Weak Null Condition Research Articles

Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations

For the 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (\nabla u)\partial _{ij}^2u = 0$, whose coefficients are independent of the solution $u$, the blowup result of small data solution has been established in [1,2] when the null condition does not hold as well as a generic nondegenerate condition of initial data is assumed. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (u,\nabla u)\partial _{ij}^2u = 0$, whose coefficients depend on $u$ and $\nabla u$ simultaneously. When the first weak null condition is not fulfilled and a suitable nondegenerate condition of initial data is assumed, we shall show that the small data smooth solution $u$ blows up in finite time, moreover, an explicit expression of lifespan and blowup mechanism are also established.

The Small Data Solutions of General 3D Quasilinear Wave Equations. I

For the three-dimensional (3D) quasilinear wave equation $\sum_{i,j=0}^3g^{ij}(\partial u)\partial_{ij}^2u=0$ with coefficients independent of the solution $u$, a blowup result or global existence result has been completely established provided that the null condition does not hold or holds, respectively. In this paper and a subsequent paper (see [D. Bingbing, W. Ingo, and Y. Huicheng, The Small Data Solutions of General $3$-D Quasilinear Wave Equations. II, preprint, 2014]), we will systematically study the more general 3D quasilinear wave equation $\sum_{i,j=0}^3g^{ij}(u, \partial u)\partial_{ij}^2u=0$ with coefficients depending simultaneously on $u$ and $\partial u$. When the weak null condition holds, we show that the smooth small data solution exists globally in the paper.

The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates

In this article, we study the coupling of the Einstein field equations of general relativity to a family of models of nonlinear electromagnetic fields. The family comprises all covariant electromagnetic models that satisfy the following criteria: they are derivable from a sufficiently regular Lagrangian, they reduce to the linear Maxwell model in the weak-field limit, and their corresponding energy-momentum tensors satisfy the dominant energy condition. Our main result is a proof of the global nonlinear stability of the 1 + 3-dimensional Minkowski spacetime solution to the coupled system for any member of the family, which includes the linear Maxwell model. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in our wave coordinate gauge. Our analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows us to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. Our analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor developed in the article. We work directly with the electromagnetic fields, and thus avoid the use of electromagnetic potentials.

Semilinear hyperbolic systems violating the null condition

We consider systems of semilinear wave equations in three space dimensions with quadratic nonlinear terms not satisfying the null condition. We prove small data global existence of the classical solution under a new structural condition related to the weak null condition. For two-component systems satisfying this condition, we also observe a new kind of asymptotic behavior: Only one component is dissipated and the other one behaves like a non-trivial free solution in the large time.

ASYMPTOTIC POINTWISE BEHAVIOR FOR SYSTEMS OF SEMILINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS

In connection with the weak null condition, Alinhac introduced a sufficient condition for global existence of small amplitude solutions to systems of semilinear wave equations in three space dimensions. We introduce a slightly weaker sufficient condition for the small data global existence, and we investigate the asymptotic pointwise behavior of global solutions for systems satisfying this condition. As an application, the asymptotic behavior of global solutions under the Alinhac condition is also derived.

Asymptotic behavior of solutions to semilinear systems of wave equations

We consider the Cauchy problem for a class of systems of semilinear wave equations, which is closely connected to the weak null condition and Alinhac's condition. We show that the energy of some global solutions to these systems grows to infinity as time tends to infinity and consequently these solutions never approach any free solutions.

Semilinear hyperbolic systems with blowup at infinity

We investigate semilinear hyperbolic systems in three-space dimensions with quadratic terms in gradient of the solution, not satisfying the null condition. Under some algebraic conditions ensuring global existence for solutions of the reduced systems (the so-called weak null condition of Lindblad and Rodnianski), we prove global existence of classical solutions for these systems.

Plane waves and spacelike infinity

In an earlier paper, we showed that the causal boundary of any homogeneous plane wave satisfying the null convergence condition consists of a single null curve. In Einstein–Hilbert gravity, this would include any homogeneous plane wave satisfying the weak null energy condition. For conformally flat plane waves such as the Penrose limit of AdS5 × S5, all spacelike curves that reach infinity also end on this boundary and the completion is Hausdorff. However, the more generic case (including, e.g., the Penrose limits of AdS4 × S7 and AdS7 × S4) is more complicated. In one natural topology, not all spacelike curves have limit points in the causal completion, indicating the need to introduce additional points at ‘spacelike infinity’—the endpoints of spacelike curves. We classify the distinct ways in which spacelike curves can approach infinity, finding a two-dimensional set of distinct limits. The dimensionality of the set of points at spacelike infinity is not, however, fixed from this argument. In an alternative topology, the causal completion is already compact, but the completion is non-Hausdorff.

The weak null condition for Einstein's equations

We show that Einstein's equations of General Relativity expressed in wave coordinates satisfy a ‘weak null condition’. In a forthcoming article we will use this to prove a global existence result for Einstein's equations in wave coordinates with small initial data. To cite this article: H. Lindblad, I. Rodnianski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).