We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, $ dλ_t^i=\frac{1}{\sqrt{N}} dW_t^i - V'(λ_t^i) dt+ \fracβ{2N} \sum_{j\not=i} \frac{dt}{λ^i_t-λ^j_t}, \qquad i=1,\ldots,N, $ with $β>1$, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $β$-ensemble, with sufficiently regular convex potential $V$. The limit $N\to\infty$ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown by the author to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation. We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $ρ_t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.