  We introduce two flow approaches to the Loewner-Nirenberg problem on comapct
Riemannian manifolds $(M^n,g)$ with boundary and establish the convergence of
the corresponding Cauchy-Dirichlet problems to the solution of the
Loewner-Nirenberg problem. In particular, when the initial data $u_0$ is a
subsolution to $(1.1)$, the convergence holds for both the direct flow
$(1.3)-(1.5)$ and the Yamabe flow $(1.6)$. Moreover, when the background metric
satisfies $R_g\geq0$, the convergence holds for any positive initial data
$u_0\in C^{2,\alpha}(M)$ for the direct flow; while for the case the first
eigenvalue $\lambda_1<0$ for the Dirichlet problem of the conformal Laplacian
$L_g$, the convergence holds for $u_0>v_0$ where $v_0$ is the largest solution
to the homogeneous Dirichlet boundary value problem of $(1.1)$ and $v_0>0$ in
$M^{\circ}$. We also give an equivalent description between the existence of a
metric of positive scalar curvature in the conformal class of $(M,g)$ and
$\inf_{u\in C^1(M)-\{0\}}Q(u)>-\infty$ when $(M,g)$ is smooth, provided that
the positive mass theorem holds, where $Q$ is the energy functional (see
$(3.2)$) of the second type Escobar-Yamabe problem.
