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Global well-posedness for the Maxwell-Klein-Gordon equation in \(4+1\) dimensions: small energy. (English) Zbl 1329.35209

Let \(\phi: \mathbb{R}^{4+1}\rightarrow \mathbb{C}\), and \(A_{\alpha}: \mathbb{R}^{4+1}\rightarrow \mathbb{R}\), \(\alpha = 0\dots,4\), with Minkowski metric \(m =\text{diag}(1,-1,-1,-1,-1)\). Introducing the curvature tensor \(F_{\alpha\beta}: = \partial_{\alpha}A_{\beta}- \partial_{\beta}A_{\alpha}\) as well as the covariant derivative \(D_{\alpha}\phi:= (\partial_{\alpha} + iA_{\alpha})\phi \), the (massless) Maxwell-Klein-Gordon (MKG) equation is \[ \begin{aligned} \partial^{\beta}F_{\alpha\beta} = \mathrm{Im}(\phi\overline{D_\alpha \phi}),\\ m^{\alpha\beta}D_{\alpha} D_{\beta}\phi = 0.\end{aligned} \] It is known that MKG is gauge invariant, and we can impose a Coulomb Gauge condition: \[ \sum_{j=1}^4\partial_j A_j = 0. \] Note that the energy \[ E(A,\phi): = \int_{\mathbb{R}^4}\big(\frac{1}{4}\sum_{\alpha,\beta}F_{\alpha\beta}^2 + \frac{1}{2}\sum_{\alpha}|D_{\alpha}\phi|^2\big)\,dx \] is preserved under the flow, and it is also invariant under the natural scaling \[ \phi(t, x) \rightarrow \lambda \phi(\lambda t, \lambda x),\,A_{\alpha}(t, x)\rightarrow \lambda A(\lambda t, \lambda x), \] which means the (4+1)-MKG system is energy critical. In this paper, the authors prove that the energy critical MKG equation on \(\mathbb{R}^{4+1}\) is globally well-posed for smooth initial data which are small in the energy.
The same analysis and result applies in all higher dimensions for small data in the scale invariant space \(\dot H^{\frac{n}2-1} \times \dot H^{\frac{n}2-2}\). This has already been known in dimensions \(n\geq 6\) from the work of I. Rodnianski and T. Tao [Commun. Math. Phys. 251, No. 2, 377–426 (2004; Zbl 1106.35073)]. On the other hand, it is still not clear whether a similar result holds in dimension \(n = 3\), despite that we have the global well-posedness results (even for large data) in \(H^{s}\) with \(s>\sqrt{3}/2\) from the works of Eardley-Moncrief, Klainerman-Machedon, Keel-Roy-Tao (recalling that the critical regularity \(s_{c}=1/2\) for \(n=3\), which is the energy-subcritical case).

MSC:

35L71 Second-order semilinear hyperbolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
35Q61 Maxwell equations

Citations:

Zbl 1106.35073
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References:

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