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We first introduce a weak type of Zariski decomposition in higher dimensions:an $\R$-Cartier divisor has a weak Zariski decomposition if birationally and ina 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 betweenZariski decompositions and log minimal models which has long been expected: weprove 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 aweak Zariski decomposition$/Z$.

On existence of log minimal models and weak Zariski decompositions