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Volume 17, Issue 2
Analysis and Optimal Control of a System of Hemivariational Inequalities Arising in Non-Stationary Navier-Stokes Equation with Thermal Effects

Hailing Xuan, Xiaoliang Cheng & Xilu Wang

DOI: 10.4208/nmtma.OA-2023-0124

Numer. Math. Theor. Meth. Appl., 17 (2024), pp. 351-378.

Published online: 2024-05

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  • Abstract

In this paper, we primarily investigate the existence, dependence and optimal control results related to solutions for a system of hemivariational inequalities pertaining to a non-stationary Navier-Stokes equation coupled with an evolution equation of temperature field. The boundary conditions for both the velocity field and temperature field incorporate the generalized Clarke gradient. The existence and uniqueness of the weak solution are established by utilizing the Banach fixed point theorem in conjunction with certain results pertaining to hemivariational inequalities. The finite element method is used to discretize the system of hemivariational inequalities and error bounds are derived. Ultimately, a result confirming the existence of a solution to an optimal control problem for the system of hemivariational inequalities is elucidated.

  • Keywords

Non-stationary Navier-Stokes equation, hemivariational inequalities, thermal effects, optimal control, existence and uniqueness.

  • AMS Subject Headings

65M15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-17-351, author = {Xuan , HailingCheng , Xiaoliang and Wang , Xilu}, title = {Analysis and Optimal Control of a System of Hemivariational Inequalities Arising in Non-Stationary Navier-Stokes Equation with Thermal Effects}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2024}, volume = {17}, number = {2}, pages = {351--378}, abstract = {

In this paper, we primarily investigate the existence, dependence and optimal control results related to solutions for a system of hemivariational inequalities pertaining to a non-stationary Navier-Stokes equation coupled with an evolution equation of temperature field. The boundary conditions for both the velocity field and temperature field incorporate the generalized Clarke gradient. The existence and uniqueness of the weak solution are established by utilizing the Banach fixed point theorem in conjunction with certain results pertaining to hemivariational inequalities. The finite element method is used to discretize the system of hemivariational inequalities and error bounds are derived. Ultimately, a result confirming the existence of a solution to an optimal control problem for the system of hemivariational inequalities is elucidated.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2023-0124}, url = {http://global-sci.org/intro/article_detail/nmtma/23104.html} }
TY - JOUR T1 - Analysis and Optimal Control of a System of Hemivariational Inequalities Arising in Non-Stationary Navier-Stokes Equation with Thermal Effects AU - Xuan , Hailing AU - Cheng , Xiaoliang AU - Wang , Xilu JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 351 EP - 378 PY - 2024 DA - 2024/05 SN - 17 DO - http://doi.org/10.4208/nmtma.OA-2023-0124 UR - https://global-sci.org/intro/article_detail/nmtma/23104.html KW - Non-stationary Navier-Stokes equation, hemivariational inequalities, thermal effects, optimal control, existence and uniqueness. AB -

In this paper, we primarily investigate the existence, dependence and optimal control results related to solutions for a system of hemivariational inequalities pertaining to a non-stationary Navier-Stokes equation coupled with an evolution equation of temperature field. The boundary conditions for both the velocity field and temperature field incorporate the generalized Clarke gradient. The existence and uniqueness of the weak solution are established by utilizing the Banach fixed point theorem in conjunction with certain results pertaining to hemivariational inequalities. The finite element method is used to discretize the system of hemivariational inequalities and error bounds are derived. Ultimately, a result confirming the existence of a solution to an optimal control problem for the system of hemivariational inequalities is elucidated.

Xuan , HailingCheng , Xiaoliang and Wang , Xilu. (2024). Analysis and Optimal Control of a System of Hemivariational Inequalities Arising in Non-Stationary Navier-Stokes Equation with Thermal Effects. Numerical Mathematics: Theory, Methods and Applications. 17 (2). 351-378. doi:10.4208/nmtma.OA-2023-0124
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