On existence of log minimal models and weak Zariski decompositions

We first introduce a weak type of Zariski decomposition in higher dimensions: an -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 -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 (in our sense) if and only if K (X) + B has a weak Zariski decomposition/Z.