AI 中文总结
研究随机mKdV方程和随机三次KdV - Benjamin - Ono方程的几乎必然非线性平滑、空间衰减及一致收敛性。通过证明局部适定性，建立非线性平滑，得出积分项的相关性质，给出特定条件下局部路径解满足的概率结论。
AI 中文摘要
本文研究了随机mKdV方程和随机三次KdV - Benjamin - Ono方程的几乎必然非线性平滑、几乎必然空间衰减和几乎必然一致收敛性。首先，对于初始数据\(g\in H^{s}(\mathbb{R})(s\geq\frac{1}{4})\)和\(\Phi_{2}\in L_{2}^{0,s}\)，证明了随机三次KdV - Benjamin - Ono方程的局部适定性。其次，建立了随机mKdV方程和随机三次KdV - Benjamin - Ono方程的几乎必然非线性平滑。最后，利用几乎必然非线性平滑得到了随机mKdV方程和随机三次KdV - Benjamin - Ono方程路径解中积分项的几乎必然空间衰减和几乎必然一致收敛。具体给出了两个方程在特定条件下局部路径解满足的概率等式。
英文摘要
In this paper, we consider the almost sure nonlinear smoothing, the almost sure spatial decay and the almost sure uniform convergence of the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. Firstly, for initial data $g\in H^{s}(\mathbb{R})(s\geq\frac{1}{4})$ and $Φ_{2}\in L_{2}^{0,s}$, we prove the local well-posedness for the stochastic cubic KdV-Benjamin-Ono equation. Secondly, we establish the almost sure nonlinear smoothing of the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. Finally, by using the almost sure nonlinear smoothing, we obtain the almost sure spatial decay and the almost sure uniform convergence of the integral term in the pathwise solutions to the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. More precisely, we have the following results: for the stochastic mKdV equation, let $s>\frac{1}{3}$, $f\in H^{s}(\mathbb{R})$ and $Φ_{1}\in L_{2}^{0,s}$. Then, the local pathwise solution $u$ satisfies \begin{eqnarray*} &&\mathbb{P}\Big(\Big\{ω: \lim_{t\rightarrow0}\Big\|u-U(t)f-\int_{0}^{t}U(t-s)Φ_{1}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\ &&\mathbb{P}\Big(\Big\{ω: \forall t\in[0,T_ω], \lim_{|x|\rightarrow\infty}\Big(u-U(t)f-\int_{0}^{t}U(t-s)Φ_{1}dW(s)\Big)=0\Big\}\Big)=1. \end{eqnarray*} For the stochastic cubic KdV-Benjamin-Ono equation, let $s>\frac{1}{3}$, $g\in H^{s}(\mathbb{R})$ and $Φ_{2}\in L_{2}^{0,s}$. Then, the local pathwise solution $v$ satisfies \begin{eqnarray*} &&\mathbb{P}\Big(\Big\{ω: \lim_{t\rightarrow0}\Big\|v-V(t)g-\int_{0}^{t}V(t-s)Φ_{2}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\ &&\mathbb{P}\Big(\Big\{ω: \forall t\in[0,T_ω],\lim_{|x|\rightarrow\infty}\Big(v-V(t)g-\int_{0}^{t}V(t-s)Φ_{2}dW(s)\Big)=0\Big\}\Big)=1. \end{eqnarray*}
