  We first introduce a weak type of Zariski decomposition in higher dimensions:
an $\R$-Cartier divisor has a weak Zariski decomposition if birationally and in
a numerical sense it can be written as the sum of a nef and an effective
$\R$-Cartier divisor. We then prove that there is a very basic relation between
Zariski decompositions and log minimal models which has long been expected: we
prove that assuming the log minimal model program in dimension $d-1$, a lc pair
$(X/Z,B)$ of dimension $d$ has a log minimal model if and only if $K_X+B$ has a
weak Zariski decomposition$/Z$.
