Stochastic Korteweg–de Vries-type systems: Local and global theory

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 ${\mathcal{F}}_0$-measurable initial data $\left({\varphi }_0,{\phi }_0\right)$ in the Sobolev space $H^s(\mathbb{R})\times H^s(\mathbb{R})$ with $s>-5/8$, and with the noise operator $\mathrm{\Xi }$ belonging to the intersection of Hilbert-Schmidt spaces $L_2^{0,s}\cap L_2^{0,s,-\frac{3}{8}}$, there exists a unique local solution. Furthermore, we demonstrated global well-posedness in the energy space $L^2(\mathbb{R})\times L^2(\mathbb{R})$ for $L^2$-valued initial data and with $\mathrm{\Xi }\in L_2^{0,0}\cap L_2^{0,0,-\frac{3}{8}}$. 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.