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We prove the existence of a contracting invariant topological foliation in a full neighbourhood for partially hyperbolic attractors. Under certain bunching conditions, it can then be shown that this stable foliation is smooth. Specialising to sectional hyperbolic attractors, we give a verifiable condition for bunching. In particular, we show that the stable foliation for the classical Lorenz equation (and nearby vector fields) is better than    C  1   , which is crucial for recent results on exponential decay of correlations. In fact, the foliation is at least    C   1.278    .

Existence and smoothness of the stable foliation for sectional hyperbolic attractors