Strong relaxation limit and uniform time asymptotics of the Jin-Xin model in the $L^{p}$ framework

We investigate the time-asymptotic stability of the Jin-Xin model and its diffusive relaxation limit toward viscous conservation laws in $\mathbb{R}^d$ for $d\geq 1$. First, we establish a priori estimates that are uniform with respect to both the time and the relaxation parameter $\varepsilon>0$, for initial data in hybrid Besov spaces based on $L^{p}$-norms. This uniformity enables us to derive $\mathcal{O}(\varepsilon)$ bounds on the difference between solutions of the viscous conservation law and its associated Jin-Xin approximation, thus justifying the strong convergence of the relaxation process. Furthermore, under an additional condition on the initial data, for instance, that the low frequencies belong to $L^{p/2}(\mathbb{R}^{d})$, we show that the $L^{p}(\mathbb{R}^d)$-norm of the solution to the Jin-Xin model decays at the optimal rate $(1+t)^{-d/{2p}}$, and the $L^{p}(\mathbb{R}^d)$-norm of its difference with the solution of the associated viscous conservation law decays at the enhanced rate $\varepsilon(1+t)^{-d/{2p}-1/2}$.