In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\"ahler) metric equation \cite{cc1} to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of K\"ahler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\mathcal{E}^1, d_1)$ are smooth. The continuity path proposed in \cite{chen15} is instrumental in this proof.

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We consider a L 2-contraction (a L 2-type stability) of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift by using the relative entropy methods. Quite different from the previous results, we find a new way to determine the shift function, which depends both on the time and space variables and solves a viscous Hamilton-Jacobi type equation with source terms. Moreover, we do not impose any conditions on the anti-derivative variables of the perturbation around the shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in L 2-norm, then the L 2-contraction holds true for the viscous shock wave up to a suitable shift function. Note that BV-norm or the L-norm of the initial perturbation and the shock wave strength can be arbitrarily large. Furthermore, as the time t tends to infinity, the L 2-contraction holds

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