Abstract
We consider the global behaviour for large solutions of the Dirac–Klein–Gordon system in critical spaces in dimension$1+3$. In particular, we show that bounded solutions exist globally in time and scatter, provided that a controlling space–time Lebesgue norm is finite. A crucial step is to prove nonlinear estimates that exploit the dichotomy between transversality and null structure, and furthermore involve the controlling norm.
Highlights
The Dirac–Klein–Gordon system for a spinor ψ : R1+3 → C4 and a scalar field φ : R1+3 → R is given as−i γ μ∂μψ + Mψ = φψ (1)2φ + m2φ = ψψ, for the Dirac matrices γ μ ∈ C4×4, using the summation convention with respect to μ = 0, . . . , 3, where ∂0 = ∂t and ∂ j = ∂x j for j = 1, 2, 3
In [1] we treated the subcritical Sobolev spaces, and for arbitrary M, m > 0 we proved this for small initial data in the critical Sobolev space with some small amount of additional angular regularity in [6]
Fourier, x )e−i(t transform of,x)·(τ,ξ) d x dt f. to be the space–time Fourier transform of u. We extend these transforms to tempered distributions in the usual manner
Summary
The Dirac–Klein–Gordon system for a spinor ψ : R1+3 → C4 and a scalar field φ : R1+3 → R is given as. The particular result from [4] we will exploit here is summarized in Theorem 6 These have the elementary yet crucial property that they can be made arbitrarily small by shrinking the time interval. This has been used successfully to prove global well-posedness and scattering results for wave and Schrodinger equations with polynomial nonlinearities; see for example, [3, 8, 14] and the references therein. The estimates proved in this paper have further applications They are applied in [5] to prove scattering results for solutions that approximately satisfy a so-called Majorana condition, which defines an open set of large initial data yielding global solutions that scatter.
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