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2024. Doi: 10.3934/cpaa.2024068
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Optimal control for non-isothermal binary viscoelastic and incompressible flows with stress diffusion

  • Department of Mathematics and Computer Science, University of the Philippines Baguio, Philippines

Received:  February 2024
Revised:  July 2024
Early access:  August 2024

The first author is supported in part by the One UP Professorial Chair Grant (2022-2024) SN 2019-101366

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  • Distributed optimal control problems for a binary mixture of non-isothermal, incompressible, and non-Newtonian fluids under the framework of diffusive Johnson–Segalman models will be discussed. The flow is governed by a coupling of the two-dimensional Cahn–Hilliard equation for the order-parameter and chemical potential, the biharmonic heat equation with Voigt-type damping for the temperature, the incompressible Navier–Stokes equation for the mean velocity, and a Jeffreys-type differential constitutive equation for the viscoelastic stress tensor. The total Cauchy stress tensor for the model is given by the sum of the viscous stress, the contribution due to surface tension, and a quadratic function of the viscoelastic stress. The latter is based on a recent non-standard constitutive law for the Helmholtz free energy. The coefficients pertaining to diffusion processes depend on the concentration and temperature. We provide regularity results for the optimal control with various objective cost functionals. Such results rely on careful analysis of the corresponding linearized and adjoint problems. In particular, we study the strong, weak, and very weak solutions of the linearized and adjoint systems, and present the function spaces for the distributional time-derivatives.

    Mathematics Subject Classification: Primary: 35B65, 35Q35, 35Q93; Secondary: 49J20, 76D55.

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