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Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D

Andrea Poiatti, Andrea Signori

Abstract

In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The control enters the governing state system in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas the cost functional to be minimized is of tracking-type. The novelties of the present paper are twofold: in addition to the control application, the intrinsic difficulties of the optimization problem forced us to first establish new regularity results on the nonlocal Cahn-Hilliard equation that were unknown even without the coupling with a velocity field and are therefore of independent interest. This happens to be shown using the recently proved separation property along with ad hoc Hölder regularities and a bootstrap method. For the control problem, the existence of an optimal strategy as well as first-order necessary conditions are then established.

Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D

Abstract

In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The control enters the governing state system in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas the cost functional to be minimized is of tracking-type. The novelties of the present paper are twofold: in addition to the control application, the intrinsic difficulties of the optimization problem forced us to first establish new regularity results on the nonlocal Cahn-Hilliard equation that were unknown even without the coupling with a velocity field and are therefore of independent interest. This happens to be shown using the recently proved separation property along with ad hoc Hölder regularities and a bootstrap method. For the control problem, the existence of an optimal strategy as well as first-order necessary conditions are then established.
Paper Structure (16 sections, 5 theorems, 147 equations)

This paper contains 16 sections, 5 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Notation and Main Results
  3. Notation
  4. Assumptions and main results
  5. Mathematical Analysis of the State System
  6. Proof of Theorem \ref{['ExistCahn']}
  7. Uniqueness and continuous dependence estimate
  8. Existence of weak solutions
  9. Existence of strong solutions: parts (i)-(ii)
  10. Extra regularity results for separated initial data: part (iii)
  11. $L^4_tW^{1,6}_x$ regularity on $\varphi$ for separated initial data: part (iv)
  12. $L^4_tH^{2}_x\cap L^3_t W^{1,\infty}_x$ regularity on $\varphi$ for separated initial data: part (v)
  13. Continuous dependence results
  14. The Control Problem
  15. Existence of optimal controls
  16. ...and 1 more sections

Key Result

Theorem 2.2

Let the assumptions h1-h2 be fulfilled. Assume that ${\boldsymbol{v}}\in L^{4}(0,T;\textbf{L}^6_\sigma)$, $\varphi _{0}\in H$ with $F(\varphi_0) \in L^{1}(\Omega)$ and $|(\varphi _{0})_\Omega|<1$. Then, there exists a unique weak solution $(\varphi,\mu)$ to eq:1-eq:4 in the sense that and it satisfies and $\varphi ({0})=\varphi _{{0}}$ almost everywhere in $\Omega$. The weak solution fulfills the

Theorems & Definitions (16)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 6 more