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(\phi_0, \varphi_0\right) $ in the Sobolev space $ H^s(\mathbb{R}) \times H^s(\mathbb{R}) $ with $ s > -5 / 8 $, and with the noise operator $ \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 $ \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.