Well-posedness result for the flow of Maxwell viscoelastic fluid of White-Metzner type

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In this paper, we investigate the flow characteristics of an unsteady quasistatic Maxwell viscoelastic fluid of White-Metzner type within a three-dimensional bounded domain. Our objective is to establish the well-posedness of the resulting partial differential equations initial boundary value problem, both locally and globally in time, under the condition of small initial data. To demonstrate the existence of a solution, we employ an iterative method that transforms our problem into a sequence of linear subproblems. At each iteration, we tackle both elliptic and hyperbolic problems. The convergence of this iterative process provides the solution to the original problem.