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On the global well-posedness and scattering of the 3D Klein–Gordon–Zakharov system

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Abstract

In this paper we are interested in the global well-posedness of the 3D Klein–Gordon–Zakharov equations with small non-compactly supported initial data. We show the uniform boundedness of the energy for the global solution without any compactness assumptions on the initial data. The main novelty of our proof is to apply a modified Alinhac’s ghost weight method together with a newly developed normal-form type estimate to remedy the lack of the space-time scaling vector field; moreover, we give a clear description of the smallness conditions on the initial data.

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Acknowledgements

X. Cheng was partially supported by the Shanghai “Super Postdoc" Incentive Plan (No. 2021014), the International Postdoctoral Exchange Fellowship Program (No. YJ20220071) and China Postdoctoral Science Foundation (Grant No. 2022M710796, 2022T150139). J. Xu is supported by the Basic and Applied Basic Research Foundation of Guangzhou.

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Appendix A: An alternative approach to to estimate \(\Vert \Gamma ^{\beta }n^0\Vert _{L^2}^2\).

Appendix A: An alternative approach to to estimate \(\Vert \Gamma ^{\beta }n^0\Vert _{L^2}^2\).

In this section we give an alternative approach to estimate \(\Vert \Gamma ^{\beta }n^0\Vert _{L^2}^2\) as in Lemma 3.1. It is clear that \(\Box \Gamma ^\beta n^0=0.\) Therefore it is known that we can write \(\Gamma ^\beta n^0\) in the following mild form:

$$\begin{aligned} \Gamma ^\beta n^0(x)= (\cos tD\ v_0)(x)+\left( \frac{\sin tD}{D}v_1\right) (x), \end{aligned}$$
(A.1)

where \(D=|\nabla |\) and \((v_0,v_1)=(\Gamma ^\beta n^0,\partial _t\Gamma ^\beta n^0)|_{t=0}\). Then it follows that \(\Vert \Gamma ^{\beta } n^0\Vert _2\le \Vert \cos tD\ v_0\Vert _2+\Vert \frac{\sin tD}{D}v_1\Vert _2\). One can easily show from the Fourier side that

$$\begin{aligned} \Vert \cos tD\ v_0\Vert _{L^2_x({\mathbb {R}}^3)}\sim \Vert \cos (t|\xi |) \hat{v_0}(\xi )\Vert _{L^2_\xi ({\mathbb {R}}^3)}\lesssim \Vert \hat{v_0}\Vert _{L^2_\xi ({\mathbb {R}}^3)}\lesssim \Vert v_0\Vert _{L^2_x({\mathbb {R}}^3)}. \end{aligned}$$
(A.2)

Similarly by the Hardy’s inequality,

$$\begin{aligned} \left\| \frac{\sin tD}{D}v_1\right\| _{L^2_x({\mathbb {R}}^3)}^2\lesssim \int _{{\mathbb {R}}^3} \frac{(\sin (t|\xi |)^2}{|\xi |^2} |\hat{v_1}|^2\ d\xi \lesssim \int _{{\mathbb {R}}^3} \frac{|\hat{v_1}|^2}{|\xi |^2} \ d\xi \lesssim \Vert \hat{v_1}\Vert ^2_{H^1_\xi ({\mathbb {R}}^3)}\lesssim \Vert \langle x\rangle v_1\Vert ^2_{L^2_x({\mathbb {R}}^3)}. \end{aligned}$$
(A.3)

Remark A.1

Note that compared to the \(X(\partial )\) trick (conformal energy estimates), this propagator estimate needs an additional assumption:

$$\begin{aligned} \Vert v_0\Vert _2+\Vert \langle x\rangle v_1\Vert _2\lesssim \sum _{j=0}^K\Vert \langle x\rangle ^{j+1}\nabla ^j(\nabla n_0,n_1)\Vert _2\le C\varepsilon . \end{aligned}$$
(A.4)

One clearly sees the difference between (3.14) and (A.4).

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Cheng, X., Xu, J. On the global well-posedness and scattering of the 3D Klein–Gordon–Zakharov system. Calc. Var. 63, 17 (2024). https://doi.org/10.1007/s00526-023-02620-5

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