Global existence and convergence of a density heat flow for strictly subcritical Liouville systems

We study a density heat flow associated with normalized Liouville systems on a closed connected Riemannian surface, and we show that this flow leads to the existence of solutions for the corresponding subcritical Liouville systems. The coupling matrix is symmetric and entrywise nonnegative, with no invertibility, irreducibility or inverse-matrix sign assumption. Under the strict subcriticality conditions $8\pi {\sum }_{i\in J}{\rho }_i>{\sum }_{i,j\in J}{a}_{ij}{\rho }_i{\rho }_j$ for every nonempty set of components $J$, every smooth positive initial density of masses ${\rho }_i>0$ generates a unique global smooth positive solution converging in every $C^m$ norm to a stationary density. The limiting potentials solve the stationary Liouville system. Our proof uses entropy coercivity and a gradient inequality in the density variables, which also yields algebraic or exponential convergence rates.