Modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition

The equation (1.1) is a special case for a general scalar nonlinear wave equation in $\mathbb{R}^{1+3}_{t,x}$

Here

The sum in (1.5) is taken over all multiindices $\alpha,\beta$ with $|\alpha|\leq|\beta|\leq 2$, $|\beta|\geq 1$ and $|\alpha|+|\beta|\leq 3$.

Since 1980s, there have been many results on the lifespan of the solutions to the Cauchy problem (1.4) with initial data (1.2). In [john, john2], John proved that (1.4) does not necessarily have a global solution for all $t\geq 0$: any nontrivial solution to $\square u=u_{t}\Delta u$ or $\square u=u_{t}^{2}$ blows up in finite time. In contrast, (1.4) in $\mathbb{R}^{1+d}$ for $d\geq 4$ has small data global existence, proved by Hörmander [horm3]. For arbitrary nonlinearities in three space dimensions, the best result on the lifespan is the almost global existence: the solution exists for $t\leq e^{c/\varepsilon}$, for sufficiently small $\varepsilon$ and some constant $c>0$. The almost global existence for (1.4) was proved by Lindblad [lind3]. We also refer to John and Klainerman [johnklai], Klainerman [klai], and Hörmander [horm2, horm] for some earlier work on almost global existence.

In contrast to the finite-time blowup in John’s examples, it was proved by Klainerman [klai3] and by Christodoulou [chri] that if the null condition is satisfied, then (1.4) has small data global existence. The null condition was first introduced by Klainerman [klai2]. It states that for each $0\leq m\leq n\leq 2$ with $m+n\leq 3$, we have

Note that the null condition is sufficient but not necessary for the small data global existence. For example, the null condition fails for (1.1) in general, but (1.1) still has small data global existence.

Later, in [lindrodn, lindrodn2], Lindblad and Rodnianski introduced the weak null condition. To state the weak null condition, we start with the asymptotic equations first introduced by Hörmander in [horm2, horm, horm3]. We make the ansatz

Plug this ansatz into (1.4) and we can derive the following asymptotic PDE for $U(s,q,\omega)$

Here $A_{mn}$ is defined in (1.6) and the sum is taken over $0\leq m\leq n\leq 2$ with $m+n\leq 3$. We say that the weak null condition is satisfied if (1.8) has a global solution for all $s\geq 0$ and if the solution and all its derivatives grow at most exponentially in $s$, provided that the initial data decay sufficiently fast in $q$. In the same papers, Lindblad and Rodnianski made a conjecture that the weak null condition is sufficient for small data global existence. To the best of the author’s knowledge, this conjecture remains open until today.

There are three remarks about the weak null condition and the corresponding conjecture. First, the weak null condition is weaker than the null condition. In fact, if the null condition is satisfied, then (1.8) becomes $\partial_{s}\partial_{q}U=0$. Second, though the conjecture remains open, there are many examples of (1.4) satisfying the weak null condition and admitting small data global existence at the same time. The equation (1.1) is one of several such examples: the small data global existence for (1.1) has been proved by Lindblad [lind]; meanwhile, the asymptotic equation (1.8) now becomes

where

whose solutions exist globally in $s$ and satisfy the decay requirements, so (1.1) satisfies the weak null condition. There are also many examples violating the weak null condition and admitting finite-time blowup at the same time. Two of such examples are $\square u=u_{t}\Delta u$ and $\square u=u_{t}^{2}$: the corresponding asymptotic equations are $(2\partial_{s}-U_{q}\partial_{q})U_{q}=0$ (Burger’s equation) and $\partial_{s}U_{q}=U_{q}^{2}$, respectively, whose solutions are known to blow up in finite time. Third, in the recent years, Keir has made important progress. In [keir], he proved the small data global existence for a large class of quasilinear wave equations satisfying the weak null condition, significantly enlarging upon the class of equations for which global existence is known. His proof also applies to (1.1). In [keir2], he proved that if the solutions to the asymptotic system are bounded (given small initial data) and stable against rapidly decaying perturbations, then the corresponding system of nonlinear wave equations admits small data global existence.

Instead of working with Hörmander’s asymptotic system (1.9) directly, in this paper we will construct a new system of asymptotic equations. Our analysis starts as in Hörmander’s derivation in [horm2, horm, horm3], but diverges at a key point: the choice of $q$ is different. One may contend from the paper that this new system is more accurate than (1.9), in that it both describes the long time evolution and contains full information about it. In addition, if we choose the initial data appropriately, our reduced system will reduce to linear first order ODEs on $\mu$ and $U_{q}$, so it is easier to solve it than to solve (1.9).

To derive the new equations, we still make the ansatz (1.7), but now we replace $q=r-t$ with a solution $q(t,r,\omega)$ to the eikonal equation related to (1.1)

In other words, $q(t,r,\omega)$ is an optical function. There are two reasons why we choose $q$ in this way. First, if we plug $u=\varepsilon r^{-1}U(s,q,\omega)$ in (1.1) where $q(t,r,\omega)$ is an arbitrary function, we get two terms in the expansion

All the other terms either decay faster than $\varepsilon^{2}r^{-2}$ for $t\approx r\to\infty$, or do not contain $U$ itself (but may contain $U_{q},U_{qq},U_{sq}$ and etc.). If $q$ satisfies the eikonal equation, then the second term vanishes. From the eikonal equation, we can also prove that the first term is approximately equal to a function depending on $U_{q}$ but not on $U$. Thus, in contrast to the second-order PDE (1.9) for $U$, we expect to get a first-order ODE for $U_{q}$ which is simpler.

Second, the eikonal equations have been used in the previous works on the small data global existence for (1.1). In [alin2], Alinhac followed the method used in Christodoulou and Klainerman [chriklai], and adapted the vector fields to the characteristic surfaces, i.e. the level surfaces of solutions to the eikonal equations. In [lind], Lindblad considered the radial eikonal equations when he derived the pointwise bounds of solutions to (1.1). When they derived the energy estimates, both Alinhac and Lindblad considered a weight $w(q)$ where $q$ is an approximate solution to the eikonal equation. Their works suggest that the eikonal equation plays an important role when we study the long time behavior of solutions to (1.1).

Since $u$ is unknown, it is difficult to solve (1.10) directly. Instead, we introduce a new auxiliary function $\mu=\mu(s,q,\omega)$ such that $q_{t}-q_{r}=\mu$. From (1.10), we can express $q_{t}+q_{r}$ in terms of $\mu$ and $U$, and then solve for all partial derivatives of $q$, assuming that all the angular derivatives are negligible. Then from (1.1), we can derive the following asymptotic equations for $\mu(s,q,\omega)$ and $U(s,q,\omega)$:

The derivation of these two equations is given in Section 3.

To solve (1.11), we need to assign the initial data at $s=0$. To choose $\mu|_{s=0}$, we use the gauge freedom. Note that if $q_{t}-q_{r}=\mu$ and if $\widetilde{q}=F(q,\omega)$, then we have $\widetilde{q}_{t}-\widetilde{q}_{r}=(\partial_{q}F)\mu$. Thus, by choosing the function $F$ appropriately, we can prescribe $\mu|_{s=0}$ freely. We set $\mu|_{s=0}=-2$ since we expect $q\approx r-t$. The initial data of $U_{q}$ can be chosen arbitrarily, so we set $U_{q}(0,q,\omega)=A(q,\omega)$ for an arbitrary $A(q,\omega)\in C_{c}^{\infty}(\mathbb{R}\times\mathbb{S}^{2})$. Here $A(q,\omega)$ is defined as the scattering data for our result. Note that (1.11) implies that $\partial_{s}(\mu U_{q})=0$, so $(\mu U_{q})(s,q,\omega)=-2A(q,\omega)$. Then, (1.11) is reduced to a linear first order ODE system

To uniquely solve $U$ from $U_{q}$, we also assume that $\lim_{q\to-\infty}U(s,q,\omega)=0$. Now we obtain an explicit solution $(\mu,U)$ to our reduced system (1.11).

To construct an approximate solution, we make a change of coordinates. For a small $\varepsilon>0$, we set $s=\varepsilon\ln(t)-\delta$, where $\delta>0$ is a sufficiently small constant to be chosen. We remark that this choice of $s$ is related to the almost global existence, since now $s=0$ if and only if $t=e^{\delta/\varepsilon}$. In fact, when $t\leq e^{\delta/\varepsilon}$, we expect the solution to (1.1) behaves as a solution to $\Box u=0$, so our asymptotic equations play a role only when $t\geq e^{\delta/\varepsilon}$. Let $q(t,r,\omega)$ be the solution to

We can use the method of characteristics to solve this equation. Then, any function of $(s,q,\omega)$ induces a new function of $(t,r,\omega)$. With an abuse of notation, we set

Here $U(t,r,\omega)$ is the asymptotic profile. We can prove that, near the light cone $\{t=r\}$, $\varepsilon r^{-1}U(t,r,\omega)$ is an approximate solution to (1.1), and $q(t,r,\omega)$ is an approximate optical function, i.e. an approximate solution to the eikonal equation corresponding with the metric $\widetilde{g}^{\alpha\beta}(\varepsilon r^{-1}U)$. See Section 4 for the explicit formulas and the estimates for $q$ and $U$.

Given the asymptotic equations (1.11), we can ask the following two questions. First, given a scattering data $A(q,\omega)$, can we use (1.11) to construct an exact solution to (1.1) which has this scattering data at infinite time? Second, as time goes to infinity, can any small global solution to the Cauchy problem (1.1) with (1.2) be well approximated by a solution to our reduced system (1.11)? For example, can we recover the scattering data $A(q,\omega)$, approximate optical function $q(t,r,\omega)$ and asymptotic profile $U(t,r,\omega)$ from an exact solution? In scattering theory, the first problem is the existence of the (modified) wave operators, and the second one is asymptotic completeness. We remark that these two questions have also been formulated and studied for many other nonlinear PDEs. For example, in the setting of nonlinear Schrödinger equations, the existence of wave operators and asymptotic completeness have been formulated in many texts such as [ginivelo]. Scattering theory for NLS has also been studied; we refer to [tao, caze] for a collection of such results.

For the equation (1.1), there are some previous results on these two questions. In [lindschl], Lindblad and Schlue proved the existence of the wave operators for the semilinear models of Einstein’s equations. In [dengpusa], Deng and Pusateri used the original Hörmander’s asymptotic system (1.9) to prove a partial scattering result for (1.1). In their proof, they applied the spacetime resonance method; we refer to [pusashat, pusa2] for some earlier applications of this method to the first order systems of wave equation. To the author’s knowledge, there is no previous result on the modified wave operators for (1.1).

In this paper, we will answer the first question, i.e. concerning the existence of the modified wave operators. Let $Z$ be one of the commuting vector fields: translations $\partial_{\alpha}$, scaling $t\partial_{t}+r\partial_{r}$, rotations $x_{i}\partial_{j}-x_{j}\partial_{i}$ and Lorentz boosts $x_{i}\partial_{t}+t\partial_{i}$. Our main theorem is the following.

Here we outline the main idea of the construction of $u$ in Theorem 1. Roughly speaking, our starting point is the ideas from both Lindblad [lind] and Lindblad-Schlue [lindschl]. To construct a matching global solution, we follow the idea in Lindblad-Schlue [lindschl]: we solve a backward Cauchy problem with some initial data at $t=T$ and then send $T$ to infinity. However, the backward Cauchy problems in [lindschl] are of simpler form, and their solutions can be constructed by Duhamel’s formula explicitly. Here, our backward Cauchy problem is quasilinear, and it is necessary to prove that the solution does exist for all $0\leq t\leq T$. We follow the proof of the small data global existence in [lind]: we use a continuity argument with the help of the adapted energy estimates and Poincar$\acute{\rm e}$’s lemma.

We now provide more detailed descriptions of the proof. First, we construct an approximate solution to (1.1). Let $q(t,r,\omega)$ and $U(t,r,\omega)$ be the approximate optical function and asymptotic profile associated to some scattering data $A(q,\omega)$. We set

for all $t\geq 0$ and $x\in\mathbb{R}^{3}$. Here $\psi\equiv 1$ when $|r-t|\leq t/4$ and $\psi\equiv 0$ when $|r-t|\geq t/2$, which is used to localize $\varepsilon r^{-1}U$ near the light cone $\{r=t\}$; $\eta$ is a cutoff function such that $\eta\equiv 0$ for $t\leq 2R$, which is used to remove the singularity at $|x|=0$ and $t=0$. We can check that $u_{app}$ is a good approximate solution to (1.1) in the sense that

Next we seek to construct an exact solution matching $u_{app}$ at infinite time. Fixing a large time $T$, we consider the following equation

Here $\chi\in C^{\infty}(\mathbb{R})$ satisfies $\chi(t/T)=1$ for $t\leq T$ and $\chi(t/T)=0$ for $t\geq 2T$. Note that $u_{app}+v$ is now an exact solution to (1.1) for $t\leq T$. In Section LABEL:smp we prove that, if $\varepsilon$ is sufficiently small, then (1.14) has a solution $v=v^{T}$ for all $t\geq 0$ which satisfies some decay in energy as $t\to\infty$. To prove this, we use a continuity argument. The proof relies on the energy estimates and Poincar$\acute{\rm e}$’s lemma, which are established in Section LABEL:sep. Note that the small constant $\delta>0$ is not chosen until the proof of the Poincar$\acute{\rm e}$’s lemma, and we remark that $\delta$ depends only on the scattering data $A(q,\omega)$. We also remark that the energy estimates and Poincar$\acute{\rm e}$’s lemma in our paper are closely related to those in [lind, alin2].

Finally we prove in Section LABEL:slim that $v^{T}$ does converge to some $v^{\infty}$ in suitable function spaces, as $T\to\infty$. Thus we obtain a global solution $u_{app}+v^{\infty}$ to (1.1) for $t\geq 0$, such that it “agrees with” $u_{app}$ at infinite time, in the sense that the energy of $v^{\infty}$ tends to $0$ as $t\to\infty$. By the Klainerman-Sobolev inequality, we can derive the pointwise bounds in the main theorem from the estimates for the energy of $v^{\infty}$.

Note that to obtain a candidate for $v^{\infty}$, we have a more natural choice of PDE than (1.14). We may consider the Cauchy problem (1.1) for $t\leq T$ with initial data $(u_{app}(T),\partial_{t}u_{app}(T))$. The problem with such a choice is that for $u_{app}$ constructed above, $Z^{I}(u-u_{app})(T)$ does not seem to have a good decay in $T$ if $Z^{I}$ only contains the scaling $S=t\partial_{t}+r\partial_{r}$ and Lorentz boosts $\Omega_{0i}=t\partial_{i}+x_{i}\partial_{t}$. For example, we can consider the linear wave equation $\square u=0$. We set $v=u-u_{app}$, then $v=v_{t}=0$ at $t=T$. Then, at $t=T$ we have $S^{2}v=t^{2}v_{tt}=-t^{2}\square u_{app}$. However, in the linear case, $u_{app}=\varepsilon r^{-1}F_{0}(r-t,\omega)$ for $t\approx r$ and thus $\square u_{app}=O(\varepsilon r^{-3})$. The power $-3$ cannot be improved, so we can only get $S^{2}v=O(\varepsilon r^{-1})$ for $t\approx r$, while we expect $S^{2}v=O(\varepsilon r^{-3/2+C\varepsilon})$ for $t\approx r$ from Theorem 1. Similarly, the same applies for $S^{k}v$ if $k\geq 3$. In the linear case, one possible way to deal with this difficulty is to consider more terms in the asymptotic expansion of the solutions, say take

where $F_{0}$ is the usual Friedlander radiation field, and $F_{n}$ satisfies some PDE based on $F_{n-1}$. This method was used by Lindblad and Schlue in their construction. However, it does not seem to work in the quasilinear case, since we do not have such a good asymptotic expansion for a solution to (1.1). In this paper, we avoid such a difficulty by considering a variant (1.14) of (1.1). Such a difficulty does not appear in (1.14), since $v\equiv 0$ for all $t\geq 2T$.

The author would like to thank his advisor, Daniel Tataru, for suggesting this problem and for many helpful discussions. The author would also like to thank the anonymous reviewers for their valuable comments and suggestions on this paper. This research was partially supported by a James H. Simons Fellowship and by the NSF grant DMS-1800294.

We use $C$ to denote universal positive constants. We write $A\lesssim B$ or $A=O(B)$ if $|A|\leq CB$ for some $C>0$. We write $A\sim B$ if $A\lesssim B$ and $B\lesssim A$. We use $C_{v}$ or $\lesssim_{v}$ if we want to emphasize that the constant depends on a parameter $v$. The values of all constants in this paper may vary from line to line.

In this paper, $R$ is reserved for the radius of the scattering data in $q$, i.e. $A(q,\omega)=0$ unless $|q|\leq R$. Unless specified otherwise, we always assume that $t\geq T_{R}>1$ for some sufficiently large constant $T_{R}$ depending on $R$ (denoted by $T_{R}\gg 1$, or $t\gg_{R}1$). We also assume $0<\varepsilon<1$ is sufficiently small (denoted by $\varepsilon\ll 1$). $T_{R}$ and $\varepsilon$ are allowed to depend on all other constants, and $\varepsilon$ can also depend on $T_{R}$.

We always assume that the latin indices $i,j,l$ take values in $\{1,2,3\}$ and the greek indices $\alpha,\beta$ take values in $\{0,1,2,3\}$. We use subscript to denote partial derivatives, unless specified otherwise. For example, $u_{\alpha\beta}=\partial_{\alpha}\partial_{\beta}u$, $q_{r}=\partial_{r}q=\sum_{i}\omega_{i}\partial_{i}q$, $A_{q}=\partial_{q}A$ and etc. For a fixed integer $k\geq 0$, we use $\partial^{k}$ to denote either a specific $k$-th partial derivative, or the collection of all $k$-th partial derivatives.

To prevent confusion, we will only use $\partial_{\omega}$ to denote the angular derivatives under the coordinate $(s,q,\omega)$, and will never use it under the coordinate $(t,r,\omega)$. We use $\partial_{\omega}^{c}$ to denote $\partial_{\omega_{1}}^{c_{1}}\partial_{\omega_{2}}^{c_{2}}\partial_{\omega_{3}}^{c_{3}}$ for a multiindex $c=(c_{1},c_{2},c_{3})$.

Let $Z$ be any of the following vector fields:

For any multiindex $I$ with length $|I|$, let $Z^{I}$ denote the product of $|I|$ such vector fields. Then we have Leibniz’s rule

The vector fields $Z$ have many good properties. First, we have the commutation properties.

We have the pointwise estimates for partial derivatives.

Finally, we have the Klainerman-Sobolev inequality.

We also state the Gronwall’s inequality.