Global well-posedness for the Cauchy problem of a system of convection–diffusion equations in the critical uniformly local space

We establish the time global well-posedness of the Cauchy problem for a convection–diffusion system of diagonal type in uniformly local Lebesgue spaces $L_{\mathrm{uloc}}^r(R^n;R^m)$. Our well-posedness result also demonstrates the existence of almost periodic solutions or non-zero asymptotic boundary conditions within the functional analytic framework, extending the scalar case presented in [18]. For local well-posedness, we apply uniformly local $L_{\mathrm{uloc}}^p$- $L_{\mathrm{uloc}}^q$ estimate for the heat evolution operator and the Banach-Caccioppoli fixed point theorem, following the approach in [18]. The time global well-posedness of the system, including the scaling critical case, is established by demonstrating that the solution of a single equation derived from the system satisfies a uniform bounded estimate, despite the absence of conservation laws. The proof is based on the Bernstein type argument for deriving the global a priori estimate (cf. [19]).