Regularity and Optimal Control of Non Local Cahn Hilliard Brinkman system with Singular Potential
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| Format: | Preprint |
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2023
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| _version_ | 1866915069169762304 |
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| author | Dharmatti, Sheetal K, Greeshma |
| author_facet | Dharmatti, Sheetal K, Greeshma |
| contents | The evolution of two incompressible, immiscible, isothermal fluids in a bounded domain and a porous media is described by the coupled Cahn-Hilliard-Brinkman (CHB) system. The CHB system consists of the Cahn-Hilliard equation describing the dynamics of the relative concentration of fluids and the Brinkman equation for velocity. This work addresses the optimal control problem for a two-dimensional nonlocal CHB system with a singular-type potential. The existence and regularity results are obtained by approximating the singular potential by a sequence of regular potentials and introducing a sequence of mobility terms to resolve the blow-up due to the singularity of the potential. Further, we prove the existence of a strong solution under higher regularity assumptions on the initial data and the uniqueness of the solution using the weak-strong uniqueness technique. By considering the external forcing term in the velocity equation as a control, we prove the existence of an optimal control for a tracking type cost functional. The differentiability properties of the control-to-state operator are studied to establish the first-order necessary optimality conditions. Moreover, the optimal control is characterised in terms of the adjoint variable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_05008 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regularity and Optimal Control of Non Local Cahn Hilliard Brinkman system with Singular Potential Dharmatti, Sheetal K, Greeshma Analysis of PDEs Optimization and Control 35D35, 35Q35, 49J20, 49J50, 49K20, 76S05, 76D99, 76T99 The evolution of two incompressible, immiscible, isothermal fluids in a bounded domain and a porous media is described by the coupled Cahn-Hilliard-Brinkman (CHB) system. The CHB system consists of the Cahn-Hilliard equation describing the dynamics of the relative concentration of fluids and the Brinkman equation for velocity. This work addresses the optimal control problem for a two-dimensional nonlocal CHB system with a singular-type potential. The existence and regularity results are obtained by approximating the singular potential by a sequence of regular potentials and introducing a sequence of mobility terms to resolve the blow-up due to the singularity of the potential. Further, we prove the existence of a strong solution under higher regularity assumptions on the initial data and the uniqueness of the solution using the weak-strong uniqueness technique. By considering the external forcing term in the velocity equation as a control, we prove the existence of an optimal control for a tracking type cost functional. The differentiability properties of the control-to-state operator are studied to establish the first-order necessary optimality conditions. Moreover, the optimal control is characterised in terms of the adjoint variable. |
| title | Regularity and Optimal Control of Non Local Cahn Hilliard Brinkman system with Singular Potential |
| topic | Analysis of PDEs Optimization and Control 35D35, 35Q35, 49J20, 49J50, 49K20, 76S05, 76D99, 76T99 |
| url | https://arxiv.org/abs/2311.05008 |