@article{Boukarou2025, 
author = {Aissa Boukarou and Mohammadi Begum Jeelani and Nouf Abdulrahman Alqahtani},
title = {Stochastic Korteweg–de Vries-type systems: Local and global theory},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {11},
pages = {27560-27580},
keywords = {white noise, stochastic, Bourgain space, KdV equation, bilinear estimates},
url = {https://www.sciopen.com/article/10.3934/math.20251212},
doi = {10.3934/math.20251212},
abstract = {This paper studied the Cauchy problem for a system of coupled Korteweg-de Vries (KdV) equations driven by multiplicative space-time white noise. We established local well-posedness for the system, proving that for              F        0  -measurable initial data        (                  ϕ        0            ,              φ        0              )   in the Sobolev space        H    s    (      R    )  ×      H    s    (      R    ) with    s  >  −  5      /    8, and with the noise operator    Ξ belonging to the intersection of Hilbert-Schmidt spaces        L    2          0      ,      s        ∩      L    2          0      ,      s      ,      −              3        8            , there exists a unique local solution. Furthermore, we demonstrated global well-posedness in the energy space        L    2    (      R    )  ×      L    2    (      R    ) for        L    2  -valued initial data and with    Ξ  ∈      L    2          0      ,      0        ∩      L    2          0      ,      0      ,      −              3        8            . The analysis employed Fourier restriction norm methods, utilizing Bourgain-type spaces        X          s      ,      b       and        Y                  s        1            ,              s        2            ,      b      . Key to the proofs was the establishment of crucial linear and bilinear estimates within these spaces and a detailed analysis of the stochastic convolution via Itô calculus. A fixed-point argument was then applied to obtain the local solution, while global existence followed from an invariance property (conservation) of the        L    2   norm, a martingale inequality, and an approximation procedure. The work extends previous results on single stochastic KdV equations to a more complex coupled system, providing a robust framework for analyzing nonlinear wave propagation subject to random perturbations, with applications in plasma physics and fluid dynamics.}
}