On the non-blow up of energy critical nonlinear massless scalar fields in ‘$3+1$’ dimensional globally hyperbolic spacetimes: light cone estimates

IF 0.4 Q4 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Annals of Mathematical Sciences and Applications Pub Date : 2021-07-05 DOI:10.4310/amsa.2021.v6.n2.a5
P. Mondal
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引用次数: 4

Abstract

Here we prove a global existence theorem for the solutions of the semi-linear wave equation with critical non-linearity admitting a positive definite Hamiltonian. Formulating a parametrix for the wave equation in a globally hyperbolic curved spacetime, we derive an apriori pointwise bound for the solution of the nonlinear wave equation in terms of the initial energy, from which the global existence follows in a straightforward way. This is accomplished by two steps. First, based on Moncrief’s light cone formulation we derive an expression for the scalar field in terms of integrals over the past light cone from an arbitrary spacetime point to an ‘initial’, Cauchy hypersurface and additional integrals over the intersection of this cone with the initial hypersurface. Secondly, we obtain apriori estimates for the energy associated with three quasi-local approximate time-like conformal Killing and one approximate Killing vector fields. Utilizing these naturally defined energies associated with the physical stress-energy tensor together with the integral equation, we show that the spacetime L∞ norm of the scalar field remains bounded in terms of the initial data and continues to be so as long as the spacetime remains singularity/Cauchy-horizon free.
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“$3+1$”维全局双曲时空中能量临界非线性无质量标量场的非爆炸:光锥估计
本文证明了具有临界非线性的半线性波动方程解的全局存在性定理。在全局双曲曲面时空中,我们为波动方程建立了一个参数,导出了非线性波动方程解的初始能量先验逐点界,由此可以直接得出全局存在性。这需要两个步骤来完成。首先,基于Moncrief的光锥公式,我们导出了标量场的表达式,表示为从任意时空点到“初始”柯西超曲面的过去光锥的积分,以及该锥与初始超曲面相交处的附加积分。其次,我们得到了与三个拟局部近似类时间共形Killing和一个近似Killing矢量场相关的能量的先验估计。利用这些与物理应力-能量张量相关的自然定义的能量以及积分方程,我们证明了标量场的时空L∞范数在初始数据方面保持有界,并且只要时空保持奇异性/柯西视界自由,就一直保持有界。
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Annals of Mathematical Sciences and Applications
Annals of Mathematical Sciences and Applications MATHEMATICS, INTERDISCIPLINARY APPLICATIONS-
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