Global Existence for a System of Quasi-Linear Wave Equations in 3D Satisfying the Weak Null Condition

Abstract

We discuss the Cauchy problem for a system of semilinear wave equations in three space dimensions with multiple wave speeds. Though our system does not satisfy the standard null condition, we show that it admits a unique global solution for any small and smooth data. This generalizes a preceding result due to Pusateri and Shatah.The proof is carried out by the energy method involving a collection of generalized derivatives. The multiple wave speeds disable the use of the Lorentz boost operators, and our proof therefore relies upon the version of Klainerman and Sideris. Due to the presence of nonlinear terms violating the standard null condition, some of components of the solution may have a weaker decay as t → ∞, which makes it difficult even to establish a mildly growing (in time) bound for the high energy estimate. We overcome this difficulty by relying upon the ghost weight energy estimate of Alinhac and the Keel-Smith-Sogge type L 2 weighted space-time estimate for derivatives.

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

“…Specifically, we identify a large class of systems that satisfy, in addition to the weak null condition, an extra hierarchical condition on the nonlinearities. This hierarchical condition generalises the algebraic condition on the semilinear terms identified in [Ali06] and extended in [HY18], in that our semilinear hierarchy is allowed to consist of more than two "layers", and (perhaps more significantly) in that we allow for quasilinear wave equations.…”

Energy decay for small solutions to semilinear wave equations with weakly dissipative structure

“…Recently, a lot of efforts have been made for finding weaker structural conditions than the null conditions which ensure the small data global existence (see e.g., [2], [4], [10], [11], [16], [20], [21], [22], [23], [24], [25], [27], [31], [32], [33], [34], [35] and so on). The Agemi condition, which we are interested in, is one of them.…”

Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition

“…Remark 1. In Section 3 of [7], thanks to compactness of the support of initial data together with the finite speed of propagation, the proof of Theorem 1.1 was able to employ the standard local existence theorem in solving locally (in time) the Cauchy problem with data given at t = 0 and in continuing the local solutions to a larger strip, though some partial differential operators with "weight" (see just below) were naturally used. We should remark that the constant ε in the above theorem is independent of the "radius" of the support of given data (u i (0), ∂ t u i (0)) = (f i , g i ) (i = 1, 2), that is, R * := inf r > 0 : supp {f 1 , g 1 , f 2 , g 2 } ⊂ {x ∈ R 3 : |x| < r} .…”

“…Actually, such an attempt of combining these two methods has been already made in [25]. With the help of some observations in [5] and [7], we adjust the machinery thereby assembled in [25], in order to reduce the amount of regularity of initial data, and also to discuss the system (1) violating the standard null condition but satisfying the weak null condition. We hope that in the future, this machinery will be useful in discussing the Cauchy problem for a nonrelativistic system satisfying the weak null condition or the initial-boundary value problems in a domain exterior to an obstacle.…”

“…Before entering into the energy estimate in the next section, we must refer to an elementary result concerning point-wise estimates for u 1 and u 2 . It compensates for the absence of ∂ i t v(t, x) (i = 2, 3, 4) in the definition of the norms N 4 (v(t)), M 4 (v(t)), G(v(t)), and L(v(t)) (see (7), (27), (84)-(85)).…”