The weak null condition and global existence using the p-weighted energy method

Abstract

We prove global existence for solutions arising from small initial data for a large class of quasilinear wave equations satisfying the `weak null condition' of Lindblad and Rodnianski, significantly enlarging upon the class of equations for which global existence is known. In addition to the usual weak null condition, we require a certain hierarchical structure in the semilinear terms. Included in this class are the Einstein equations in harmonic coordinates, so a special case of our results is a new proof of the stability of Minkowski space. Our proof also applies to the coupled Einstein-Maxwell system in harmonic coordinates and Lorenz gauge, as well as to various model scalar wave equations which do not satisfy the null condition. Our proof also applies to the Einstein(-Maxwell) equations if, after writing the equations as a set of nonlinear wave equations, we then `forget' about the gauge conditions. The methods we use allow us to treat initial data which only has a small `degenerate energy', involving a weight that degenerates at null infinity, so the usual (unweighted) energy might be unbounded. We also demonstrate a connection between the weak null condition and geometric shock formation, showing that equations satisfying the weak null condition can exhibit `shock formation at infinity', of which we provide an explicit example. The methods that we use are very robust, including a generalisation of the p-weighted energy method of Dafermos and Rodnianski, adapted to the dynamic geometry. This means that our proof applies in a wide range of situations, including those in which the metric remains close to, but never approaches the flat metric in some spatially bounded domain, and those in which the `geometric' null infinity and the `background' null infinity differ dramatically, for example, when the solution exhibits shock formation at null infinity.

Figures (9)

• Figure 1: A diagram of the null frame vector fields. The vector fields $L$ and $\underline{L}$ are both null, future directed, and orthogonal to the spheres of constant $r$ and $t$. The vector field $L$ points away from the origin $r = 0$ while the vector field $\underline{L}$ points toward the origin. The vector fields $X_{A}$ are tangent to the spheres of constant $r$ and $t$. Note that we have suppressed a dimension, so these spheres appear as circles. Note also that in the null frame that we actually use, the time $t$ must be replaced by the "geometric retarded time" $\tau$.
• Figure 2: A figure illustrating the structure of the semilinear hierarchy in a case where the hierarchy consists of three levels. On the left, we illustrate the three levels, with the pairs of arrows between the levels representing possible pairs of "bad" semilinear terms. On the right, we show a diagram of a particular system of wave equations satisfying the hierarchical null condition, also with three levels, and where each pair of "legs" represents a bad semilinear term. Note that each pair of legs either has the property that both legs point to fields at a lower level of the hierarchy, or it has the property that one leg points to a field at the bottom level and one leg points to a field at the same level. The system illustrated is of the form \tilde{\Box}_g \phi_{(1)}= 0\tilde{\Box}_g \phi_{(2)}= 0\tilde{\Box}_g \phi_{(3)}= (\partial \phi_{(1)})(\partial \phi_{(3)})\tilde{\Box}_g \phi_{(4)}= (\partial \phi_{(1)})(\partial \phi_{(2)})\tilde{\Box}_g \phi_{(5)}= (\partial \phi_{(1)})(\partial \phi_{(6)})\tilde{\Box}_g \phi_{(6)}= (\partial \phi_{(3)})(\partial \phi_{(4)})\tilde{\Box}_g \phi_{(7)}= (\partial \phi_{(4)})^2 + (\partial \phi_{(2)})(\partial \phi_{(7)})
• Figure 3: "Penrose diagrams" illustrating the geometric and non-geometric foliations with respect to different compactifications. In all three figures, the red lines depict the leaves of the "background" foliation, while the blue lines represent the leaves of the "geometric" foliation. The solid black line on the left of each figure is the line $r = 0$, and the dotted black line in the interior of each figure represents the hypersurface $r = r_0$. Note that all of the foliations agree in the region $r \leq r_0$. In subfigure i), we have compactified with respect to the "background" geometry, so that the lines of constant $(t-r)$ are straight lines at forty five degrees. In other words, this is a conformal compactification with respect to the "background" metric $m$, and not the metric $g$. Note that, if we reduce the space to a two dimensional space by quotienting out by the usual action of $SO(3)$ on the spheres of constant $(t-r)$ and $r$, then the "geometric" foliation is not invariant under this action. Hence the red lines cannot really represent the geometric foliation - they can be taken either as an indication of the type of behaviour of this foliation, or alternatively, as representing a single angle on the sphere. Subfigure ii) shows the same foliations, but this time we have performed a conformal compactification with respect to the dynamic metric $g$, so that this time the hypersurfaces of constant $u$ are drawn as straight lines at forty five degrees. In other words, this is a true Penrose diagram for the manifold $(\mathcal{M}, g)$. Note that the hypersurfaces of constant $(t-r)$ (i.e. the red lines) can be timelike in some regions (though not in the region $r \leq r_0$), which is one reason why this foliation is unsuitable. The same comments apply as above: this time, the blue lines can only really be taken as representing a single angle on the leaves of the background foliation. Subfigure iii) shows another possible scenario. We have again compactified with respect to the dynamic metric $g$. However, this time, the leaves of constant $t-r$ all approach the same value of $u$. We expect to see energy decay through the geometric foliation, but, in this case, it is clear that we cannot expect the energy to decay through the background foliation.
• Figure 4: A more detailed diagram of our foliation, suppressing only one dimension. We have drawn this from the point of view of the "background geometry", i.e. the coordinate functions $x^a$ would define a square grid on this diagram. The line $r = 0$ is drawn as a dashed black line in the centre of the diagram. The red cylinder represents a surface of constant $r$, while the blue "cone" represents a surface of constant $\tau$. Note that, since the curves of constant $\tau$ are defined "geometrically" (in the region $r \geq r_0$), this cone is deformed relative to the standard cone. The blue "cone" and the red cylinder intersect along the green curve, which represents a "sphere" of constant $\tau$ and $r$. If we had not suppressed one dimension, then the green deformed circle would be replaced by a deformed sphere. The "background" angular momentum operators act by rigidly rotating the (red) cylinder of constant $r$ around the axis $r = 0$. It should be clear, from this diagram, that the vector fields which generate these rotations are not tangent to the (green) "sphere". In other words, this "sphere" is not invariant under the action of the background angular momentum operators.
• Figure 5: This figure shows several conformal compactifications with respect to the background geometry, together with (in blue) a geometric foliation at a single fixed angle on the sphere (see the comments in the caption of figure \ref{['figure penrose diagrams 1']}). Subfigure i) shows the kind of foliation which arises if the wave equation actually satisfies the classical null condition but is nevertheless quasilinear - see Yang2013 for examples of this kind of equation. Note that no shocks can form. Subfigure ii) shows a possible behaviour for systems of the kind we are studying, i.e. equations satisfying the weak null condition. Note that, after some time, the characteristics can converge, signalling that a shock is forming. However, this can only happen at null infinity. In other words, the shock cannot form at any finite value of $r$. This is a crucial fact which allows us to prove the global existence result. Finally, subfigure iii) shows the possible behaviour of a system which does not satisfy the weak null condition, and for which shocks form at a finite value of $r$. Note that it is not possible to extend the solution classically past the point where the shock forms, since certain derivatives of the solution (actually, the "bad" derivatives) blow up there. See Christodoulou2007e or Speck2016a for more details.

Theorems & Definitions (528)

• Definition 1.2.1: The weak null condition
• Definition 1.2.2: The semilinear hierarchy
• Theorem 1.2.3: Main theorem, first version
• Theorem 1.6.1: Main theorem, second version
• Definition 2.1.1: Rectangular coordinates and the radial function
• Definition 2.1.2: The eikonal function
• Definition 2.1.3: The hyperboloidal time
• Remark 2.1.4: Continuity of $\tau$
• Definition 2.1.5: Hypersurfaces and the foliations
• Definition 2.1.6: Induced metrics