论文标题
通过几何方法关于非辐射线性波的不平等
An inequality regarding non-radiative linear waves via a geometric method
论文作者
论文摘要
在这项工作中，我们考虑操作员 \ [ （\ Mathbf {t} g）（x）= \ int _ {\ mathbb {s}^2} g（x \ cdotω，ω）dΩ，\ quad x \ in \ mathbb {r}^3，\; g \ in l^2（\ mathbb {r} \ times \ mathbb {s}^2）。 \] 这是ra transform的伴随操作员。如果功能$ g $紧凑，我们设法给出了Infinity方法附近$ \ mathbf {t} g $的最佳$ l^6 $衰减估计。作为应用程序，我们对外部区域$ \ \ {（x，t）\ in \ Mathbb {r}^3 \ times \ times \ times \ mathbb {r}：| x | x | x |> r+| t | \} $进行了对3D线性波方程的衰减估计。这种衰减估计值在波方程的能量方法中很有用
In this work we consider the operator \[ (\mathbf{T} G) (x)= \int_{\mathbb{S}^2} G(x\cdot ω, ω) dω, \quad x\in \mathbb{R}^3, \; G\in L^2(\mathbb{R}\times \mathbb{S}^2). \] This is the adjoint operator of the Radon transform. We manage to give an optimal $L^6$ decay estimate of $\mathbf{T} G$ near the infinity by a geometric method, if the function $G$ is compactly supported. As an application we give decay estimate of non-radiative solutions to the 3D linear wave equation in the exterior region $\{(x,t)\in \mathbb{R}^3 \times \mathbb{R}: |x|>R+|t|\}$. This kind of decay estimate is useful in the channel of energy method for wave equations
