Global Behavior of Small Data Solutions for The 2D Dirac-Klein-Gordon Equations

Abstract

In this paper, we are interested in the two-dimensional Dirac-Klein-Gordon system, which is a basic model in particle physics. We investigate the global behaviors of small data solutions to this system in the case of a massive scalar field and a massless Dirac field. More precisely, our main result is twofold: 1) we show sharp time decay for the pointwise estimates of the solutions which imply the asymptotic stability of this system; 2) we show the linear scattering result of this system which is a fundamental problem when it is viewed as dispersive equations. Our result is valid for general small, highregular initial data, in particular, there is no restriction on the support of the initial data.

Global Solution to the Klein--Gordon--Zakharov Equations with Uniform Energy Bounds

“…As for the 2D case, the case (m, M ) = (1, 1) can be reduced to coupled Klein-Gordon equations, which were tackled by Simon-Taflin [37], by Ozawa-Tsutaya-Tsutsumi [35], and by Delort-Fang-Xue [14] with pointwise decay result, and for all combinations of masses, Grünrock-Pecher [21] first proved the global existence result. Recently, the pointwise decay and scattering for the 2D case with (m, M ) = (1, 0) were established in [16,18]. We also remind one of some interesting results regarding low-regular initial data [5,13,40].…”

“…Some of these are not issues if we only want to show global existence and pointwise decay, but they turn into obstacles when we establish a uniform boundedness result. One finds a detailed discussion in recent results [16,18] for instance. In what follows, we list the three key points in the proof that can be used to overcome the difficulties listed in the above.…”

“…First, for the case of (m, M ) = (1, 1), we recall that the system (1.1) is reduced to coupled Klein-Gordon equations, and thus the classical results of Klainerman [28] and Shatah [36] imply the uniform boundedness of the total energy. Second, for the case of (m, M ) = (1, 0), thanks to the faster decay rates of the 3D case than the 2D case, we expect that the same strategy used in [16,18] can lead to the uniform boundedness for the total energy of (1.1). Therefore, what is still open is the remaining four cases for (m, M ) = (0, 0) and (m, M ) = (0, 1).…”

“…Note that, this rate is a non-integrable quantity in time t, which further makes it nontrivial to show a uniform boundedness result for the total energy of (1.3)-(1.4) and (1.7)-(1.8). Fortunately, the ghost weight method by Alinhac [1] (possibly adapted to Klein-Gordon setting) together with some special structure of the Dirac equation and the nonlinearities used in [16,18] allows one to turn t − r decay into t + r decay, which solves the problem of insufficient time decay of the solution φ. Second, one key issue is how to get sufficient bounds for the undifferentiated wave component φ (as it appears in the nonlinearities of the Dirac equation), both in the L 2 and L ∞ levels.…”

Global stability of the Dirac-Klein-Gordon system in two and three space dimensions

“…Dong and Wyatt [13] first proved sharp asymptotic decay results for (1.1) with ( , ) = (2, 0) and small, regular initial data with compact support, using the method of hyperboloidal foliation of space-time. Recently Dong, Li, Ma and Yuan [12] removed the compactness condition on the support of the initial data in [13]. Precisely, by applying the ghost weight method introduced by Alinhac [1], they proved global existence, sharp time decay and linear scattering for (1.1) with ( , ) = (2, 0) and small, regular initial data.…”