On the quasilinear wave equations in time dependent inhomogeneous media 111This work is part of the author’s Ph.D. thesis at Princeton University.
We consider the problem of small data global existence for quasilinear wave equations with null condition on a class of Lorentzian manifolds with time dependent inhomogeneous metric. We show that sufficiently small data give rise to a unique global solution for metric which is merely close to the Minkowski metric inside some large cylinder and approaches the Minkowski metric weakly as . Based on this result, we give weak but sufficient conditions on a given large solution of quasilinear wave equations such that the solution is globally stable under perturbations of initial data.
In this paper, we study the Cauchy problem for the quasilinear wave equations
on a Lorentzian manifold , where is the covariant wave operator for the metric . The nonlinearities are assumed to satisfy the null condition: , are constants such that , whenever and is at least cubic in terms of , for small , .
The Cauchy problem for nonlinear wave equations with general quadratic nonlinearities on has been studied extensively. In or higher dimensions, the decay rate of the solution to a linear wave equation is sufficient to obtain the small data global existence result, see e.g. , , ,  and reference therein. However, in dimensions, one can only show the almost global existence result , . In fact, in , F. John showed that any nontrivial solution of the equation
with compactly supported initial data blows up in finite time. Nevertheless, a sufficient condition on the quadratic nonlinearities, which guarantees the small data global existence result, is the celebrated null condition introduced by S. Klainerman . Under this condition, D. Christodoulou  and S. Klainerman  independently proved the small data global existence result.
The approach of  used the conformal method, which relies on the conformal embedding of Minkowski space to the Einstein cylinder . S. Klainerman used the vector field method , which connects the symmetries of the flat with the quantitative decay properties of solutions of linear wave equations. The vector fields, used as commutators or multipliers, are the killing and conformal killing vector fields in and can be given explicitly
Based on this vector field method, there have been an extensive literature on generalizations and variants of D. Christodoulou and S. Klainerman’s work, in particular on the multiple speed problems e.g. , ,  and obstacle problems e.g. , , , . All these works used the scaling vector field .
Another application of the vector field method is to the wave equations on a Lorentzian manifold with metric , which may also depend on the solution of the equation. The motivation for studying such problems arises from studying the problem of global nonlinear stability of Minkowski space in wave coordinates. The stability of Minkowski space was first established by Christodoulou-Klainerman by recasting the problem as a system of Bianchi equations for the curvature tensor . Later, Lindblad-Rodnianski  obtained a different proof in wave coordinates, in which the problem was formulated as a system of quasilinear wave equations for the components of the metric perturbation. This is an example that the background metric depends on the solution of the wave equation, where (the Minkowski metric). The quasilinear part of such equations never satisfies the null condition defined in the original work of S. Klainerman , . Nevertheless, besides the global nonlinear stability of Minkowski space aforementioned , the nonlinear wave equations
has been investigated by S. Alinhac .
The linear and nonlinear wave equations on a Lorentzian manifold with given metric have also received considerable attention, in particular on black hole spacetimes. For the linear wave equation on , S. Alinhac  showed that the solution has the decay properties similar to those of a solution of a linear wave equation on Minkowski space provided that the metric approaches the Minkowski metric suitably as . In , D. Tataru proved the local decay of the solution but with the assumption that the background metric is stationary or time independent. For the decay of solution of linear wave equations on Kerr spacetimes (including Schwarzschild spacetimes), we refer the readers to , , ,  and references therein. For the nonlinear equations, J. Luk  proved the small data global existence result for semilinear wave equations with derivatives on slowly rotating Kerr spacetimes. In a recent work , Wang-Yu proved the small data global existence result for quasilinear wave equations on static spacetimes, which are more restrictive than stationary ones.
A common feature of these problems is that the background metric settles down to a stationary metric either by the assumptions (Kerr spacetimes are stationary) or, for the case when the metric depends on the solution, by the assumption and the expected convergence as . The need for such convergence, or at least convergence of the time derivative of the metric to , is dictated by the vector field method. All applications of the vector field method require commutations with some generators of the conformal symmetries of Minkowski space. In particular, we note that all the applications have used the scaling vector field or the conformal killing vector field as commutators. For the problem
the error term coming from the commutation with or would be of the form or which leads to the requirement that is at least bounded and thus the time derivative of the metric decays to as .
To our knowledge, the first work on nonlinear wave equations on time dependent inhomogeneous background is the author’s work . It was shown that if the background metric is merely close to the Minkowski metric inside the cylinder and is identical to the Minkowski metric outside, then the semilinear wave equations with derivatives satisfying the null condition admit small data global solutions. This result relies on a new method for proving the decay of the solutions of linear wave equations developed by Dafermos-Rodnianski in . This new approach is a blend of an integrated local energy inequality and a -weighted energy inequality in a neighborhood of the null infinity, also see applications in , , .
The aim of the present work is to extend the result in  to quasilinear wave equations on time dependent inhomogeneous backgrounds with metric which is not merely a perturbation of the Minkowski metric inside some cylinder but can be a perturbation of the Minkowski metric on the whole spacetime. We show that if the metric is merely close to the Minkowski metric inside some large cylinder with radius and approaches the Minkowski metric outside with some weak rates and if the initial data are sufficiently small, then the solutions of the quasilinear wave equations satisfying the null condition are global in time. In particular, the metric does not necessarily settle down to any particular stationary metric.
Before stating the main theorems, we now introduce some necessary notations. We use the coordinate system of Minkowski space. We may also use the standard polar local coordinate system and the null coordinates
Let denote the induced covariant derivative, the induced Laplacian on the spheres of constant , the angular momentum with components . Here is the partial derivative . We may use to abbreviate . The vector fields, used as commutators, are
We use the convention that Greek indices run from 0 to 3 while the Latin indices run from 1 to 3.
Following the setup in , we now introduce a null frame , which is locally a basis of the tangent space at any point of the Minkowski space for . We let
We then let , be an orthonormal basis of the spheres with constant radius . We use to denote the “good” derivatives
For any symmetric two tensor , relative to the null frame , we have
In our argument, we estimate the decay of the solution with respect to the foliation , defined as follows:
where , . The radius is a to-be-fixed constant. The corresponding energy flux is
We now give the assumptions on the metric . Relative to the coordinates , assume
We assume are given smooth functions satisfying the following conditions
for some small positive constant and some large constant (will be the radius of the foliation . Hence in the sequel the foliation is fixed). will be a small constant depending only on . Here we denote
and is the parameter of the foliation which can be defined as for all .
Without loss of generality (see the Remark 5), we assume that the initial data are smooth and are supported on . The initial energy is defined to be
We see that is uniquely determined by the initial data together with the equation (1).
We have the following small data global existence result for quasilinear wave equations.
Consider the quasilinear wave equation (1) satisfy the null condition. Assume that the background metric satisfy condition (3) for small positive constants , . Assume the initial data are smooth and are support on . Then there exist , depending only on , and , depending on , , , such that for all , , there exists a unique global smooth solution of equation (1) with the following properties
where the constant depends only on , , and also depends on .
We give several remarks
A similar result can be obtained in higher dimensions without null condition.
Inside the cylinder with radius , the null condition on the nonlinearities is not necessary. The nonlinearities can be any quadratic terms of the solution and its derivatives .
As in , the smallness assumption ( appeared in (3)) on the metric inside the cylinder can be replaced by assuming two integrated local energy estimates. When , the small constant in the assumption (3) can be removed as the smallness can be obtained by choosing sufficiently large and shrinking to be .
For simplicity, we merely considered the scalar equations in this paper. However, minor modifications of our approach can also be applied to system of quasilinear wave equations satisfying the null condition.
It is not necessary to require that the initial data have compact support. The general assumption on the initial data can be that the following quantity
is sufficiently small. In particular the constant in the assumptions on the metric can be different from the radius of the support of the initial data. A more general discussion on the initial data will appear in the author’s forthcoming paper .
We remark here that the special case when the metric approaches the Minkowski metric in the spatial directions with a rate has been discussed in the recent work . But in that work there is an extra condition that the metric is static and is independent of time .
We now apply the above result to the problem of global stability of solutions to quasilinear wave equations initiated by S. Alinhac in . He studied the quasilinear wave equations
on Minkowski space, where are constants satisfying the null condition. Suppose is a smooth global solution of the above equation when . He showed that if satisfies the condition
for some positive constants and , then the solution of the above quasilinear wave equation (5) exists globally in time for all sufficiently small . Here denotes the collection of vector fields given in line (2) except the conformal killing vector field .
We give weaker conditions than (6) on the solution to guarantee the global stability. We assume the initial data are smooth and are supported on for some large constant . Let be a smooth solution of (5) when . Before some large time , we assume the metric is hyperbolic and
for some constant . After time , we assume satisfies the following weak decay estimates
where is chosen to be radius of the foliation . We let be the initial energy for defined in (4).
We have the following global stability of solutions to quasilinear wave equations.
Assume the constants in (5) satisfy the null condition. Assume the initial data are smooth and are supported on for some large constant . Let be a smooth solution of (5) when satisfying the conditions (7), (8). Then there exist two small positive constants , depending on , and , depending on , , , such that for all , , there exists a unique global smooth solution of equation (5) with the property that for the foliation with radius , the difference satisfies the same estimates as given in Theorem 1 but with the constant depending on , , , .
The problem of global stability of solutions to semilinear wave equations has been discussed in .
Compared to the condition (6) imposed in , we do not require the given solution to decay in time uniformly. In fact, we can even allow to be independent of in the cylinder . Moreover, since the initial data are supported on , the finite speed of propagation for wave equations shows that the solution vanishes when . Hence condition (6) implies (8). Finally, the collection of vector fields used in the condition (8) is a subset of the collection in (6). In particular, we avoid the use of the scaling vector field or the Lorentz rotations which grow in time .
Our argument relies on a new method developed by Dafermos-Rodnianski in . Based on an integrated local energy inequality, which is usually obtained by using the vector fields , , where is some appropriate function of , as multipliers, and a -weighted energy inequality in a neighborhood of the null infinity, the new approach leads to the decay, in particular, of the energy flux for solutions of linear wave equations. The integrated local energy inequality has been well studied on various backgrounds, including black hole spacetimes, see e.g.  and references mentioned above. When the metric is flat in a neighborhood of the null infinity, the -weighted energy inequality can be derived by multiplying the equation with and then integrating by parts. This is the situation in  as there the metric is identical to the Minkowski metric when . For the backgrounds considered in this paper, the metric is merely asymptotically flat in the spatial directions. As mentioned in the original work  of Dafermos-Rodnianski, a much more flexible and robust way to derive the -weighted energy inequality is to use the vector fields as multipliers. However, these vector fields can not be applied directly to asymptotically flat backgrounds. We may need to modify the vector fields as
see details in Section 3.2.
Nevertheless, for the general metrics in this paper, there is another difficulty arising from the error terms on the boundary . Those error terms can not be controlled without losing any derivatives. We hence are not able to show the decay of the energy flux as we did in . However, using the boundedness of the integrated energy on the whole spacetime together with a pigeon hole argument, we still can show the decay of the integrated energy on the region bounded by and , see details in Section 4. The pointwise decay of the solution then follows by commuting the equation with the vector fields , .
The plan of this paper is as follows: we will review the energy method for wave equations and define the notations in Section 2. In Section 3, we establish an integrated local energy inequality on the region bounded by and and two -weighted energy inequalities. In Section 4, we show the decay of the integrated energy for solutions of linear wave equations. In the last section, we use bootstrap argument to prove the main theorems.
Acknowledgements The author is deeply indebted to his advisor Igor Rodnianski for his continuous support on this problem. He thanks Igor Rodnianski for sharing numerous valuable thoughts. The author would also like to thank Beijing International Center for Mathematical Research for the wonderful hospitality when he was visiting there and part of this work was carried out there.
2 Preliminaries and Energy Method
Given any Lorentzian metric
on , we let denotes the components of the inverse of the metric . Throughout this paper, we let , be any vector fields in and be any vector fields in . Relative to the null frame, the metric components are . The inverse is . We denote
At any fixed point , we may choose , such that
This helps to compute those geometric quantities which are independent of the choice of the local coordinates. Denote the incoming null hypersurface
We simply use to denote the future null infinity (part of) where . Define the energy flux through the null infinity as the limit infimum of the energy flux through as , that is,
We define the modified energy flux
We now review the energy method for wave equations. For a Lorentzian space with metric , we denote the volume form. In the local coordinate system , we have
Here we have chosen to be the time orientation for the Lorentzian space . We recall the energy-momentum tensor of the scalar field on the Lorentzian space with metric
Throughout this paper, we raise and lower indices of any tensor relative to the given metric , e.g., . Given a vector field , we define the currents
where is the deformation tensor of the vector field . We denote as the vector field
where is the covariant wave operator and is the covariant derivative of the metric .
Take any function . We have the following identity
Modify the vector field to be
We then have the identity
For any bounded region in , using Stoke’s formula, we have the following energy identity
denotes the boundary of the domain and denotes the contraction of the volume form with the vector field which gives the surface measure of the boundary. For example, for any basis , we have . Here we have chosen to be the time orientation. For more details on this formula, we refer to the appendix of .
Throughout this paper, the domain will be regular regions bounded by the -constant slices, the outgoing null hypersurfaces or the incoming null hypersurfaces . We now compute on these three kinds of hypersurfaces. We now compute on or on the -constant incoming null hypersurfaces (with respect to the Minkowski metric). We have the following three cases.
On , the surface measure is a function times . Recall the volume form
Here note that is a -form. We thus can show that
On the null hypersurface with respect to the Minkowski metric, we can write the volume form
Here , are the null coordinates. Notice that . We can compute
Similarly, on the -constant incoming null hypersurfaces , we have
We remark here that the above three formulae hold for any vector field and any function .
The following several lemmas, which have been proven in , will be used later on.
Let be a smooth function on . Assume
Then in the polar coordinates , we have
Moreover, if , then
For solutions of linear wave equations, the good derivative of the solution decays better. In that case, we have
Let . Assume satisfies the condition in Lemma 1. Then we have
where is a constant depending only on , .
Let . By Lemma 1, we have
Multiply the above inequality by and then integrate from to infinity. We obtain
We will also frequently use the following simple lemma.
Suppose is smooth. Then we have the identity
Integration by parts gives the lemma. ∎
We also need the following analogue of Hardy’s inequality to control by the energy.
Let satisfy the same conditions as in the previous lemma. Then
Here we simply use to denote the integral region .
By Lemma 1, if is finite, then all the above statements hold if we replace with .
Finally, for , we define several notations:
where . Similarly, we have the notation for . We remark here that this notation is different from which is the energy flux through the null infinity.
3 Weighted Energy Estimates
Our approach relies on two estimates: integrated local energy inequality and -weighted energy inequality. In this section, we use the multiplier method to establish an integrated energy inequality and two -weighted energy inequalities for quasilinear wave equations. The integrated energy inequality was first proven by C. Morawetz in . We follow the method developed in  to obtain the integrated energy inequality here. In , Dafermos-Rodnianski introduced the -weighted energy inequalities in a neighborhood of null infinity. These estimates have been established in  for semilinear wave equation. As mentioned in the original work of Dafermos-Rodnianski, we can use the robust multiplier method to show the -weighted energy inequality on general backgrounds.
In this section, we prove a general integrated energy inequality for solutions of the linear wave equations
on the Lorentzian manifold . Here is a linear term and is a vector field on with components .
Fix a large constant so that we can determine the foliation with radius . Recall that . We assume , satisfy the following weak decay estimates