2026/07/20 by Xiangqian Yan, Yongsheng Li, Jianhua Huang +1
#math.AP
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∈ Hs(ℝ)(s≥(1)/(4)) and Φ2∈ L20,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>(1)/(3), f∈ Hs(ℝ) and Φ1∈ L20,s. Then, the local pathwise solution u satisfies &ℙ(\ω: limt→0‖u-U(t)f-∫0tU(t-s)Φ1dW(s)‖_Lx∞=0\)=1,
&ℙ(\ω: ∀ t∈[0,Tω], lim|x|→∞(u-U(t)f-∫0tU(t-s)Φ1dW(s))=0\)=1. For the stochastic cubic KdV-Benjamin-Ono equation, let s>(1)/(3), g∈ Hs(ℝ) and Φ2∈ L20,s. Then, the local pathwise solution v satisfies &ℙ(\ω: limt→0‖v-V(t)g-∫0tV(t-s)Φ2dW(s)‖_Lx∞=0\)=1,
&ℙ(\ω: ∀ t∈[0,Tω],lim|x|→∞(v-V(t)g-∫0tV(t-s)Φ2dW(s))=0\)=1.