半线性波动方程弱解的非唯一性的尖锐性
Sharp non-uniqueness of weak solutions to the semilinear wave equation
- School of Mathematical Sciences & Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学数学科学学院与当代应用数学上海市重点实验室）
- School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对半线性波动方程，在低正则性下构造无穷多个零初值弱解，证明非唯一性，并指出该结果与经典唯一性理论互补，达到尖锐界限。
AI中文摘要：
本文建立了半线性波动方程\\[ \partial_{tt}u-\partial_{xx} u= \Theta_1(\partial_t u)^2 + \Theta_2(\partial_x u)^2 \\]的尖锐非唯一性结果。对于任意$\Theta_1 \Theta_2 < 0$，$\beta<1$，我们构造了具有零初始数据的无穷多个弱解$u \in C_{t,x}^{\beta}$。据我们所知，这是半线性波动方程的第一个非唯一性结果。在这个低正则性水平$C_{t,x}^{\beta}(\beta<1)$下，二次项通过Littlewood–Paley分解在$C_t H_x^{-s}$中定义。我们的构造基于频率局部化的凸积分方案。结合经典的$C_{t,x}^{1}$局部唯一性理论，该非唯一性结果是尖锐的。
英文摘要：
In this paper, we establish a sharp non-uniqueness result to the semilinear wave equation \[ \partial_{tt}u-\partial_{xx} u= Θ_1(\partial_t u)^2 + Θ_2(\partial_x u)^2. \] For any $Θ_1 Θ_2 < 0$, $β<1$, we construct infinitely many weak solutions $u \in C_{t,x}^β$ with zero initial data. To our knowledge, this is the first non-uniqueness result for a semilinear wave equation. At this low level of regularity $C_{t,x}^β(β<1)$, the quadratic term is defined by Littlewood--Paley decomposition in $C_t H_x^{-s}$. Our construction is based on a frequency-localized convex integration scheme. Together with the classical local uniqueness theory in $C_{t,x}^{1}$, this non-uniqueness result is sharp.
