  Given a $k$-graph $H$ a complete blow-up of $H$ is a $k$-graph $\hat{H}$
formed by replacing each $v\in V(H)$ by a non-empty vertex class $A_v$ and then
inserting all edges between any $k$ vertex classes corresponding to an edge of
$H$. Given a subgraph $G\subseteq \hat{H}$ and an edge $e\in E(H)$ we define
the density $d_e(G)$ to be the proportion of edges present in $G$ between the
classes corresponding to $e$. The density Tur\'an problem for $H$ asks:
determine the minimal value $d_{crit}(H)$ such that any subgraph $G\subseteq
\hat{H}$ satisfying $d_e(G)> d_{crit}(H)$ for every $e\in E(H)$ contains a copy
of $H$ as a transversal, i.e. a copy of $H$ meeting each vertex class of
$\hat{H}$ exactly once. We give upper bounds for this hypergraph density
Tur\'an problem that generalise the known bounds for the case of graphs due to
Csikv\'ari and Nagy, [Combinatorics, Probability and Computing, 21(4):531-553,
2012] although our methods are different, employing an entropy compression
argument.
