Existence of Optimal Control for Dirichlet Boundary Optimization in a Phase Field Problem

Abstract

Phase field modeling finds utility in various areas. In optimization theory in particular, the distributed control and Neumann boundary control of phase field models have been investigated thoroughly. Dirichlet boundary control in parabolic equations is commonly addressed using the very weak formulation or an approximation by Robin boundary conditions. In this paper, the Dirichlet boundary control for a phase field model with a non-singular potential is investigated using the Dirichlet lift technique. The corresponding weak formulation is analyzed. Energy estimates and problem-specific embedding results are provided, leading to the existence and uniqueness of the solution for the state equation. These results together show that the control to state mapping is well defined and bounded. Based on the preceding findings, the optimization problem is shown to have a solution.

1 Introduction

Phase field models (PFM) find utility in various areas, where interface tracking is of interest [7, 9]. Applications include, but are not limited to, the modeling of phase transitions in materials science [8, 9, 12, 21, 22, 25, 28, 30, 33], crack propagation [11] or tumor growth [14]. The first of the three mentioned applications is addressed in this article. Namely, the controlled solidification of a pure substance by means of supercooling. To achieve this goal, the methods of PDE-constrained optimization are applied, where the PFM serves as the state equation.

This class of problems has been studied thoroughly since the 1990s. One of the first contributions addresses the distributed control (DC) of the phase field model [20]. Since then, the DC of PFM has seen many theoretical developments and applications, including the simulation of isothermal alloy solidification [3], tumor growth modeling [26], inductive heating [35], and many others. Some of the theoretical advancements in the DC for the PFM include results on existence of the control and the derivation of optimality conditions for specific variants of the model that feature singular potentials [12, 14].

In all of the contributions discussed so far, the problem formulations are typically supplemented by homogeneous Neumann (natural) or Dirichlet boundary conditions. As a consequence, the resulting state equation is of a variational type [23, 29, 32]. Another type of control that may be considered for PFM is the Neumann or Robin Control (NoR) [13, 34]. This type of control naturally results in a variational formulation of the problem as well. To supplement the existing theoretical results, this paper focuses on the Dirichlet boundary control for the PFM.

A particular PFM that governs the solidification of a supercooled melt [4, 27, 28] is considered as the state equation of the system. It consists of the heat equation with a latent heat term and the phase field equation. In the optimization problem, a Dirichlet boundary condition for the heat equation that results in a particular solid shape (expressed by the phase field) at final time is found. The problem is described fully in Section 2. The state equation is shown to have a unique solution and the necessary auxiliary results, such as the boundedness of the solution operator with respect to the control and problem specific embedding statements are provided. These are then used to prove the existence of solution of the optimal control problem.

2 Dirichlet Control for the PFM Using the Dirichlet Lift

To the authors’ best knowledge, there exist only a handful of publications addressing the Dirichlet boundary control of parabolic differential equations. The Dirichlet boundary condition optimization for a single parabolic PDE has been treated in the past by using a Robin boundary condition approximation [2]. The same class of problems was addressed in [16, 23], where the very weak formulation is utilized for the state equation. Dirichlet lifts have been used in the so called energy approach to optimization and first detailed for an elliptic problem in [10]. More recently, this technique has been used for the optimization of a Dirichlet boundary condition for a parabolic PDE as well [17].

The presented method, although developed independently, bears some resemblances to [10, 17]. We use the technique of the Dirichlet lift to give a direct correspondence between the original state equation and the “lifted” reformulation.

For reader’s convenience, a summary of notation used for Bochner spaces and fundamental statements that find utility in the proof are included in the Appendix. Also, note that the differential symbols dx and dS in volume and surface integrals are left out throughout the text to improve readability.

The state equation of the problem in question is a phase field system that can be used to simulate the solidification of a supercooled pure melt. It is composed of the heat equation and the phase field (Allen-Cahn) equation. The model is introduced in dimensionless form in accordance with the related works [6, 27, 28]. The goal of the optimization, which is described by the cost functional J, is to achieve a certain shape of the solid body at final time T. To formulate the problem, we let ${\varOmega }\subset \mathbb {R}^{n}$ for n = 1,2,3 be a bounded domain with a Lipschitz boundary and assume T > 0. Let $u:\overline {{\varOmega }}\times \left [0,T\right ]\rightarrow \mathbb {R}$ represent the temperature field and let $p:\overline {{\varOmega }}\times \left [0,T\right ]\rightarrow \mathbb {R}$ be the phase field which determines the shape of the solid subdomain by

$$ {\varOmega}_{s}\left( t\right)=\left\{ \left|\boldsymbol{x}\in{\varOmega}\right|p\left( \boldsymbol{x},t\right)>\frac{1}{2}\right\} . $$

Then, the formal statement of the problem reads

$$ \begin{array}{@{}rcl@{}} \underset{\left( \left( u,p\right),\theta\right)}{{\min}}J\left( u,p,\theta\right)&\equiv & \frac{1}{2}\underset{{\varOmega}}{\int}\left|p\left( T\right)-p_{\text{f}}\right|^{2}+\frac{\gamma}{2}\underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}\left|\theta\right|^{2} \end{array} $$

(1)

$$ \begin{array}{@{}rcl@{}} \text{s.t. {\qquad} \quad}u_{t}&= & {\Delta} u+Qp_{t} \qquad\qquad\qquad \text{on }\left( 0,T\right)\times{\varOmega}, \end{array} $$

(2)

$$ \begin{array}{@{}rcl@{}} u|_{\partial{\varOmega}}&= & \theta \qquad\qquad~~~~~~~\qquad ~~~~~~\text{on }\left[0,T\right]\times\partial{\varOmega}, \end{array} $$

(3)

$$ \begin{array}{@{}rcl@{}} u|_{t=0}&= & u_{\text{ini}} \qquad\qquad\qquad\qquad\qquad\qquad~~~ \text{on }{\varOmega}, \end{array} $$

(4)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}p_{t}&= & \xi^{2}{\Delta} p+f\left( u,p,\xi\right), \end{array} $$

(5)

$$ \begin{array}{@{}rcl@{}} f\left( u,p,\xi\right)&= & p\left( 1-p\right)\left( p-\frac{1}{2}\right)-b\beta\xi u, \end{array} $$

(6)

$$ \begin{array}{@{}rcl@{}} p|_{\partial{\varOmega}}&= & 0 \qquad\qquad\qquad~~\qquad\quad \text{on }\left[0,T\right]\times\partial{\varOmega}, \end{array} $$

(7)

$$ \begin{array}{@{}rcl@{}} p|_{t=0}&= & p_{\text{ini}} \qquad\qquad\qquad\qquad\qquad\qquad\quad \text{on }{\varOmega}, \end{array} $$

(8)

where β > 0 specifies the initial supercooling and α,Q are material-specific constants. The parameter ξ > 0 determines the phase interface thickness and b > 0 is a constant calculated so that the Gibbs-Thomson relation [18] is asymptotically recovered as ξ → 0. For this particular model, we get $b=\frac {1}{6}$ (see, e.g. [28]). The initial conditions for the temperature and phase field are given by $u_{\text {ini}}\in H^{1}\left ({\varOmega }\right )$, $p_{\text {ini}}\in H^{1}\left ({\varOmega }\right )$. The solution of the minimization problem (1)–(8) is a function $\theta :\partial {\varOmega }\times \left [0,T\right ]\rightarrow \mathbb {R}$ which controls the temperature at the boundary of Ω in such a way that p is as close as possible to the target phase field profile $p_{\text {f}}\in L_{2}\left ({\varOmega }\right )$ at t = T. If necessary, the magnitude of the control 𝜃 can be reduced by increasing the regularization parameter γ ≥ 0 in the cost functional (1). Derivation of the dimensionless model and the meaning of the original dimensional quantities can be found in [28].

Next, we cast the problem into weak form, define the control and solution spaces and impose restrictions on the control set. To formulate the problem using the Dirichlet lift, consider the control space

$$ X_{C}\equiv H^{4}\left( {\varOmega}\times\left( 0,T\right)\right). $$

(9)

This high level of regularity is a consequence of the compact embedding statements used in the analysis that follows. Let 𝜃 in (3) be such that there exists η ∈ X_C for which

$$ \text{Tr}\left( \eta\right)=\theta. $$

(10)

Using the concept of a weak solution, we look for (see Section A.1 in the Appendix for Bochner space related notation)

$$ \left( u,p\right)\in X_{S}\equiv W\left( 0,T;2,2;H^{1}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right)\times W\left( 0,T;2,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right) $$

such that (1)–(8) is satisfied in the distributional sense. To this end, it is purposeful to define

$$ \hat{u}\equiv u-\eta\in W\left( 0,T;2,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right). $$

(11)

Using the principles of Dirichlet lift and (11), we can reformulate the optimization problem (1)–(8). Let W_ad ⊂ X_C be closed, convex and bounded, then the problem in question reads

$$ \begin{array}{@{}rcl@{}} \underset{\left( \left( \hat{u},\hat{p}\right),\eta\right)\in X_{S}\times W_{\text{ad}}}{{\min}}J\left( \hat{u},\hat{p},\eta\right)&\equiv & \frac{1}{2}\underset{{\varOmega}}{\int}|\hat{p}\left( T\right)-p_{\text{f}}|^{2}+\frac{\gamma}{2}\underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}|\text{Tr }\left( \eta\right)|^{2} \end{array} $$

(12)

$$ \begin{array}{@{}rcl@{}} \text{s.t. {\quad}}\hat{u}_{t}-{\Delta}\hat{u}&= & Q\hat{p}_{t}-\eta_{t}+{\Delta}\eta \qquad ~~~\text{on }\left( 0,T\right)\times{\varOmega}, \end{array} $$

(13)

$$ \begin{array}{@{}rcl@{}} \hat{u}|_{\partial{\varOmega}}&= & 0 \qquad\qquad\qquad\qquad~ \text{on }\left[0,T\right]\times\partial{\varOmega}, \end{array} $$

(14)

$$ \begin{array}{@{}rcl@{}} \hat{u}|_{t=0}&= & u_{\text{ini}}-\eta|_{t=0} \qquad \qquad\qquad\qquad\text{on }{\varOmega}, \end{array} $$

(15)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}\hat{p}_{t} &= & \xi^{2}{\Delta}\hat{p}+f\left( \hat{u},\hat{p},\eta,\xi\right), \end{array} $$

(16)

$$ \begin{array}{@{}rcl@{}} f\left( \hat{u},\hat{p},\xi\right)&= & \hat{p}\left( 1-\hat{p}\right)\left( \hat{p}-\frac{1}{2}\right)-b\beta\xi\left( \hat{u}+\eta\right), \end{array} $$

(17)

$$ \begin{array}{@{}rcl@{}} \hat{p}|_{\partial{\varOmega}}&= & 0 \qquad \qquad\qquad~\qquad\text{on }\left[0,T\right]\times\partial{\varOmega}, \end{array} $$

(18)

$$ \begin{array}{@{}rcl@{}} \hat{p}|_{t=0}&= & p_{\text{ini}} \qquad\qquad\qquad\qquad\qquad ~~~~~~\text{on }{\varOmega}, \end{array} $$

(19)

where the equations (13) and (16) are in the sense of distributions. A pair of functions

$$ \left( \hat{u},\hat{p}\right)\in\hat{X}_{S}\equiv W\left( 0,T;2,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right)\times W\left( 0,T;8,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right) $$

(20)

that satisfies (13)–(19) is called a solution of the state equation (13)–(19). It follows from the principles of the Dirichlet lift that a solution of (13)-(19) is also a solution to (2)–(8) when (10) holds. The statement that we aim to prove follows.

Theorem 1

Let the control space X_C and solution space $\hat {X}_{S}$ be given by (9) and (20), respectively, and let W_ad ⊂ X_C be closed, convex and bounded. Then, there exists a (not necessarily unique) solution of the optimization problem (12)–(19).

The rest of the text is dedicated to the proof of Theorem 1. In Section 3, problem specific embedding results are provided. These are then used in Section 4 to show that there exists a unique solution of the state equation (13)–(19) for any η ∈ X_C. The results of Sections 3 and 4 are then used to show the existence of optimal control for (12)–(19) in Section 5, which concludes the proof of Theorem 1.

3 Specific Embedding-Based Results

In this section, two specific results are derived, which find utility in the analysis of the solution operator of (13)–(19) and in the proof of the existence of optimal control.

Lemma 1

Let ${\varOmega }\subset \mathbb {R}^{n}$ be a bounded domain with a Lipschitz boundary for $n\in \left \{ 1,2,3\right \}$ and T > 0. Let $f\in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )$. Then,

$$ \left\Vert f\left( t\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}<+\infty\text{ for any }t\in\left[0,T\right]. $$

(21)

Proof

Since Ω has a Lipschitz boundary, so does ${\varOmega }\times \left (0,T\right )$. Using Theorem 3 and setting m = 4, p = 2 and k = 1 leads to

$$ H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)\hookrightarrow C^{1,\beta}\left( \overline{{\varOmega}}\times\left[0,T\right]\right)\text{ for all }\beta\in\left( 0,1\right]. $$

This means that there exists a Hölder continuous function $g\in C^{1,\beta }\left (\overline {{\varOmega }}\times \left [0,T\right ]\right )$ such that f = g in $H^{4}\left ({\varOmega }\times \left (0,T\right )\right )$ and specifically $g\left (t\right )\in C^{1,\beta }\left (\overline {{\varOmega }}\right )$ for any $t\in \left [0,T\right ]$. Since $\left \Vert g\left (t\right )\right \Vert _{H^{1}\left ({\varOmega }\right )}=\left \Vert f\left (t\right )\right \Vert _{H^{1}\left ({\varOmega }\right )}$, the conclusion (21) holds. □

Lemma 2

Let ${\varOmega }\subset \mathbb {R}^{n}$ be a bounded domain (with an arbitrary boundary) for $n\in \left \{ 1,2,3\right \}$ and T > 0. Let $f_{n}\rightharpoonup f$ in $W\left (0,T;8,2;{H_{0}^{1}}\left ({\varOmega }\right ),L_{2}\left ({\varOmega }\right )\right )$. Then,

$$ \begin{array}{@{}rcl@{}} {f_{n}^{3}}\rightarrow f^{3} & \text{ in }L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right), \end{array} $$

(22)

$$ \begin{array}{@{}rcl@{}} {f_{n}^{2}}\rightarrow f^{2} & \text{ in }L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right). \end{array} $$

(23)

Proof

With the help of Theorem 4 (setting m = 1, k = 0, p = 2 and q = 5), the Gelfand triple

$$ {H_{0}^{1}}\left( {\varOmega}\right)\hookrightarrow\hookrightarrow L_{5}\left( {\varOmega}\right)\hookrightarrow L_{2}\left( {\varOmega}\right) $$

can be constructed. Applying Theorem 5 (Aubin-Lions) gives

$$ f_{n}\rightarrow f\text{ in }L_{8}\left( 0,T;L_{5}\left( {\varOmega}\right)\right). $$

It is sufficient to prove strong convergence of ${f_{n}^{3}}\rightarrow f^{3}$ in $L_{2}\left (0,T;L_{\frac {5}{3}}\left ({\varOmega }\right )\right )$, since the reflexivity of $L_{2}\left (0,T;L_{\frac {5}{3}}\left ({\varOmega }\right )\right )$ and the embedding

$$ {H_{0}^{1}}\left( {\varOmega}\right)\hookrightarrow L_{\frac{5}{2}}\left( {\varOmega}\right) $$

(24)

give rise to

$$ L_{2}\!\left( \!0,T;L_{\frac{5}{3}}\left( {\varOmega}\right)\!\right)\!\cong L_{2}\!\left( \!0,T;L_{\frac{5}{3}}\left( {\varOmega}\right)\!\right)^{**}\cong L_{2}\!\left( 0,T;L_{\frac{5}{2}}\left( {\varOmega}\right)^{*}\right)\hookrightarrow L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right). $$

(25)

For any two numbers $a,b\in \mathbb {R}$, the estimate

$$ |b^{3}-a^{3}|\leq3\left( a^{2}+b^{2}\right)|b-a| $$

(26)

may be derived. Using (26), the estimate

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left\Vert {f_{n}^{3}}-f^{3}\right\Vert_{L_{\frac{5}{3}}\left( {\varOmega}\right)} & \leq&3\!\left( \underset{{\varOmega}}{\int}\left( {f_{n}^{2}}\left( x\right)+f^{2}\left( x\right)\right)^{\frac{5}{3}}|f_{n}\left( x\right)-f\left( x\right)|^{\frac{5}{3}}\mathrm{d} x\right)^{\frac{3}{5}} \\ & \leq&3\!\left\Vert {f_{n}^{2}}\left( f_{n} - f\right)\!\right\Vert_{L_{\frac{5}{3}}\left( {\varOmega}\right)}+3\!\left\Vert f^{2}\left( f_{n} - f\right)\right\Vert_{L_{\frac{5}{3}}\left( {\varOmega}\right)}\text{ for all }t\!\in\!\left[0,T\right] \end{array} $$

(27)

can be constructed. For any $v,w\in L^{5}\left ({\varOmega }\right )$, Hölder’s inequality (Lemma 5) can be used with exponents $\frac {3}{2}$ and 3 to arrive at

$$ \left\Vert w^{2}v\right\Vert_{L_{\frac{5}{3}}\left( {\varOmega}\right)}=\left( \underset{{\varOmega}}{\int}w^{\frac{10}{3}}v^{\frac{5}{3}}\right)^{\frac{3}{5}}\leq\left( \underset{{\varOmega}}{\int}w^{5}\right)^{\frac{2}{5}}\left( \underset{{\varOmega}}{\int}v^{5}\right)^{\frac{1}{5}}=\left\Vert w\right\Vert_{L_{5}\left( {\varOmega}\right)}^{2}\left\Vert v\right\Vert_{L_{5}\left( {\varOmega}\right)}. $$

(28)

Combining (27) and (28) yields

$$ \left\Vert {f_{n}^{3}}-f^{3}\right\Vert_{L_{\frac{5}{3}}\left( {\varOmega}\right)}\leq3\left\Vert f_{n}\right\Vert_{L_{5}\left( {\varOmega}\right)}^{2}\left\Vert f_{n}-f\right\Vert_{L_{5}\left( {\varOmega}\right)}+3\left\Vert f\right\Vert_{L_{5}\left( {\varOmega}\right)}^{2}\left\Vert f_{n}-f\right\Vert_{L_{5}\left( {\varOmega}\right)}. $$

Squaring, integrating over $\left [0,T\right ]$ and using Young’s inequality (Lemma 6) then yields

$$ \begin{array}{@{}rcl@{}} \left\Vert {f_{n}^{3}}-f^{3}\right\Vert_{L_{2}\left( 0,T;L_{\frac{5}{3}}\left( {\varOmega}\right)\right)}^{2} & \leq6\underset{0}{\overset{T}{\int}}\left\Vert f_{n}\left( t\right)\right\Vert_{L_{5}\left( {\varOmega}\right)}^{4}\left\Vert f_{n}\left( t\right)-f\left( t\right)\right\Vert_{L_{5}\left( {\varOmega}\right)}^{2} \\ & +6\underset{0}{\overset{T}{\int}}\left\Vert f\left( t\right)\right\Vert_{L_{5}\left( {\varOmega}\right)}^{4}\left\Vert f_{n}\left( t\right)-f\left( t\right)\right\Vert_{L_{5}\left( {\varOmega}\right)}^{2}. \end{array} $$

(29)

Applying the standard Hölder’s inequality (Lemma 5) to the right-hand side of (29) gives

$$ \begin{array}{@{}rcl@{}} \left\Vert {f_{n}^{3}}-f^{3}\right\Vert_{L_{2}\left( 0,T;L_{\frac{5}{3}}\left( {\varOmega}\right)\right)}^{2} & \leq6\left( \underset{0}{\overset{T}{\int}}\left\Vert f_{n}\left( t\right)\right\Vert_{L_{5}\left( {\varOmega}\right)}^{8}\mathrm{d} t\right)^{\frac{1}{2}}\left( \underset{0}{\overset{T}{\int}}\left\Vert f_{n}\left( t\right)-f\left( t\right)\right\Vert_{L_{5}\left( {\varOmega}\right)}^{4}\mathrm{d} t\right)^{\frac{1}{2}}\\ & +6\left( \underset{0}{\overset{T}{\int}}\left\Vert f\left( t\right)\right\Vert_{L_{5}\left( {\varOmega}\right)}^{8}\mathrm{d} t\right)^{\frac{1}{2}}\left( \underset{0}{\overset{T}{\int}}\left\Vert f_{n}\left( t\right)-f\left( t\right)\right\Vert_{L_{5}\left( {\varOmega}\right)}^{4}\mathrm{d} t\right)^{\frac{1}{2}}. \end{array} $$

Note that $f_{n}\rightarrow f$ in $L_{8}\left (0,T;L_{5}\left ({\varOmega }\right )\right )$, $\left \Vert f_{n}\right \Vert _{L_{8}\left (0,T;L_{5}\left ({\varOmega }\right )\right )}^{4}$ is bounded and $\left \Vert f\right \Vert _{L_{8}\left (0,T;L_{5}\left ({\varOmega }\right )\right )}^{4}$ is finite. Furthermore,

$$ \begin{array}{@{}rcl@{}} f_{n}\rightarrow f & \text{ in }L_{4}\left( 0,T;L_{5}\left( {\varOmega}\right)\right),\\ f_{n}\rightarrow f & \text{ in }L_{2}\left( 0,T;L_{5}\left( {\varOmega}\right)\right), \end{array} $$

since $L_{8}\left (0,T;L_{5}\left ({\varOmega }\right )\right )\hookrightarrow L_{4}\left (0,T;L_{5}\left ({\varOmega }\right )\right )$. Altogether,

$$ {f_{n}^{3}}\rightarrow f^{3}\text{ in }L_{2}\left( 0,T;L_{\frac{5}{3}}\left( {\varOmega}\right)\right)\Rightarrow {f_{n}^{3}}\rightarrow f^{3}\text{ in }L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right) $$

which completes the proof of (22). To get the result (23), one just replaces the estimate (26) with

$$ |b^{2}-a^{2}|\leq\left( |a|+\left|b\right|\right)\left|b-a\right| $$

and uses the inequality

$$ \left\Vert w\right\Vert_{L_{\frac{5}{2}}\left( {\varOmega}\right)}\leq c\left\Vert w\right\Vert_{L_{5}\left( {\varOmega}\right)} $$

in the estimate analogous to (28). □

4 Analysis of the State Equation

We start by stating the weak formulation of the problem (13)–(19). To simplify the notation, we use u and p instead of $\hat {u}$ and $\hat {p}$, respectively. Using standard methods, we arrive at

$$ \begin{array}{@{}rcl@{}} \frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}u\left( t\right)v+\underset{{\varOmega}}{\int}\nabla u\left( t\right)\cdot\nabla v+\underset{{\varOmega}}{\int}\nabla\eta\left( t\right)\cdot\nabla v & =&Q\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}p\left( t\right)v-\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}\eta\left( t\right)v \\ && \text{ for all }v\in {H_{0}^{1}}\left( {\varOmega}\right)\text{ a.e. in }\left[0,T\right], \end{array} $$

(30)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}p\left( t\right)w+\xi^{2}\underset{{\varOmega}}{\int}\nabla p\left( t\right)\cdot\nabla w & =&\underset{{\varOmega}}{\int}f_{0}\left( p\left( t\right)\right)w-b\beta\xi\underset{{\varOmega}}{\int}\left( u\left( t\right)+\eta\left( t\right)\right)w \\ && \text{ for all }w\in {H_{0}^{1}}\left( {\varOmega}\right)\text{ a.e. in }\left[0,T\right], \end{array} $$

(31)

$$ \begin{array}{@{}rcl@{}} u\left( 0\right) & =&u_{\text{ini}}-\eta|_{t=0}, \end{array} $$

(32)

$$ \begin{array}{@{}rcl@{}} p\left( 0\right) & =&p_{\text{ini}}, \end{array} $$

(33)

where $f_{0}:\mathbb {R}\rightarrow \mathbb {R}$ is defined as $f_{0}\left (y\right )\equiv y\left (1-y\right )\left (y-\frac {1}{2}\right )$.

The following theorem states the existence and boundedness of the solution operator of the system (13)–(19). This result is used later in Section 5 to prove Theorem 1.

Theorem 2

Let the control space X_C and solution space X_S (the hat is omitted for the sake of readability) be given by (9) and (20), respectively, and let η ∈ X_C. Then, (30)–(33) has a unique solution in X_S.

Proof

The steps of the proof are as follows. First, the m th Galerkin approximation of the problem (30)–(33) is defined (part A). Then, the key energy estimate is derived (part B). Lastly, the existence and uniqueness of the solution is proven (parts C and D).

Part A - Galerkin Approximation

Consider a countable set of functions $w_{i}:{\varOmega }\rightarrow \mathbb {R}$ such that

$$ \begin{array}{@{}rcl@{}} \left( w_{i}\right)_{i\in\mathbb{N}} & \text{ form an orthonormal basis of }L_{2}\left( {\varOmega}\right), \end{array} $$

(34)

$$ \begin{array}{@{}rcl@{}} \left( w_{i}\right)_{i\in\mathbb{N}} & \text{ form an orthogonal basis of }{H_{0}^{1}}\left( {\varOmega}\right). \end{array} $$

(35)

An example of functions that satisfy properties (34)–(35) are the eigenfunctions of the Laplacian on ${H_{0}^{1}}\left ({\varOmega }\right )$ [15]. For each $m\in \mathbb {N}$, let there be two vector valued functions $u_{m},p_{m}:\left [0,T\right ]\rightarrow {H_{0}^{1}}\left ({\varOmega }\right )$ such that

$$ u_{m}\left( t\right)\equiv\sum\limits_{k=1}^{m}{\alpha_{m}^{k}}\left( t\right)w_{k},\quad p_{m}\left( t\right)\equiv\sum\limits_{k=1}^{m}{\beta_{m}^{k}}\left( t\right)w_{k}. $$

(36)

Additionally, let (36) be such that the problem

$$ \begin{array}{@{}rcl@{}} \underset{{\varOmega}}{\int}\dot{u}_{m}\left( t\right)w_{k}+\underset{{\varOmega}}{\int}\nabla u_{m}\left( t\right)\cdot\nabla w_{k}+\underset{{\varOmega}}{\int}\nabla\eta\left( t\right)\cdot\nabla w_{k}&= & Q\underset{{\varOmega}}{\int}\dot{p}_{m}\left( t\right)w_{k}-\underset{{\varOmega}}{\int}\dot{\eta}\left( t\right)w_{k} \\ & &\text{for{\quad}}k=1,\ldots,m,\text{ a.e. in }\left[0,T\right], \end{array} $$

(37)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}\underset{{\varOmega}}{\int}\dot{p}_{m}\left( t\right)w_{k}+\xi^{2}\underset{{\varOmega}}{\int}\nabla p_{m}\left( t\right)\cdot\nabla w_{k}&= & \underset{{\varOmega}}{\int}f_{0}\left( p_{m}\left( t\right)\right)w_{k}-b\beta\xi\underset{{\varOmega}}{\int}\left( u_{m}\left( t\right)+\eta\left( t\right)\right)w_{k} \\ && \text{for{\quad}}k=1,\ldots,m,\text{ a.e. in }\left[0,T\right], \end{array} $$

(38)

with the initial conditions

$$ \begin{array}{@{}rcl@{}} {\alpha_{m}^{k}}\left( 0\right) & =&\underset{{\varOmega}}{\int}\left( u_{\text{ini}}-\eta\left( 0\right)\right)w_{k}\text{{\quad}{for}}\quad k=1,\ldots,m, \end{array} $$

(39)

$$ \begin{array}{@{}rcl@{}} {\beta_{m}^{k}}\left( 0\right) & =&\underset{{\varOmega}}{\int}p_{\text{ini}}w_{k}\text{{\quad}{for}}\quad k=1,\ldots,m, \end{array} $$

(40)

is satisfied for all $m\in \mathbb {N}$. Standard ODE theory can be applied to show that (37)–(40) has a unique solution [5, 15]. Thus the functions (36) are well defined. We call the problem (37)–(40) the m th Galerkin approximation of (30)–(33).

Part B - Energy Estimates

Many of the techniques featured in this section have been inspired by previous works on PDE analysis [5, 15].

To simplify the notation, consider

$$ \begin{array}{@{}rcl@{}} \left( f,g\right) & \equiv&\underset{{\varOmega}}{\int}fg, \end{array} $$

(41)

$$ \begin{array}{@{}rcl@{}} \left\langle f,g\right\rangle & \equiv&\underset{{\varOmega}}{\int}fg+\underset{{\varOmega}}{\int}\nabla f\cdot\nabla g, \end{array} $$

(42)

$$ \begin{array}{@{}rcl@{}} B\left[f,g\right] & \equiv&\underset{{\varOmega}}{\int}\nabla f\cdot\nabla g, \end{array} $$

(43)

for any $f,g\in H^{1}\left ({\varOmega }\right )$. Additionally, the time derivative of the vector valued function $f:\left [0,T\right ]\rightarrow H^{1}\left ({\varOmega }\right )$ is denoted by $\dot {f}$ and the evaluation at a point (for any function) is also omitted, i.e. instead of $f\left (t\right )=g\left (t\right )\text { a.e. in }\left [0,T\right ]$, we write $f=g\text { a.e. in }\left [0,T\right ]$. In the following, the symbols $c_{1},c_{2},\text {{\dots }}$ denote positive constants.

Consider a fixed but arbitrary $m\in \mathbb {N}$. Then, multiply (37) by $\dot {\alpha }_{m}^{k}$and (38) by $\dot {\beta }_{m}^{k}$ for each k = 1,…,m. Adding up the equations multiplied by $\dot {\alpha }_{m}^{k}$ and $\dot {\beta }_{m}^{k}$ separately gives

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left( \dot{u}_{m},\dot{u}_{m}\right) + B\!\left[u_{m},\dot{u}_{m}\right] + B\!\left[\eta,\dot{u}_{m}\!\right] & = &Q\!\left( \dot{p}_{m},\dot{u}_{m}\right) - \left( \dot{\eta},\dot{u}_{m}\right)\text{ \!a.e. in \!}\left[0,T\right], \end{array} $$

(44)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}\!\left( \dot{p}_{m},\dot{p}_{m}\right) + \xi^{2}B\!\left[p_{m},\dot{p}_{m}\right] & = &\left( f_{0}\!\left( p_{m}\right),\dot{p}_{m}\right)-b\beta\xi\left( u_{m} + \eta,\dot{p}_{m}\right)\text{ \!a.e. in }\!\left[0,T\right]\!. \end{array} $$

(45)

Some of the terms in (45) can be rewritten as

$$ \begin{array}{@{}rcl@{}} B\left[p_{m},\dot{p}_{m}\right] & =&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla p_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}, \end{array} $$

(46)

$$ \begin{array}{@{}rcl@{}} f_{0}\left( p_{m}\right)\dot{p}_{m} & =&-\frac{\mathrm{d}}{\mathrm{d} t}\omega_{0}\left( p_{m}\right), \end{array} $$

(47)

where $\omega _{0}\left (y\right )=\frac {1}{4}\left (\left (y-\frac {1}{2}\right )^{2}-\frac {1}{4}\right )^{2}$ for any $y\in \mathbb {R}$. Using (46) and (47), the system (44)–(45) may be reformulated as

$$ \begin{array}{@{}rcl@{}} \left\Vert \dot{u}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+B\left[u_{m},\dot{u}_{m}\right]+B\left[\eta,\dot{u}_{m}\right]&= & Q\left( \dot{p}_{m},\dot{u}_{m}\right)-\left( \dot{\eta},\dot{u}_{m}\right)\text{ a.e. in }\left[0,T\right], \end{array} $$

(48)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}\left\Vert \dot{p}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla p_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\right)&= & -b\beta\xi\left( \dot{u}_{m}+\eta,\dot{p}_{m}\right)\text{ a.e. in }\left[0,T\right]. \end{array} $$

(49)

The bilinearity of $B\left [\cdot ,\cdot \right ]$ gives rise to the identity

$$ \begin{array}{@{}rcl@{}} B\left[u_{m}+\eta,\dot{u}_{m}\right] & =B\left[u_{m}+\eta,\dot{u}_{m}+\dot{\eta}\right]-B\left[u_{m}+\eta,\dot{\eta}\right]. \end{array} $$

(50)

Using (50), we can rewrite (48) as

$$ \begin{array}{@{}rcl@{}} \left\Vert \dot{u}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+B\left[u_{m}+\eta,\dot{u}_{m}+\dot{\eta}\right]= & Q\left( \dot{p}_{m},\dot{u}_{m}\right)-\left( \dot{\eta},\dot{u}_{m}\right)+B\left[u_{m}+\eta,\dot{\eta}\right]\text{ a.e. in }\left[0,T\right]. \end{array} $$

(51)

Applying the Young’s (Lemma 6) and Schwarz inequalities on the right-hand side of (51), we get the estimates

$$ \begin{array}{@{}rcl@{}} \left|Q\left( \dot{p}_{m},\dot{u}_{m}\right)\right| & \leq&\frac{\delta_{1}}{2}\left\Vert \dot{u}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{Q^{2}}{2\delta_{1}}\left\Vert \dot{p}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\text{ a.e. in }\left[0,T\right],\\ \left|-\left( \dot{\eta},\dot{u}_{m}\right)\right| & \leq&\frac{1}{2\delta_{2}}\left\Vert \dot{\eta}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\delta_{2}}{2}\left\Vert \dot{u}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\text{ a.e. in }\left[0,T\right],\\ \left|B\left[u_{m}+\eta,\dot{\eta}\right]\right| & \leq&\frac{c_{1}\delta_{3}}{2}\left\Vert u_{m}+\eta\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+\frac{c_{2}}{2\delta_{3}}\left\Vert \dot{\eta}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\text{ a.e. in }\left[0,T\right], \end{array} $$

where δ₁,δ₂ > 0. Setting $\delta _{1}=\delta _{2}=\frac {1}{2}$ leads to the estimate

$$ \begin{array}{@{}rcl@{}} \frac{1}{2}\left\Vert \dot{u}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} & \leq& Q^{2}\left\Vert \dot{p}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\left\Vert \dot{\eta}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{c_{1}\delta_{3}}{2}\left\Vert u_{m}+\eta\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2} \\ & &+\frac{c_{2}}{2\delta_{3}}\left\Vert \dot{\eta}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\text{ a.e. in }\left[0,T\right]. \end{array} $$

(52)

Estimating the right-hand side of (49) using analogous methods with δ₄ > 0 yields

$$ \begin{array}{@{}rcl@{}} \left|-b\beta\xi\left( u_{m}+\eta,\dot{p}_{m}\right)\right| & \leq b\beta\xi\left( \frac{\delta_{4}}{2}\left\Vert u_{m}+\eta\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{1}{2\delta_{4}}\left\Vert \dot{p}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\right)\text{ a.e. in }\left[0,T\right]. \end{array} $$

Setting $\delta _{4}=\frac {b\beta \xi }{\alpha \xi ^{2}}$ gives rise to the estimate (see (49))

$$ \begin{array}{@{}rcl@{}} \frac{\alpha\xi^{2}}{2}\left\Vert \dot{p}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla p_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\right) & \leq\frac{\left( b\beta\right)^{2}}{\alpha}\left\Vert u_{m}+\eta\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\text{ a.e. in }\left[0,T\right]. \end{array} $$

(53)

Multiplying (52) by $\frac {\alpha \xi ^{2}}{4Q^{2}}$ and adding the result to (53) gives

$$ \begin{array}{@{}rcl@{}} && \frac{\alpha\xi^{2}}{8Q^{2}}\left\Vert \dot{u}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\alpha\xi^{2}}{4}\left\Vert \dot{p}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\alpha\xi^{2}}{8Q^{2}}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla p_{m}\right\Vert +\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\right) \\ &\leq & c_{3}\left( \delta_{3}\left\Vert u_{m}+\eta\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+\left\Vert u_{m}+\eta\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\right.\left.+\left\Vert \dot{\eta}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{1}{\delta_{3}}\left\Vert \dot{\eta}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\right)\text{ a.e. in }\left[0,T\right]. \end{array} $$

(54)

Moreover, the generalized Poincaré’s inequality (Lemma 4) is applied to the right-hand side of (54) along with the continuity of the trace operator $\text {Tr}:H^{1}\left ({\varOmega }\right )\rightarrow L_{2}\left (\partial {\varOmega }\right )$ and $u_{m}\in {H_{0}^{1}}\left ({\varOmega }\right )$ to give the estimate

$$ \begin{array}{@{}rcl@{}} && c_{3}\left( \delta_{3}\left\Vert u_{m}+\eta\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+\left\Vert u_{m}+\eta\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\right.+\left.\left\Vert \dot{\eta}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{1}{\delta_{3}}\left\Vert \dot{\eta}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\right) \\ &\leq & c_{4}\left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+c_{5}\underset{\partial{\varOmega}}{\int}\left|\text{Tr}\left( u_{m}+\eta\right)\right|^{2}+c_{6}\left\Vert \dot{\eta}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2} \\ &\leq & c_{7}\left( \left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\left\Vert \eta\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\right.\left.+\left\Vert \dot{\eta}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\right)\text{ a.e. in }\left[0,T\right]. \end{array} $$

(55)

Note that $\left \Vert \eta \right \Vert _{H^{1}\left ({\varOmega }\right )}^{2}$ and $\left \Vert \dot {\eta }\right \Vert _{H^{1}\left ({\varOmega }\right )}^{2}$ must be finite almost everywhere in $\left (0,T\right )$ since $\eta \in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )$. Dropping the positive terms $\frac {\alpha \xi ^{2}}{8Q^{2}}\left \Vert \dot {u}_{m}\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2},\frac {\alpha \xi ^{2}}{4}\left \Vert \dot {p}_{m}\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2}$ from the left hand side of (54) and applying (55) leads to

$$ \begin{array}{@{}rcl@{}} && \frac{\alpha\xi^{2}}{8Q^{2}}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\frac{\mathrm{d}}{\mathrm{d} t}\left( \left\Vert \nabla p_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\right)+\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\right) \\ &\leq & c_{7}\left( \left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\left\Vert \eta\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\right.\left.+\left\Vert \dot{\eta}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\right)\text{ a.e. in }\left[0,T\right]. \end{array} $$

(56)

Adding the non-negative terms $\left \Vert \nabla p_{m}\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2}$ and $\underset {{\varOmega }}{\int \limits }\omega _{0}\left (p_{m}\right )$ to the right-hand side of (56) and adjusting the constants yields

$$ \begin{array}{@{}rcl@{}} & \frac{\alpha\xi^{2}}{8Q^{2}}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\frac{\mathrm{d}}{\mathrm{d} t}\left( \left\Vert \nabla p_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\right)+\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\right) \\ \leq & c_{8}\left( \frac{\alpha\xi^{2}}{8Q^{2}}\left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\left\Vert \nabla p_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\right)\right) \\ + & c_{7}\left\Vert \eta\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+c_{7}\left\Vert \dot{\eta}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\text{ a.e. in }\left[0,T\right] \end{array} $$

(57)

Using Grönwall’s lemma (Lemma 3) with the setting

$$ \begin{array}{@{}rcl@{}} {\varrho}\left( t\right) & \equiv&\frac{\alpha\xi^{2}}{8Q^{2}}\left\Vert \nabla\left( u_{m}\left( t\right)+\eta\left( t\right)\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\left\Vert \nabla p_{m}\left( t\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\left( t\right)\right), \\ \phi\left( t\right) & \equiv& c_{8}, \\ \psi\left( t\right) & \equiv& c_{7}\left\Vert \eta\left( t\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+c_{7}\left\Vert \dot{\eta}\left( t\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2} \end{array} $$

(58)

leads to

$$ \begin{array}{@{}rcl@{}} && \frac{\alpha\xi^{2}}{8Q^{2}}\left\Vert \nabla\left( u_{m}\left( t\right)+\eta\left( t\right)\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\left\Vert \nabla p_{m}\left( t\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\left( t\right)\right) \\ &\leq & e^{tc_{8}}\left( \frac{\alpha\xi^{2}}{8Q^{2}}\left\Vert \nabla\left( u_{m}\left( 0\right)+\eta\left( 0\right)\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\left\Vert \nabla p_{m}\left( 0\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\left( 0\right)\right)\right) \\ &&+ e^{tc_{8}}\overset{t}{\underset{0}{\int}} c_{7}\left( \left\Vert \eta\left( r\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+\left\Vert \dot{\eta}\left( r\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\right)\mathrm{d} r, \end{array} $$

(59)

where $t\in \left [0,T\right ]$. Taking t = T on the right-hand side of (59) (this overestimates the right-hand side for any $t\in \left [0,T\right ]$) gives the estimate

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!e^{Tc_{8}}\underset{0}{\overset{T}{\int}}c_{7}\left( \left\Vert \eta\left( r\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+\left\Vert \dot{\eta}\left( r\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\right)\mathrm{d} r & \!\leq\! c_{9}\left\Vert \eta\right\Vert_{H^{2}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}\!\leq c_{10}\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}. \end{array} $$

(60)

The initial condition for the Galerkin approximation (39) gives

$$ \begin{array}{@{}rcl@{}} \left\Vert \nabla\left( u_{m}\left( 0\right)+\eta\left( 0\right)\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} & \leq&\left\Vert u_{m}\left( 0\right)+\eta\left( 0\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2} \\ & =&\left\Vert \sum\limits_{k=1}^{m}\left( u_{\text{ini}}-\eta\left( 0\right),w_{k}\right)w_{k}+\sum\limits_{k=1}^{\infty}\left( \eta\left( 0\right),w_{k}\right)w_{k}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2} \\ & =&\left\Vert \sum\limits_{k=1}^{m}\left( u_{\text{ini}},w_{k}\right)w_{k}+\sum\limits_{k=m+1}^{\infty}\left( \eta\left( 0\right),w_{k}\right)w_{k}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2} \\ & \leq&2\left\Vert \sum\limits_{k=1}^{\infty}\left( u_{\text{ini}},w_{k}\right)w_{k}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+2\left\Vert \sum\limits_{k=1}^{\infty}\left( \eta\left( 0\right),w_{k}\right)w_{k}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2} \\ & =&2\left\Vert u_{\text{ini}}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+2\left\Vert \eta\left( 0\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}. \end{array} $$

(61)

Taking $\eta \in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )$ into account, (61) may be refined further. From Lemma 1, it follows that $\left \Vert \eta \left (0\right )\right \Vert _{H^{1}\left ({\varOmega }\right )}^{2}<\infty$. To obtain an estimate in the $H^{4}\left ({\varOmega }\times \left (0,T\right )\right )$-norm, one can write

$$ \left\Vert \eta\left( 0\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\leq c_{11}\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}, $$

(62)

where the continuous embedding from Lemma 1 was used once again. The term $\left \Vert \nabla p_{m}\left (0\right )\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2}$ on the right-hand side of (59) can be estimated using (40) as

$$ \begin{array}{@{}rcl@{}} \left\Vert \nabla p_{m}\left( 0\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\leq & \left\Vert p_{\text{ini}}\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{2}. \end{array} $$

(63)

Lastly, the term $\underset {{\varOmega }}{\int \limits }\omega _{0}\left (p_{m}\left (0\right )\right )$ in (59) is estimated as follows. Viewing $\omega _{0}\left (y\right )$ as a function from $\mathbb {R}$ to $\mathbb {R}$ and using Young’s inequality (Lemma 6), constants c₁₂,c₁₃,c₁₄,c₁₅ can be found such that

$$ c_{12}y^{4}-c_{13}\leq\frac{1}{a}\omega_{0}\left( y\right)\leq c_{14}y^{4}+c_{15}. $$

This estimate gives

$$ \left|\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\left( 0\right)\right)\right|\leq c_{14}\underset{{\varOmega}}{\int}p_{m}\left( 0\right)^{4}+c_{15}\mu\left( {\varOmega}\right). $$

(64)

The continuous embedding given by Theorem 4 for n = 1,2,3 yields

$$ {H_{0}^{1}}\left( {\varOmega}\right)\hookrightarrow L_{4}\left( {\varOmega}\right). $$

Using this embedding along with (63) leads to

$$ \begin{array}{@{}rcl@{}} \left\Vert p_{m}\left( 0\right)\right\Vert_{L_{4}\left( {\varOmega}\right)}^{4}\leq c_{16}\left\Vert p_{m}\left( 0\right)\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{4} & \leq c_{17}\left\Vert p_{\text{ini}}\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{4}. \end{array} $$

(65)

Applying (60), (61), (62), (64), and (65) gives rise to the estimate

$$ \begin{array}{@{}rcl@{}} & \frac{\alpha\xi^{2}}{8Q^{2}}\left\Vert \nabla\left( u_{m}\left( t\right)+\eta\left( t\right)\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\left\Vert \nabla p_{m}\left( t\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\left( t\right)\right) \\ & \!\!\!\!\!\!\!\!\leq c_{18}\left( \left\Vert u_{\text{ini}}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+\left\Vert p_{\text{ini}}\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{2}+\left\Vert p_{\text{ini}}\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{4}+\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}+1\right)\text{ a.e. in }\left[0,T\right]. \end{array} $$

(66)

Since

$$ \begin{array}{@{}rcl@{}} u_{\text{ini}} & \in& H^{1}\left( {\varOmega}\right),\\ p_{\text{ini}} & \in& {H_{0}^{1}}\left( {\varOmega}\right),\\ \eta & \in& H^{4}\left( {\varOmega}\times\left( 0,T\right)\right), \end{array} $$

and ω₀ is positive (see (47)) taking the essential supremum of (66) shows that

$$ \begin{array}{@{}rcl@{}} \left( \nabla\left( u_{m}+\eta\right)\right)_{m\in\mathbb{N}} && \text{ \text{ is bounded in }}L_{\infty}\left( 0,T;L_{2}\left( {\varOmega}\right)\right),\\ \left( \nabla p_{m}\right)_{m\in\mathbb{N}} && \text{ \text{ is bounded in }}L_{\infty}\left( 0,T;L_{2}\left( {\varOmega}\right)\right). \end{array} $$

Using Poincaré’s (Lemma 4) and Young’s inequality (Lemma 6) gives the estimate

$$ \begin{array}{@{}rcl@{}} \left\Vert u_{m}\right\Vert_{L_{\infty}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right)}^{2} & \leq &c_{19}\left\Vert \nabla u_{m}\right\Vert_{L_{\infty}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2} \\ & \leq &c_{20}\left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{\infty}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2}+c_{21}\left\Vert \nabla\eta\right\Vert_{L_{\infty}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2} \\ & \leq &c_{22}\left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{\infty}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2}+c_{23}\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}. \end{array} $$

(67)

Similarly,

$$ \begin{array}{@{}rcl@{}} \left\Vert p_{m}\right\Vert_{L_{\infty}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right)}^{2} & \leq\left\Vert \nabla p_{m}\right\Vert_{L_{\infty}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2}. \end{array} $$

(68)

The estimates (67), (68) along with (66) imply that

$$ \begin{array}{@{}rcl@{}} \left( u_{m}\right)_{m\in\mathbb{N}}& & \text{ is bounded in }L_{\infty}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right), \end{array} $$

(69)

$$ \begin{array}{@{}rcl@{}} \left( p_{m}\right)_{m\in\mathbb{N}} && \text{ is bounded in }L_{\infty}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right). \end{array} $$

(70)

Furthermore, the bound on the time derivatives is provided. Letting ${\varrho }\left (t\right )$ be defined as in (58) and using a similar estimate to (60), the inequality (59) can be viewed as

$$ \begin{array}{@{}rcl@{}} {\varrho}\left( t\right) & \leq e^{tc_{8}}\left( \rho\left( 0\right)+c_{24}\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}\right)\text{ for all }t\in\left[0,T\right]. \end{array} $$

(71)

Integrating (71) with respect to t over $\left [0,T\right ]$ gives rise to

$$ \begin{array}{@{}rcl@{}} \underset{0}{\overset{T}{\int}}\rho\left( r\right)\mathrm{d} r & \leq\frac{1}{c_{8}}\left( \rho\left( 0\right)+c_{24}\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}\right)\left( e^{Tc_{8}}-1\right). \end{array} $$

(72)

Returning to the inequality (54), applying (55) and retaining all the terms on the left-hand side (as opposed to the estimate 66), one arrives at

$$ \begin{array}{@{}rcl@{}} && \frac{\alpha\xi^{2}}{8Q^{2}}\left\Vert \dot{u}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\alpha\xi^{2}}{4}\left\Vert \dot{p}_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} \\ &+ & \frac{\alpha\xi^{2}}{8Q^{2}}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\xi^{2}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert \nabla p_{m}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}\omega_{0}\left( p_{m}\right) \\ &\leq & c_{7}\left( \left\Vert \nabla\left( u_{m}+\eta\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\left\Vert \eta\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+\left\Vert \dot{\eta}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\right)\text{ a.e. in }\left[0,T\right]. \end{array} $$

(73)

Adding the non-negative terms $\frac {8Q^{2}c_{7}}{\alpha }\left \Vert \nabla p_{m}\right \Vert _{L_{2}\left ({\varOmega }\right )}^{2}$ and $\frac {8Q^{2}c_{7}}{\alpha \xi ^{2}}\underset {{\varOmega }}{\int \limits }\omega _{0}\left (p_{m}\right )$ to the right-hand side of (73), adjusting the constants and integrating with respect to t over $\left [0,T\right ]$ yields

$$ \begin{array}{@{}rcl@{}} && \frac{\alpha\xi^{2}}{8Q^{2}}\underset{0}{\overset{T}{\int}}\left\Vert \dot{u}_{m}\left( t\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\mathrm{d} t+\frac{\alpha\xi^{2}}{4}\underset{0}{\overset{T}{\int}}\left\Vert \dot{p}_{m}\left( t\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\mathrm{d} t+\rho\left( T\right)-\rho\left( 0\right) \\ & \leq&\frac{1}{c_{8}}\frac{8Q^{2}c_{7}}{\alpha\xi^{2}}\left( {\varrho}\left( 0\right)+c_{24}\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}\right)\left( e^{Tc_{8}}-1\right) \\ & +&c_{7}\underset{0}{\overset{T}{\int}}\left\Vert \eta\left( t\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\mathrm{d} t+c_{7}\underset{0}{\overset{T}{\int}}\left\Vert \dot{\eta}\left( t\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\mathrm{d} t, \end{array} $$

(74)

where (72) was used. Since $\left (e^{Tc_{8}}-1\right )$ is positive, the right hand side of (74) can be further estimated by

$$ \begin{array}{@{}rcl@{}} && \frac{1}{c_{8}}\frac{8Q^{2}c_{7}}{\alpha\xi^{2}}\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}\left( e^{Tc_{8}}-1\right)+c_{7}\underset{0}{\overset{T}{\int}}\left\Vert \eta\left( t\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\mathrm{d} t+c_{7}\underset{0}{\overset{T}{\int}}\left\Vert \dot{\eta}\left( t\right)\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\mathrm{d} t\\ &\leq & c_{25}\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}. \end{array} $$

Since c₇ can be chosen such that $\frac {1}{c_{8}}\frac {8Q^{2}c_{7}}{\alpha \xi ^{2}}>1$ (see (57)), adding $\rho \left (0\right )$ to both sides of (74) and dropping the positive term $\rho \left (T\right )$ leads to

$$ \begin{array}{@{}rcl@{}} && \frac{\alpha\xi^{2}}{8Q^{2}}\underset{0}{\overset{T}{\int}}\left\Vert \dot{u}_{m}\left( t\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\mathrm{d} t+\frac{\alpha\xi^{2}}{4}\underset{0}{\overset{T}{\int}}\left\Vert \dot{p}_{m}\left( t\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\mathrm{d} t \\ & \leq& c_{26}\rho\left( 0\right)+c_{25}\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}. \end{array} $$

(75)

The inequality (75) can be rewritten as

$$ \begin{array}{@{}rcl@{}} && \frac{\alpha\xi^{2}}{8Q^{2}}\left\Vert \dot{u}_{m}\right\Vert_{L_{2}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2}+\frac{\alpha\xi^{2}}{4}\left\Vert \dot{p}_{m}\right\Vert_{L_{2}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2} \\ & \leq& c_{27}\left( \left\Vert u_{\text{ini}}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+\left\Vert p_{\text{ini}}\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{2}+\left\Vert p_{\text{ini}}\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{4}+\left\Vert \eta\right\Vert_{H^{4}\left( {\varOmega}\times\left( 0,T\right)\right)}^{2}+1\right), \end{array} $$

(76)

where the estimates (62), (63), (64), and (65) were used once again to estimate $\rho \left (0\right )$.

Estimate (76) implies that

$$ \begin{array}{@{}rcl@{}} (\dot{u}_{m})_{m\in\mathbb{N}},(\dot{p}_{m})_{m\in\mathbb{N}} & \text{ are bounded in }L_{2}\left( 0,T;L_{2}\left( {\varOmega}\right)\right). \end{array} $$

(77)

Combining (69), (70) with (76) yields that

$$ \begin{array}{@{}rcl@{}} (u_{m})_{m\in\mathbb{N}},(p_{m})_{m\in\mathbb{N}} & \text{ are bounded in }W\left( 0,T;q,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right), \end{array} $$

(78)

where $q\in \mathbb {N}$ is arbitrary.

Part C - Existence of a Solution

Since the space $W\left (0,T;q,2;{H_{0}^{1}}\left ({\varOmega }\right ),L_{2}\left ({\varOmega }\right )\right )$ is reflexive for any $q\in \mathbb {N}$, from (78), it follows that there exist sub-sequences such that

$$ \begin{array}{@{}rcl@{}} u_{h_{m}}\rightharpoonup u^{\left( q\right)} & \text{ in }L_{q}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right), \end{array} $$

(79)

$$ \begin{array}{@{}rcl@{}} p_{h_{m}}\rightharpoonup p^{\left( q\right)} & \text{ in }L_{q}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right), \end{array} $$

(80)

$$ \begin{array}{@{}rcl@{}} \dot{u}_{h_{m}}\rightharpoonup\dot{u}^{\left( q\right)} & \text{ in }L_{2}\left( 0,T;L_{2}\left( {\varOmega}\right)\right), \end{array} $$

(81)

$$ \begin{array}{@{}rcl@{}} \dot{p}_{h_{m}}\rightharpoonup\dot{p}^{\left( q\right)} & \text{ in }L_{2}\left( 0,T;L_{2}\left( {\varOmega}\right)\right). \end{array} $$

(82)

Picking sub-sequences $\left (u_{h_{m}}\right ),\left (p_{h_{m}}\right )$ that converge weakly in $W\left (0,T;8,2;{H_{0}^{1}}\left ({\varOmega }\right ),\right .$ $\left .L_{2}\left ({\varOmega }\right )\right )$ to $u^{\left (8\right )},p^{\left (8\right )}$, respectively, and then choosing sub-sequences of $\left (u_{h_{m}}\right ),\left (p_{h_{m}}\right )$ that converge weakly in $W\left (0,T;2,2;{H_{0}^{1}}\left ({\varOmega }\right ),L_{2}\left ({\varOmega }\right )\right )$ to $u^{\left (2\right )},p^{\left (2\right )}$, respectively, one arrives at

$$ \begin{array}{@{}rcl@{}} \left( u_{m}\right)\rightharpoonup u^{\left( 8\right)},\left( p_{m}\right)\rightharpoonup p^{\left( 8\right)} && \text{ in }W\left( 0,T;8,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right), \end{array} $$

(83)

$$ \begin{array}{@{}rcl@{}} \left( u_{m}\right)\rightharpoonup u^{\left( 2\right)},\left( p_{m}\right)\rightharpoonup p^{\left( 2\right)} && \text{ in }W\left( 0,T;2,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right), \end{array} $$

(84)

where the sub-sequence notation has been omitted for simplicity. Applying Theorem 5 with the Gelfand triple ${H_{0}^{1}}\left ({\varOmega }\right )\hookrightarrow \hookrightarrow L_{2}\left ({\varOmega }\right )\hookrightarrow {H_{0}^{1}}\left ({\varOmega }\right )^{*}$, we find that (83), (84) lead to $u^{\left (2\right )}=u^{\left (8\right )}\equiv u,p^{\left (2\right )}=p^{\left (8\right )}\equiv p$.

To show that the pair $\left (u,p\right )$ is a solution to (30)–(33), consider two functions of the form

$$ \varphi\left( t\right)\equiv\sum\limits_{i=1}^{m}a^{i}\left( t\right)w_{i},\text{ }\psi\left( t\right)\equiv\sum\limits_{i=1}^{m}b^{i}\left( t\right)w_{i}, $$

(85)

where $a^{i},b^{i}:\left [0,T\right ]\rightarrow \mathbb {R}$ are smooth coefficients. Functions of the form (85) are dense in $L_{2}\left (0,T;{H_{0}^{1}}\left ({\varOmega }\right )\right )$ (see [15]). Multiplying (37) by $a^{i}\left (t\right )$ and (38) by $b^{i}\left (t\right )$ for all $i\in \left \{ 1,\ldots ,m\right \}$ and summing each of them separately leads to

$$ \begin{array}{@{}rcl@{}} \underset{{\varOmega}}{\int}\dot{u}_{m}\varphi+\underset{{\varOmega}}{\int}\nabla u_{m}\cdot\nabla\varphi+\underset{{\varOmega}}{\int}\nabla\eta\cdot\nabla\varphi&= & Q\underset{{\varOmega}}{\int}\dot{p}_{m}\varphi-\underset{{\varOmega}}{\int}\dot{\eta}\varphi\text{ a.e. in }\left[0,T\right], \end{array} $$

(86)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}\underset{{\varOmega}}{\int}\dot{p}_{m}\psi+\xi^{2}\underset{{\varOmega}}{\int}\nabla p_{m}\cdot\nabla\psi&= & \underset{{\varOmega}}{\int}f_{0}\left( p_{m}\right)\psi-b\beta\xi\underset{{\varOmega}}{\int}\left( u_{m}+\eta\right)\psi\text{ a.e. in }\left[0,T\right]. \end{array} $$

(87)

Integrating (86)–(87) with respect to t over $\left [0,T\right ]$ gives rise to

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\!\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{u}_{m}\varphi+\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\nabla u_{m}\cdot\nabla\varphi+\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\nabla\eta\cdot\nabla\varphi&= & Q\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{p}_{m}\varphi-\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{\eta}\varphi, \end{array} $$

(88)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{p}_{m}\psi+\xi^{2}\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\nabla p_{m}\cdot\nabla\psi&= & \underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}f_{0}\left( p_{m}\right)\psi-b\beta\xi\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\left( u_{m}+\eta\right)\psi. \end{array} $$

(89)

To complete the proof of existence of the solution, the convergence of all the terms in (88)–(89) needs to be shown. The terms

$$ \begin{array}{@{}rcl@{}} \underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{u}_{m}\varphi,\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{p}_{m}\varphi & \text{ and }\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{p}_{m}\psi \end{array} $$

converge since $\varphi ,\psi \in L_{2}\left (0,T;{H_{0}^{1}}\left ({\varOmega }\right )^{*}\right )$ (using the canonical pairing). Moreover, (83) and (84) ensure that

$$ \underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{u}_{m}\varphi\rightarrow\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{u}\varphi,\quad\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{p}_{m}\varphi\rightarrow\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{p}\varphi,\quad\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{p}_{m}\psi\rightarrow\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\dot{p}\psi. $$

The convergence

$$ \underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}u_{m}\psi\rightarrow\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}u\psi $$

follows from $\psi \in L_{2}\left (0,T;{H_{0}^{1}}\left ({\varOmega }\right )^{*}\right )$ (in the sense of the canonical embedding) and (84). The remaining linear terms $\underset {0}{\overset {T}{\int \limits }}\underset {{\varOmega }}{\int \limits }\nabla u_{m}\cdot \nabla \varphi ,\underset {0}{\overset {T}{\int \limits }}\underset {{\varOmega }}{\int \limits }\nabla p_{m}\cdot \nabla \psi$ are handled similarly. It can be shown that

$$ \underset{0}{\overset{T}{\int}}B\left[\cdot,\varphi\right]dt\in L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right), $$

which results in

$$ \underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\nabla u_{m}\cdot\nabla\varphi\rightarrow\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\nabla u\cdot\nabla\varphi,\quad\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\nabla p_{m}\cdot\nabla\psi\rightarrow\underset{0}{\overset{T}{\int}}\underset{{\varOmega}}{\int}\nabla p\cdot\nabla\psi, $$

where (83)–(84) was used.

Furthermore, Lemma 2 can be applied since

$$ p^{m}\rightharpoonup p\text{ in }W\left( 0,T;8,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right), $$

which gives the convergence of the term $\underset {0}{\overset {T}{\int \limits }}\underset {{\varOmega }}{\int \limits }f_{0}\left (p_{m}\right )\psi .$ More specifically, the cubic and quadratic terms converge:

$$ \begin{array}{@{}rcl@{}} \left( p^{m}\right)^{3}\rightarrow p^{3}\text{ in } && L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right), \\ \left( p^{m}\right)^{2}\rightarrow p^{2}\text{ in } && L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right). \end{array} $$

(90)

The relations (90) imply that

$$ \begin{array}{@{}rcl@{}} \left|\left( \left( p^{m}\right)^{3}-p^{3}\right)\left( \psi\right)\right| & \leq&\left\Vert \left( p^{m}\right)^{3}-p^{3}\right\Vert_{L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right)}\left\Vert \psi\right\Vert_{L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right)}\rightarrow0,\\ \left|\left( \left( p^{m}\right)^{2}-p^{2}\right)\left( \psi\right)\right| & \leq&\left\Vert \left( p^{m}\right)^{2}-p^{2}\right\Vert_{L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right)}\left\Vert \psi\right\Vert_{L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right)}\rightarrow0, \end{array} $$

which concludes the proof that the limit functions satisfy (30)–(31). Applying integration by parts to the time derivatives in (30)–(31), equations (88)–(89) and taking the limit shows that the initial conditions are satisfied by the limit functions. This concludes the proof of existence of a solution for (30)–(33) for an arbitrary $\eta \in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )$.

Part D - Uniqueness of the Solution

To show that the solution found in part C is unique, let $\left (u_{1},p_{1}\right )$ and $\left (u_{2},p_{2}\right )$ be two solutions of (30)–(33). Define u₁₂ ≡ u₁ − u₂ and p₁₂ ≡ p₁ − p₂. Subtracting the two systems leads to

$$ \begin{array}{@{}rcl@{}} \frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}u_{12}\left( t\right)v+\underset{{\varOmega}}{\int}\nabla u_{12}\left( t\right)\cdot\nabla v & =&Q\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}p_{12}\left( t\right)v \\ && \text{ for all }v\in {H_{0}^{1}}\left( {\varOmega}\right)\text{ a.e. in }\left[0,T\right], \end{array} $$

(91)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}\frac{\mathrm{d}}{\mathrm{d} t}\underset{{\varOmega}}{\int}p_{12}\left( t\right)w+\xi^{2}\underset{{\varOmega}}{\int}\nabla p_{12}\left( t\right)\cdot\nabla w & =&\underset{{\varOmega}}{\int}\left[f_{0}\left( p_{1}\left( t\right)\right)-f_{0}\left( p_{2}\left( t\right)\right)\right]w-b\beta\xi\underset{{\varOmega}}{\int}u_{12}\left( t\right)w \\ && \text{ for all }w\in {H_{0}^{1}}\left( {\varOmega}\right)\text{ a.e. in }\left[0,T\right], \end{array} $$

(92)

$$ \begin{array}{@{}rcl@{}} u_{12}|_{t=0}=0,\quad && p_{12}|_{t=0}=0. \end{array} $$

(93)

Setting $v=u_{12}\left (t\right )-Qp_{12}\left (t\right )$ and $w=p_{12}\left (t\right )$ in (91) and (92), respectively, and using the notation (41)-(43) leads to

$$ \begin{array}{@{}rcl@{}} \left( \dot{u}_{12}-Q\dot{p}_{12},u_{12}-Qp_{12}\right)+B\left[u_{12}-Qp_{12},u_{12}-Qp_{12}\right] & +QB\left[p_{12},u_{12}-Qp_{12}\right]=0, \end{array} $$

(94)

$$ \begin{array}{@{}rcl@{}} \alpha\xi^{2}\left( \dot{p}_{12},p_{12}\right)+\xi^{2}B\left[p_{12},p_{12}\right]=\left( f_{0}\left( p_{1}\right)-f_{0}\left( p_{2}\right),p_{12}\right) & -b\beta\xi\left( u_{12},p_{12}\right)\text{ a.e. in }\left[0,T\right]. \end{array} $$

(95)

The regularity properties of the solution guarantee that there exists a constant $C_{f_{0}}$ such that

$$ \left\Vert \frac{f_{0}\left( p_{1}\right)-f_{0}\left( p_{2}\right)}{p_{1}-p_{2}}\right\Vert_{L_{2}\left( {\varOmega}\right)}\leq C_{f_{0}}. $$

Using the Young’s (Lemma 6) and Schwarz inequalities, the following estimates can be made:

$$ \begin{array}{@{}rcl@{}} \left|-QB\left[p_{12},u_{12}-Qp_{12}\right]\right| & \leq&\frac{Q^{2}}{2}\left\Vert \nabla p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{1}{2}\left\Vert \nabla\left( u_{12}-Qp_{12}\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}, \end{array} $$

(96)

$$ \begin{array}{@{}rcl@{}} \left|\left( f_{0}\left( p_{1}\right)-f_{0}\left( p_{2}\right),p_{12}\right)\right| & \leq &C_{f_{0}}\left\Vert p_{12}\right\Vert_{L_{4}\left( {\varOmega}\right)}^{2}\leq C_{f_{0}}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{\frac{1}{2}}\left\Vert p_{12}\right\Vert_{L_{6}\left( {\varOmega}\right)}^{\frac{6}{4}} \\ & \leq& C_{f_{0}}C_{\text{emb}}^{\frac{6}{4}}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{\frac{1}{2}}\left\Vert p_{12}\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{\frac{6}{4}} \\ & \leq &c_{28}\left[\frac{\varepsilon^{4}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}}{4}+\frac{\left\Vert \nabla p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}}{\frac{4}{3}\varepsilon^{\frac{4}{3}}}\right], \end{array} $$

(97)

$$ \begin{array}{@{}rcl@{}} \left|-b\beta\xi\left( u_{12},p_{12}\right)\right| & \leq&\frac{c_{29}}{2}\left[\left\Vert u_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\right], \end{array} $$

(98)

In (97), C_emb is the constant of the continuous embedding ${H_{0}^{1}}\left ({\varOmega }\right )\hookrightarrow L_{6}\left ({\varOmega }\right )$. Combining (94) with (96) results in

$$ \begin{array}{@{}rcl@{}} \frac{\mathrm{d}}{\mathrm{d} t}\left\Vert u_{12}-Qp_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\left\Vert \nabla\left( u_{12}-Qp_{12}\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} & \leq Q^{2}\left\Vert \nabla p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}. \end{array} $$

(99)

Similarly, the estimate

$$ \begin{array}{@{}rcl@{}} \frac{\alpha\xi^{2}}{2}\frac{d}{dt}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{3\xi^{2}}{4}\left\Vert \nabla p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} & \leq c_{30}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+c_{31}\left\Vert u_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} \end{array} $$

(100)

arises by blending (98), (97) with (95), where ε in (97) is set so that

$$ \frac{c_{28}}{\frac{4}{3}\varepsilon^{\frac{4}{3}}}=\frac{\xi^{2}}{4}. $$

Adding the $\frac {\xi ^{2}}{2Q^{2}}$ multiple of (99) to (100) along with using the estimate

$$ \begin{array}{@{}rcl@{}} \left\Vert u_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} & \leq&\left( \left\Vert u_{12}-Qp_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}+Q\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}\right)^{2}\\ & \leq&2\left\Vert u_{12}-Qp_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+2Q^{2}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} \end{array} $$

gives rise to

$$ \begin{array}{@{}rcl@{}} && \frac{\xi^{2}}{2Q^{2}}\left\Vert \nabla\left( u_{12}-Qp_{12}\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\xi^{2}}{4}\left\Vert \nabla p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} \\ &&+ \frac{\alpha\xi^{2}}{2}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\xi^{2}}{2Q^{2}}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert u_{12}-Qp_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} \\ &\leq & c_{32}\left\Vert u_{12}-Qp_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+c_{33}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\text{ a.e. in }\left[0,T\right]. \end{array} $$

(101)

Dropping the positive terms on the left hand side of (101) and increasing the constants on the right-hand side so that

$$ \begin{array}{@{}rcl@{}} \frac{\alpha\xi^{2}}{2}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\xi^{2}}{2Q^{2}}\frac{\mathrm{d}}{\mathrm{d} t}\left\Vert u_{12}-Qp_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\\ \leq c_{34}\left( \frac{\alpha\xi^{2}}{2}\left\Vert u_{12}-Qp_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\xi^{2}}{2Q^{2}}\left\Vert p_{12}\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\right)\text{ a.e. in }\left[0,T\right], \end{array} $$

for some constant c₃₄ > 0 sets the stage for using Grönwall’s lemma (Lemma 3). We get

$$ \begin{array}{@{}rcl@{}} \frac{\alpha\xi^{2}}{2}\left\Vert p_{12}\left( t\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\xi^{2}}{2Q^{2}}\left\Vert u_{12}\left( t\right)-Qp_{12}\left( t\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2} \end{array} $$

(102)

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\!\!\leq e^{c_{34}T}\left( \frac{\alpha\xi^{2}}{2}\left\Vert u_{12}\left( 0\right)-Qp_{12}\left( 0\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\frac{\xi^{2}}{2Q^{2}}\left\Vert p_{12}\left( 0\right)\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}\right) & \text{ a.e. in }\left[0,T\right]. \end{array} $$

(103)

Integrating (102) over $\left [0,T\right ]$ and using the initial conditions of the problem (91)–(92) gives

$$ \begin{array}{@{}rcl@{}} \left\Vert p_{12}\right\Vert_{L_{2}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2}=0 & \Rightarrow& p_{1}=p_{2},\\ \left\Vert u_{12}-Qp_{12}\right\Vert_{L_{2}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2}=0 & \Rightarrow& u_{1}=u_{2}. \end{array} $$

□

5 Existence of Optimal Solution

Utilizing the notation (9), (20) and letting W_ad ⊂ X_C be closed, convex and bounded, we provide the proof of Theorem 1.

Proof

(of Theorem 1) In order to ensure the existence of optimal control, we prove the following sufficient conditions (Theorem 7):

• (c1) The solution operator of the system $S:W_{\text {ad}}\rightarrow \hat {X}_{S}$ is a bounded operator.

• (c2) The state equation operator

  $$ e:\hat{X}_{S}\times W_{\text{ad}}\rightarrow L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)^{*}\right)^{2}\times L_{2}\left( {\varOmega}\right)^{2} $$

  is weakly sequentially continuous.

• (c3) J (defined by (12)) is convex and continuous.

Note that condition (c3) can be weakened to J being sequentially lower semi-continuous [19].

As for condition (c1), the existence of the solution operator S is ensured by Theorem 2. To prove its boundedness, the boundedness of W_ad is used. Taking the supremum of (66) while using (67) and (68) yields

$$ \begin{array}{@{}rcl@{}} & c_{35}\left\Vert u_{m}\right\Vert_{L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right)}^{2}+c_{36}\left\Vert p_{m}\right\Vert_{L_{8}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right)}^{2}\\ \leq & c_{39}\left( \left\Vert u_{\text{ini}}\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}+\left\Vert p_{\text{ini}}\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{2}+\left\Vert p_{\text{ini}}\right\Vert_{{H_{0}^{1}}\left( {\varOmega}\right)}^{4}+1+M^{2}\right), \end{array} $$

where M is the constant bounding W_ad in X_C. This means that there exists a constant M₁ dependent only on the Sobolev norms of the initial conditions u_ini,p_ini, final time T and the constant M such that

$$ c_{35}\left\Vert u_{m}\right\Vert_{L_{2}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right)}^{2}+c_{36}\left\Vert p_{m}\right\Vert_{L_{8}\left( 0,T;{H_{0}^{1}}\left( {\varOmega}\right)\right)}^{2}\leq M_{1}\left( u_{\text{ini}},p_{\text{ini}},T,M\right). $$

(104)

Similarly, using (76) leads to

$$ \begin{array}{@{}rcl@{}} c_{37}\left\Vert \dot{u}_{m}\right\Vert_{L_{2}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2}+c_{38}\left\Vert \dot{p}_{m}\right\Vert_{L_{2}\left( 0,T;L_{2}\left( {\varOmega}\right)\right)}^{2} & \leq M_{2}\left( u_{\text{ini}},p_{\text{ini}},T,M\right). \end{array} $$

(105)

Taking the limit inferior of the estimates (104), (105), adding them and adjusting the constants yields

$$ \begin{array}{@{}rcl@{}} \left\Vert \left( u,p\right)\right\Vert_{\hat{X}_{S}} & \leq M_{3}\left( u_{\text{ini}},p_{\text{ini}},T,M\right) \end{array} $$

which is the desired boundedness property of the solution operator.

To prove (c2), only the convergence of the nonlinear terms is checked. Lemma 2 provides the necessary convergence property.

To prove (c3), consider the embedding given by Theorem 6 and observe that

$$ W\left( 0,T;2,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right)\hookrightarrow C\left( \left[0,T\right];L_{2}\left( {\varOmega}\right)\right). $$

(106)

Take

$$ \left( u_{1},p_{1}\right),\left( u_{2},p_{2}\right)\in\hat{X}_{S} $$

and $\eta _{1},\eta _{2}\in H^{4}\left ({\varOmega }\times \left (0,T\right )\right )=X_{C}$ and make the estimate

$$ \begin{array}{@{}rcl@{}} && \left|J\left( u_{1},p_{1},\eta_{1}\right)-J\left( u_{2},p_{2},\eta_{2}\right)\right| \\ &\leq & \underset{{\varOmega}}{\int}\left[\left|p_{f}\right|\left|p_{1}\left( T\right)-p_{2}\left( T\right)\right|+\frac{1}{2}\left|p_{1}\left( T\right)+p_{2}\left( T\right)\right|\left|p_{1}\left( T\right)-p_{2}\left( T\right)\right|\right] \\ &&+ \frac{\gamma}{2}\underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}\left[\text{Tr }\left( \eta_{1}\right)\left( \text{Tr }\left( \eta_{1}-\eta_{2}\right)\right)+\text{Tr }\left( \eta_{2}\right)\left( \text{Tr }\left( \eta_{1}-\eta_{2}\right)\right)\right] \\ &\leq & \left( \underset{{\varOmega}}{\int}\left|p_{f}\right|^{2}\right)^{\frac{1}{2}}\left( \underset{{\varOmega}}{\int}\left|p_{1}\left( T\right)-p_{2}\left( T\right)\right|^{2}\right)^{\frac{1}{2}} \\ &&+ \left( \underset{{\varOmega}}{\int}\left|p_{1}\left( T\right)+p_{2}\left( T\right)\right|^{2}\right)^{\frac{1}{2}}\left( \underset{{\varOmega}}{\int}\left|p_{1}\left( T\right)-p_{2}\left( T\right)\right|^{2}\right)^{\frac{1}{2}} \\ &&+ \frac{\gamma}{2}\left( \underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}\text{Tr }\left( \eta_{1}\right)^{2}\right)^{\frac{1}{2}}\left( \underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}\text{Tr }\left( \eta_{1}-\eta_{2}\right)^{2}\right)^{\frac{1}{2}} \\ &&+ \frac{\gamma}{2}\left( \underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}\text{Tr }\left( \eta_{2}\right)^{2}\right)^{\frac{1}{2}}\left( \underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}\text{Tr }\left( \eta_{1}-\eta_{2}\right)^{2}\right)^{\frac{1}{2}}. \end{array} $$

(107)

Considering $\left (p_{1},u_{1},\eta _{1}\right )\in \hat {X}_{S}\times X_{C}$ such that $\left \Vert \left (p_{1},u_{1},\eta _{1}\right )-\left (p_{2},u_{2},\eta _{2}\right )\right \Vert _{\hat {X}_{S}\times X_{C}}<1$ makes

$$ \begin{array}{@{}rcl@{}} \left( \underset{{\varOmega}}{\int}\left|p_{1}\left( T\right)+p_{2}\left( T\right)\right|^{2}\right)^{\frac{1}{2}},\frac{\gamma}{2}\left( \underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}\text{Tr }\left( \eta_{1}\right)^{2}\right)^{\frac{1}{2}}, & \text{ and }\frac{\gamma}{2}\left( \underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}\text{Tr }\left( \eta_{2}\right)^{2}\right)^{\frac{1}{2}} \end{array} $$

bounded thanks to the embedding statements (106) and the continuity of the trace operator. Using the same embedding statements, it is possible to make

$$ \begin{array}{@{}rcl@{}} \left( \underset{{\varOmega}}{\int}\left|p_{1}\left( T\right)-p_{2}\left( T\right)\right|^{2}\right)^{\frac{1}{2}} & ,\left( \underset{{\varOmega}}{\int}\left|p_{1}\left( T\right)-p_{2}\left( T\right)\right|^{2}\right)^{\frac{1}{2}}, & \text{ and }\left( \underset{0}{\overset{T}{\int}}\underset{\partial{\varOmega}}{\int}\text{Tr }\left( \eta_{1}-\eta_{2}\right)^{2}\right)^{\frac{1}{2}} \end{array} $$

arbitrarily small by choosing an appropriate 0 < δ < 1 such that

$$ \left\Vert \left( p_{1},u_{1},\eta_{1}\right)-\left( p_{2},u_{2},\eta_{2}\right)\right\Vert_{\hat{X}_{S}\times X_{C}}<\delta. $$

This proves the continuity of J. Lastly, J is convex due to the linearity of the integral and convexity of the square function. □

6 Conclusion

In this paper, a weak formulation of the Dirichlet Boundary condition optimization for a phase field problem is proposed. Energy estimates along with some specific compact embedding results are provided. These results together with the proof of uniqueness of the solution for a given control give rise to a bounded solution operator. Using the aforementioned results, the existence of optimal control is proven.

It is worthwhile to note that quite strong regularity assumptions are imposed on the control ($H^{4}\left ({\varOmega }\times \left (0,T\right )\right )$). This is a consequence of the compact embedding statements used in the analysis.

Appendix

In this section, the notation used for Bochner spaces and fundamental statements that find utility in the above proof are listed.

1.1 A.1 Bochner Spaces $L_{p}\left (0,T;Y\right )$, $W\left (0,T;p,\widetilde {p},X_{0},X,X_{1}\right )$ and $W\left (0,T;p,\widetilde {p},X_{0},X_{1}\right )$

Let Y be a Banach space. Suppose that $f:\left [0,T\right ]\rightarrow Y$ is a vector valued function and define the norms

$$ \begin{array}{@{}rcl@{}} \left\Vert f\right\Vert_{L_{p}\left( 0,T;Y\right)} & \equiv&\left( \underset{0}{\overset{T}{\int}} \left\Vert f\left( t\right)\right\Vert_{Y}^{p}dt\right)^{p},\\ \left\Vert f\right\Vert_{L_{\infty}\left( 0,T;Y\right)} & \equiv&\underset{t\in\left[0,T\right]}{\text{ess sup}}\left\Vert f\left( t\right)\right\Vert_{Y}. \end{array} $$

For $1\leq p\leq \infty$, the Bochner space is defined as

$$ L_{p}\left( 0,T;Y\right)\equiv\left\{ g:\left[0,T\right]\rightarrow Y\text{ measurable}:\left\Vert g\right\Vert_{L_{p}\left( 0,T;Y\right)}<\infty\right\} . $$

Let X₀,X,X₁ be Banach Spaces, X₀,X₁ be reflexive and

$$ X_{0}\hookrightarrow\hookrightarrow X\hookrightarrow X_{1}. $$

(108)

Additionally, consider $p,\widetilde {p}\in \left (1,\infty \right )$ and T > 0. Then, define the set

$$ W\equiv W\left( 0,T;p,\widetilde{p};X_{0},X_{1}\right)\equiv\left\{ v\in L_{p}\left( 0,T;X_{0}\right),\dot{v}\in L_{\widetilde{p}}\left( 0,T;X_{1}\right)\right\} , $$

where the distributional derivative $\dot {v}$ is taken in the space $L_{1}\left (0,T;X_{1}\right )$. This is possible since for any $f\in L_{p}\left (0,T;X_{0}\right )$, we get $f\in L_{1}\left (0,T;X_{0}\right )$ and $f\in L_{1}\left (0,T;X_{1}\right )$ by (108). If $\dot {f}\in L_{1}\left (0,T;X_{1}\right )$ exists and the additional condition $\dot {f}\in L_{\widetilde {p}}\left (0,T;X_{1}\right )$ is fulfilled, then f ∈ W. In this setting, (108) is called a Gelfand (or Hilbert) triple. For any $f:\left [0,T\right ]\rightarrow X_{0}$ define the norm

$$ \left\Vert f\right\Vert_{W}\equiv\left\Vert f\right\Vert_{L_{p}\left( 0,T;X_{0}\right)}+\left\Vert \dot{f}\right\Vert_{L_{\widetilde{p}}\left( 0,T;X_{1}\right)}. $$

(109)

The set W along with (109) is a Banach space denoted $\left (W,\left \Vert \cdot \right \Vert _{W}\right )$.

Commonly, the spaces $\left (W,\left \Vert f\right \Vert _{W}\right )$ are used as the solution space for time dependent partial differential equations (PDEs). Moreover, $\left (W,\left \Vert f\right \Vert _{W}\right )\hookrightarrow \hookrightarrow L_{p}\left (0,T;X\right )$, which comes into play when dealing with non-linear PDEs (see Section 3). Lastly, an example configuration for the Gelfand triple and space W is presented.

Lastly, other spaces that are associated with vector valued functions are discussed. Let X be a Banach space and T > 0. Define the set

$$ C\left( \left[0,T\right];X\right)\equiv\left\{ g:\left[0,T\right]\rightarrow X:g\text{ is continuous}\right\} . $$

The space $C\left (\left [0,T\right ];X\right )$ together with the norm

$$ \left\Vert f\right\Vert_{C\left( \left[0,T\right];X\right)}\equiv\underset{t\in\left[0,T\right]}{\sup}\left\Vert f\left( t\right)\right\Vert_{X} $$

is a Banach space denoted $\left (C\left (\left [0,T\right ];X\right ),\left \Vert \cdot \right \Vert _{C\left (0,T;X\right )}\right )$ (since X is complete).

1.2 A.2 Inequalities

Lemma 3

(Grönwall’s lemma) Let ${\varrho }\in C\left (\left [0,T\right ],\mathbb {R}\right )$, $\phi ,\psi \in L_{1}\left (\left [0,T\right ],\mathbb {R}\right )$ be non-negative and let

$$ {\varrho}\left( t\right)^{\prime}\leq\phi\left( t\right){\varrho}\left( t\right)+\psi\left( t\right)\text{{\quad}a.e. in }\left[0,T\right]. $$

Then,

$$ {\varrho}\left( t\right)\leq e^{\overset{t}{\underset{0}{\int}}\phi\left( r\right)\mathrm{d} r}\left[{\varrho}\left( 0\right)+\overset{t}{\underset{0}{\int}}\psi\left( r\right)\mathrm{d} r\right]\text{{\quad} for all }t\in\left[0,T\right]. $$

Lemma 4

(Poincaré’s inequality) Let Ω be a bounded Lipschitz domain and $f\in H^{1}\left ({\varOmega }\right )$. Then, there exists a constant c_p such that

$$ \left\Vert f\right\Vert_{H^{1}\left( {\varOmega}\right)}^{2}\leq c_{\text{p}}\left( \left\Vert \nabla f\right\Vert_{L_{2}\left( {\varOmega}\right)}^{2}+\underset{\partial{\varOmega}}{\int}\left|\text{Tr} \left( f\right)\right|^{2}\mathrm{d} S\right). $$

Lemma 5

(Hölder’s inequality) Let Ω be a measure space, $f,g:{\varOmega }\rightarrow \mathbb {R}$ be measurable functions and p,q be Hölder conjugates (i.e. $\frac {1}{q}+\frac {1}{p}=1$). Then,

$$ \left\Vert fg\right\Vert_{L_{1}\left( {\varOmega}\right)}\leq\left\Vert f\right\Vert_{L_{p}\left( {\varOmega}\right)}\left\Vert g\right\Vert_{L_{q}\left( {\varOmega}\right)}. $$

Lemma 6

(Young’s inequality) Let a,b ≥ 0 and ε > 0. Then,

$$ ab\leq\frac{a^{2}}{2\varepsilon}+\frac{\varepsilon b^{2}}{2}. $$

1.3 A.3 Embedding Theorems

Selected statements about embeddings of functional spaces [1, 15, 31] are listed in this section. The reader should note that even though these embeddings are only discussed for bounded subdomains of $\mathbb {R}^{n}$, some of these results hold for unbounded domains as well (possibly under further technical assumptions) [1, 15].

Definition 1

Let ${\varOmega }\subset \mathbb {R}^{n}$ be a bounded domain. We say that Ω has a Lipschitz boundary if every point x ∈ ∂Ω has an open neighborhood U_x such that U_x ∩ ∂Ω is the graph of a Lipschitz continuous function.

Theorem 3

(Hölder-Sobolev embedding) Let ${\varOmega }\subset \mathbb {R}^{n}$ be a bounded domain with a Lipschitz boundary. Let $m\in \mathbb {N}$ and $p\in \left [1,\infty \right )$. Then, for any $k\in \mathbb {N}_{0}$, $\beta \in \left (0,1\right )$ such that

$$ m-\frac{n}{p}\geq k+\beta, $$

the embedding

$$ W^{m,p}\left( {\varOmega}\right)\hookrightarrow C^{k,\beta}\left( {\varOmega}\right) $$

holds.

Theorem 4

(Sobolev compact embedding) Let ${\varOmega }\subset \mathbb {R}^{n}$ be a bounded domain. Let $m\in \mathbb {N}$, $p\in \left [1,\infty \right )$, q ≥ 1, $k\in \mathbb {N}_{0}$, m > k and

$$ m-\frac{n}{p}>k-\frac{n}{q} $$

hold. Then,

$$ W_{0}^{m,p}\left( {\varOmega}\right)\hookrightarrow\hookrightarrow W_{0}^{k,q}\left( {\varOmega}\right). $$

Theorem 5

(embedding of Bochner spaces I.) Let X₀,X,X₁ form a Gelfand triple as defined by (108). Then, for any $p,\widetilde {p}\in \left (1,\infty \right )$ and T > 0,

$$ W\left( 0,T;p,\widetilde{p};X_{0},X_{1}\right)\hookrightarrow\hookrightarrow L_{p}\left( 0,T;X\right). $$

Theorem 6

(embedding of Bochner spaces II.) Let T > 0. Then,

$$ W\left( 0,T;2,2;{H_{0}^{1}}\left( {\varOmega}\right),L_{2}\left( {\varOmega}\right)\right)\hookrightarrow C\left( \left[0,T\right];L_{2}\left( {\varOmega}\right)\right). $$

1.4 A.3 Existence of Optimal Control

The following theorem can be used to prove the existence of optimal control [19, 24].

Theorem 7

(existence of optimal control) Let Z be a Banach space and X_c and X_s be reflexive Banach spaces. Suppose $J:X_{s}\times X_{c}\rightarrow \mathbb {R},e:X_{s}\times X_{c}\rightarrow Z$ are continuous maps and assume W ⊂ X_c is convex bounded and closed and V ⊂ X_s is a convex and closed. Consider the problem

$$ \begin{array}{@{}rcl@{}} \underset{\left( u,\eta\right)\in\left( X_{s},X_{c}\right)}{\min} && J\left( u,\eta\right) \end{array} $$

(110)

$$ \begin{array}{@{}rcl@{}} \text{s.t. } && e\left( u,\eta\right)=0\text{ where }u\in V,\eta\in W. \end{array} $$

(111)

If V has a feasible point, the state equation (111) has a bounded solution operator $\eta \in W\rightarrow u\left (\eta \right )\in X_{s}$, e is weakly sequentially continuous, and J is convex and continuous, the problem (110)-(111) has a solution.