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The existence of a weak solution u(x, t) , in the -sense of J. Le'ray (), is established for the initial-boundary value problem for the Navier-Stokes equations: [Formula omitted] The solution is required to satisfy the initial condition u(x, 0) = u[subscript]o (x) for x ɛΩ, and the boundary condition u(x, t) = 0 on ∂Ω x [0, T], where Ω is an open bounded domain in IR[superscript]n, with 2 ≤ n ≤ 4. Galerkin's method is employed to find a weak solution u as the limit of approximate solutionsdoi:10.14288/1.0104056 fatcat:cmuv7fclojbo3kkazdyq4ncxp4