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• W2883990480 abstract "We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field $theta$ via an active control of the flow velocity $v$, governed by the incompressible Navier--Stokes equations, in an open bounded and connected domain $OmegasubsetBbb{R}^2$. We consider the velocity field generated by a control input that acts tangentially on the boundary of the domain through the Navier slip boundary conditions. This problem is motivated by mixing the fluids within a cavity or vessel by moving the walls or stirring at the boundaries. Our main objective is to design an optimal Navier slip boundary control that optimizes mixing at a given final time $T>0$. Nondissipative scalars, both passive and active, governed by the transport equation will be addressed. In the absence of diffusion, transport and mixing occur due to pure advection. This essentially leads to a nonlinear control problem of a semidissipative system. Sobolev norm for the dual space $(H^{1}(Omega))'$ of $H^{1}( Omega)$ is adopted to quantify mixing due to the property of weak convergence. The challenge arises from the vanishing diffusivity and nonlinear coupling of the system, which results in requiring the velocity field to satisfy $ int^{T}_{0}|nabla v|_{L^{infty}(Omega)}, dtau<infty$. We present a rigorous proof to show the existence of an optimal controller for both passive and active scalars and that the compatibility conditions for initial and boundary data are not required for Navier slip boundary control. Finally, we establish the first-order necessary conditions for optimality for both cases by using a variational inequality." @default.
• W2883990480 created "2018-08-03" @default.
• W2883990480 creator A5026693129 @default.
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• W2883990480 date "2018-01-01" @default.
• W2883990480 modified "2023-10-06" @default.
• W2883990480 title "Boundary Control for Optimal Mixing via Navier--Stokes Flows" @default.
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• W2883990480 doi "https://doi.org/10.1137/17m1148049" @default.
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