+1)}-u^{(m)}$, subtract two successive equations. A representative source is

$$\bigl(g(u^{(m)})-g(u^{(m-1)})\bigr) \partial^2u^{(m)}.$$

Estimating this source at full $H^{s-1}$ would demand one derivative too many. Instead, estimate $w^{(m)}$ in the one-lower energy corresponding to $H^{s-1}\times H^{s-2}$. The uniform $H^s$ bound controls $\partial^2u^{(m)}$ in the required lower product norm, and the coefficient difference is Lipschitz in $w^{(m-1)}$. For sufficiently small $T$,

$$\|w^{(m)}\|_{X_{s-1,T}} \le q\|w^{(m-1)}\|_{X_{s-1,T}}, \qquad 1>q>0.$$

Here $\|u\|_{X_{r,T}}=\sup_{0\le t\le T}(\|u(t)\|_{H^r}+\|\partial_tu(t)\|_{H^{r-1}})$. Convergence in this lower norm plus weak compactness and the uniform top-order bound yields the candidate $H^s$ solution. A smoothing and energy argument then gives strong continuity in the top norm and continuous dependence. This is controlled loss in the iteration estimate, not loss of regularity in the solution.

Closing the nonlinear bootstrap

To make the lifespan uniform, choose a bootstrap bound $E_s(t)\le 4E_s(0)$ and restrict to metrics whose hyperbolicity constants remain in a compact admissible set. The energy inequality then bounds the coefficient $C(E_s^{1/2})$ by a number depending only on the initial controlling norm. Integrating for sufficiently small $T$ improves the assumed bound to, for example, $E_s(t)\le2E_s(0)$. Continuity closes the bootstrap.

The same argument must control the inverse metric. Since inversion is a smooth map on uniformly nondegenerate symmetric matrices, Sobolev–Moser composition estimates bound $g^{-1}$ in $H^s$ as long as the eigenvalues stay away from zero. Choosing $T$ small and using $C^0$ continuity preserves Lorentzian signature and spacelikeness of the slices. Thus hyperbolicity is propagated as part of the bootstrap rather than assumed for an unknown future solution.

For uniqueness, subtract two solutions $u$ and $v$. Their difference satisfies a linear wave equation with coefficients from one solution and a source containing $g(u)-g(v)$. A lower-order energy estimate and the local Lipschitz property of the nonlinearities give

$$\|u-v\|_{X_{s-1,t}} \le C\|u(0)-v(0)\|_{H^{s-1}\times H^{s-2}} +C\int_0^t\|u-v\|_{X_{s-1,\tau}}d\tau.$$

Identical data imply equality by Grönwall. Continuous dependence at the top norm requires the standard approximation argument because the direct difference estimate is one derivative lower. Stating only the formal top-order inequality would omit this important part of well-posedness.

A simple counterexample diagnoses the coefficient hypothesis. Let a wave speed approach zero at finite time. The algebraic energy can lose equivalence to $|\nabla u|^2$ even before its numerical value diverges. Bounded energy in that degenerating form is not sufficient for continuation because uniform hyperbolicity has failed.

Derivative bookkeeping for the metric system

For Einstein evolution the unknown is tensor-valued, but the same bookkeeping applies component by component after wave-gauge reduction. The quadratic source contains first metric derivatives, while the inverse metric supplies the principal coefficients. At top order, one never estimates an undifferentiated second derivative of the coefficient independently of the solution: it is included in the same metric energy. Commutators are organized so that the single factor placed in a pointwise norm carries fewer derivatives and the factor in the square-integrable norm carries the top derivative.

Localization is needed on a noncompact manifold. Choose a locally finite atlas and partition of unity with controlled coordinate geometry. Commuting the cutoffs through the wave operator adds lower-order sources supported where neighboring charts overlap. Finite propagation speed permits a local estimate to depend only on data in a slightly larger causal neighborhood. Uniform chart and partition bounds then combine the estimates into one lifespan. If those bounds deteriorate toward infinity, pointwise local existence does not automatically produce a uniform global development interval.

Smooth approximation closes another gap. Derive the estimates first for smooth data, where every integration by parts is justified. Approximate the $H^s\times H^{s-1}$ data by smooth data with uniformly bounded norms, solve the smooth problems on the common bootstrap interval, and use the lower-order difference estimate to pass to a limit. Weak compactness preserves the top bound, while strong lower-order convergence identifies nonlinear products. This procedure explains why the energy inequality must be uniform in the smoothing parameter.

Finally, the Einstein data include constraints. The energy argument does not project arbitrary metric data onto the constraint surface. It assumes constraint-compatible initial data, evolves the reduced equations, and relies on a separate subsidiary estimate to show that the solution remains on that surface. Keeping these estimates separate prevents circular reasoning: the metric energy proves existence, while the constraint energy proves equivalence to the full geometric system.

Limitations

The estimate presumes controlled boundary flux. On a timelike boundary, a sign-indefinite incoming flux can destroy the energy inequality unless boundary conditions select incoming characteristic fields appropriately. At lower regularity, $\partial g$ may not lie in $L^1_tL^\infty_x$ and the direct estimate fails even if a solution might exist by a more geometric method. Finally, an energy estimate for the reduced equations says nothing about the constraints until the subsidiary system is included.

Check your understanding

For $\partial_t^2u-a(t,x)^2\Delta u=0$, where does $\|\partial a\|_{L^\infty}$ enter the energy inequality?

Solution

Assume $0\lt a_*\le a\le a^*$ and sufficient decay. With $E=\tfrac12\int[(\partial_tu)^2+a^2|\nabla u|^2]dx$, differentiation and one integration by parts give the exact identity

$$E'(t)=\int a\,a_t|\nabla u|^2dx -2\int a\,u_t\nabla a\cdot\nabla u\,dx.$$

The first term comes from the changing energy density; the second comes from the spatially variable wave speed. Their absolute values are bounded by $C(a_*,a^*)\|\partial a\|_\infty E$. If $a$ is constant, both vanish and the flat wave energy is conserved. If $a$ is time independent but spatially variable, the second term remains for this nondivergence-form equation; dropping it would give a false conservation law.

References

• Friedrich, Helmut, and Alan D. Rendall. “The Cauchy Problem for the Einstein Equations.” In Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540, 127–224. Springer, 2000. arXiv:gr-qc/0002074.
• Ringström, Hans. The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, 2009. Publisher record.