AI 中文总结
本研究在$\u2119^2$和$\u211d^2$上为多类KP及修正KP方程构造无穷多零初值紧时间支集弱奇异解，明确其正则性指数与尖锐阈值，还构造周期稳态解并证明$L^2$为光滑性临界阈值。
AI 中文摘要
我们在$\u2119^2$和$\u211d^2$上，为三阶、五阶KP-I、KP-II及其修正形式构造了无穷多具有零初值和紧时间支集的弱奇异解。这些解的非线性项与截断无关，是绝对收敛的傅里叶卷积。二次解属于$C_tL^p$（$p<2$）和$C_t(H^{-σ,0}\u2229H^{-σ})$（$σ>0$）。修正解属于$C_tL^p$（$p<3$）。一类三次解族属于$C_tH^α$（$α<1/3$）；另一类具有抛物型傅里叶支集，属于$C_tH^{s,0}$（$s<1/2$）和$C_tH^α$（$α<1/4$）。指数$1/3$和$1/2$在$L^3$乘积阈值处是尖锐的。对于$\u211d^2$上的二次五阶KP方程，非唯一性范围几乎是尖锐的。我们还构造了周期稳态KP-I和KP-II解，并证明$L^2$是奇异稳态KP-I解与光滑性之间的尖锐阈值。
英文摘要
We construct infinitely many weak singular solutions with zero initial data and compact time support for third- and fifth-order KP-I, KP-II, and their modified counterparts on $\mathbb T^2$ and $\mathbb R^2$. Their nonlinearities are cutoff-independent, absolutely convergent Fourier convolutions. Quadratic solutions belong to $C_tL^p$ for $p<2$ and $C_t(H^{-σ,0}\cap H^{-σ})$ for $σ>0$. Modified solutions belong to $C_tL^p$ for $p<3$. One cubic family lies in $C_tH^α$ for $α<1/3$; another has parabolic Fourier support and lies in $C_tH^{s,0}$ for $s<1/2$ and $C_tH^α$ for $α<1/4$. The exponents $1/3$ and $1/2$ are sharp at the $L^3$ product threshold. For quadratic fifth-order KP on $\mathbb R^2$, the nonuniqueness range is almost sharp. We also construct periodic stationary KP-I and KP-II solutions and prove that $L^2$ is the sharp threshold between singular stationary KP-I solutions and smoothness.
Comments49 pages
