We prove uniform bounds on moments X_a = \sum_{ m}{ m^a f_m(x,t)} of the Smoluchowski coagulation equations with diffusion, valid in any dimension. If the collision propensities \alpha(n,m) of mass n and mass m particles grow more slowly than (n+m)(d(n) + d(m)), and the diffusion rate d(\cdot) is non-increasing and satisfies m^{ -b_1} \leq d(m) \leq m^{ -b_2} for some b_1 and b_2 satisfying 0 \leq b_2 < b_1 < \infty, then any weak solution satisfies X_a \in L^{ \infty}(\mathbb{ R}^d \times [0,T]) \cap L^1(\mathbb{ R}^d \times [0,T]) for every a \in \mathbb{ N} and T \in (0,\infty), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass.
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