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Abstract

We consider the single layer potential associated to the fundamental solution of the time-dependent Oseen system. It is shown this potential belongs to $L^2(0, \infty ,H^1( \Omega )^3)$ and to $H^1(0, \infty ,V ^{\prime} )$ if the layer function is in $L^2( \partial \Omega \times (0, \infty )^3)$. ($\Omega $ denotes the complement of a bounded Lipschitz set; $V$ denotes the set of smooth solenoidal functions in $H^1_0( \Omega )^3$.) This result means that the usual weak solution of the time-dependent Oseen function with zero initial data and zero body force may be represented by a single layer potential, provided a certain integral equation involving the boundary data may be solved.