Large Deviations for Stochastic Nematic Liquid Crystals Driven by Multiplicative Gaussian Noise

Abstract

We study a stochastic two-dimensional nematic liquid crystal model with multiplicative Gaussian noise. We prove the Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method.

Appendix: Proof of Theorem 5.5

This theorem speaks about the existence and uniqueness of the skeleton Eqs. 5.8–5.9. We will use the classical Faedo-Galerkin approximation to prove the existence results.

1.1 A.1. Faedo-Galerkin Approximation

Our proof of existence of the skeleton equation depends on the Galerkin approximation method. Let $\{ \rho _{i}\}_{i = 1}^{\infty }$ be the orthonormal basis of H composed of eigenfunctions of the stokes operator $\mathscr{A}$. Let $\{ \sigma _{i}\}_{i = 1}^{\infty }$ be the orthonormal basis of L² consisting of the eigenfunctions of the Neumann Laplacian $\mathcal {A}$. Let us define the following finite dimensional spaces for any $n \in \mathbb {N}$

$$ \begin{array}{@{}rcl@{}} \mathrm{H}_{n} &:=& Linspan\{ \rho_{1}, \dots, \rho_{n}\},\\ \mathbb{L}_{n} &:=& Linspan\{ \sigma_{1}, \dots, \sigma_{n}\}. \end{array} $$

Our aim is to derive uniform estimates for the solution of the projection of Eqs. 5.8–5.9 onto the finite dimensional space $\mathrm {H}_{n} \times \mathbb {L}_{n}$, i.e., its Galerkin approximation. For this let us denote by P_n the projection from H onto H_n and $\tilde {P}_{n}$ be the projection from L² onto $\mathbb {L}_{n}$. We consider the following locally Lipschitz mappings:

$$ \begin{array}{@{}rcl@{}} &&B_{n} : \mathrm{H}_{n} \ni \mathbf{u} \mapsto P_{n} B(\mathbf{u}, \mathbf{u}) \in \mathrm{H}_{n},\\ &&M_{n} : \mathbb{L}_{n} \ni \mathbf{d} \mapsto P_{n} M(\mathbf{d}) \in \mathrm{H}_{n},\\ &&f_{n} : \mathbb{L}_{n} \ni \mathbf{d} \mapsto \tilde{P}_{n} f(\mathbf{d}) \in \mathbb{L}_{n},\\ &&\tilde{B}_{n} : \mathrm{H}_{n} \times \mathbb{L}_{n} \ni (\mathbf{u}, \mathbf{d}) \mapsto \tilde{P}_{n} \tilde{B}(\mathbf{u}, \mathbf{d}) \in \mathbb{L}_{n}. \end{array} $$

Let P_nu₀ = u_n(0) := u_0n and $\tilde {P}_{n} \mathbf {d}_{0} = \mathbf {d}_{n}(0) := \mathbf {d}_{0n}$.

So the Galerkin approximation of the problem is:

$$ \begin{array}{@{}rcl@{}} d\mathbf{u}_{n}(t) + \left[ \mathscr{A}\mathbf{u}_{n}(t) + B_{n}(\mathbf{u}_{n}(t)) + M_{n}(\mathbf{d}_{n}(t)) \right] dt = P_{n} \sigma(\mathbf{u}_{n}(t)) \theta(t) dt, \end{array} $$

(A.1)

$$ \begin{array}{@{}rcl@{}} d\mathbf{d}_{n}(t) + \left[ \mathcal{A} \mathbf{d}_{n}(t) + \tilde{B}_{n}(\mathbf{u}_{n}(t), \mathbf{d}_{n}(t)) + f_{n}(\mathbf{d}_{n}(t)) \right] dt = \tilde{P}_{n} G(\mathbf{d}_{n}(t)) \rho(t) dt. \end{array} $$

(A.2)

Lemma A.1

For each $n \in \mathbb {N},$the problem (A.1)-(A.2) has a unique global solution.

Proof

We have the existence result due to [20]. □

The processes $\left (\mathbf {u}_{n}\right )_{n \in \mathbb {N}}$ and $\left (\mathbf {d}_{n}\right )_{n \in \mathbb {N}}$ satisfy the following estimates.

Proposition A.2

For any p ≥ 2, there exists a positive constant $\tilde {C} = \tilde {C}_{p}$, independent of ρ such that

$$ \begin{array}{@{}rcl@{}} \sup_{n \in \mathbb{N}} \left[ \sup_{s \in [0,T]} \left| \mathbf{d}_{n}(s) \right|_{L^{2}}^{p} + {{\int}_{0}^{T}} \left| \mathbf{d}_{n}(s) \right|_{L^{2}}^{p-2} \left( | \nabla \mathbf{d}_{n}(s)|_{L^{2}}^{2} + |\mathbf{d}_{n}(s)|^{2N+2}_{L^{2N+2}}\right) ds \right] \leq \tilde{C}, \end{array} $$

where $\tilde {C} := |\mathbf {d}_{0}|_{L^{2}}^{p} \left (1+CTe^{CT}\right )$ .

Proof

The proof is similar to the proof in [4]. In our case we will be using the fact that 〈G(d_n(s)), d_n(s)〉 = 0. Then the other steps of the proof will follow as in [4] (see also [20]). □

Proposition A.3

There exists a positive constant $\bar {C}$ depending on $K, T, \mathbf {h}, |\rho |_{L^{2}(0, T; \mathbb {R})},$ $|\theta |_{L^{2}(0, T; \mathrm {H}_{0})}$ such that

$$ \begin{array}{@{}rcl@{}} \sup_{n \geq 1} \left[ \sup_{s \in [0, T]} \left| \mathbf{u}_{n}(s) \right|_{\mathrm{H}}^{2} + {\Psi}(\mathbf{d}_{n}(s)) + {{\int}_{0}^{T}} \left( \big\|\mathbf{u}_{n}(s) \big\| + \frac{1}{2} \left| f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s) \right|_{L^{2}}^{2} \right) ds \right] \leq \bar{C}, \end{array} $$

where ${\Psi }(z) := \frac {1}{2} |\nabla z|^{2} + \frac {1}{2} {\int }_{\mathcal {O}} \tilde {\mathrm {F}}(|z|^{2}) dx$and K is the linear growth coefficient for σ.

Proof

Consider the approximated system (A.1)-(A.2). Now take the inner product of (A.1) with u_n and writing in integral form we obtain,

$$ \begin{array}{@{}rcl@{}} &&\left| \mathbf{u}_{n}(t) \right|^{2}_{\mathrm{H}} - \left| \mathbf{u}_{0n} \right|^{2}_{\mathrm{H}} \\ &&= - {{\int}_{0}^{t}} \left\langle \mathscr{A}\mathbf{u}_{n}(s) + B_{n}(\mathbf{u}_{n}(s)) + M_{n}(\mathbf{d}_{n}(s)), \mathbf{u}_{n}(s) \right\rangle ds + {{\int}_{0}^{t}} \left\langle P_{n} \sigma(\mathbf{u}_{n}(s)) \theta(s), \mathbf{u}_{n}(s) \right\rangle ds \\ &&= - {{\int}_{0}^{t}} | \nabla \mathbf{u}_{n}(s) |^{2} ds - {{\int}_{0}^{t}} \left\langle M_{n}(\mathbf{d}_{n}(s)), \mathbf{u}_{n}(s) \right\rangle ds + {{\int}_{0}^{t}} \left\langle P_{n} \sigma(\mathbf{u}_{n}(s)) \theta(s), \mathbf{u}_{n}(s) \right\rangle ds\\ \end{array} $$

(A.3)

Now consider the map

$$ {\Psi}(z) := \frac{1}{2} |\nabla z|^{2} + \frac{1}{2} {\int}_{\mathcal{O}} \tilde{\mathrm{F}}(|z|^{2}) dx. $$

The first Fréchet derivative is: ${\Psi }^{\prime }(z)[g] = \langle \nabla z, \nabla g \rangle + \langle f(z), g\rangle = \langle \mathcal {A} z+ f(z), g\rangle$.

From (A.2) we get,

$$ \begin{array}{@{}rcl@{}} &&{\Psi}(\mathbf{d}_{n}(t)) - {\Psi}(\mathbf{d}_{n}(0))\\ &&= - {{\int}_{0}^{t}} {\Psi}^{\prime}(\mathbf{d}_{n}(s)) \left[ \mathcal{A} \mathbf{d}_{n}(s) + \tilde{B}_{n}(\mathbf{u}_{n}(s), \mathbf{d}_{n}(s)) + f_{n}(\mathbf{d}_{n}(s)) \right] ds\\ &&\quad+ {{\int}_{0}^{t}} {\Psi}^{\prime}(\mathbf{d}_{n}(s)) \left[ \tilde{P}_{n} G(\mathbf{d}_{n}(s)) \rho(s) \right] ds\\ &&= {{\int}_{0}^{t}} - |f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s)|^{2} ds - \left\langle \tilde{B}_{n}(\mathbf{u}_{n}(s), \mathbf{d}_{n}(s)), f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s) \right\rangle ds\\ &&\quad \ + {{\int}_{0}^{t}} \left\langle \tilde{P}_{n} G(\mathbf{d}_{n}(s)) \rho(s), f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s) \right\rangle ds. \end{array} $$

(A.4)

From the proof of Proposition 5.5 in [9] we have,

$$ \begin{array}{@{}rcl@{}} &&\left\langle \tilde{B}_{n}(\mathbf{u}_{n}(s), \mathbf{d}_{n}(s)), f_{n}(\mathbf{d}_{n}(s)) \right\rangle = 0,\\ &&\text{and}\\ &&\left\langle \tilde{B}_{n}(\mathbf{u}_{n}(s), \mathbf{d}_{n}(s)), \mathcal{A} \mathbf{d}_{n}(s) \right\rangle = \left\langle M_{n}(\mathbf{d}_{n}(s)), \mathbf{u}_{n}(s) \right\rangle. \end{array} $$

Now adding (A.3) and (A.4) and rearranging we obtain,

$$ \begin{array}{@{}rcl@{}} &&| \mathbf{u}_{n}(t)|^{2} + {\Psi}(\mathbf{d}_{n}(t)) + {{\int}_{0}^{t}} \left( | \nabla \mathbf{u}_{n}(s) |^{2} + |f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s)|^{2} \right) ds\\ &&= | \mathbf{u}_{0n}|^{2} + {\Psi}(\mathbf{d}_{0n}) + {{\int}_{0}^{t}} \left\langle P_{n} \sigma(\mathbf{u}_{n}(s)) \theta(s), \mathbf{u}_{n}(s) \right\rangle ds\\ &&\quad+ {{\int}_{0}^{t}} \left\langle \tilde{P}_{n} G(\mathbf{d}_{n}(s)) \rho(s), f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s) \right\rangle ds \end{array} $$

(A.5)

We will estimate the term ${{\int }_{0}^{t}} \left | \left \langle P_{n} \sigma (\mathbf {u}_{n}(s)) \theta (s), \mathbf {u}_{n}(s) \right \rangle \right | ds$. Using the linear growth property of σ (with growth constant K), the Cauchy-Schwarz inequality, the embedding of H₀↪H and the fact that $a \leq \sqrt {1+a^{2}} \leq 1+a^{2},$ for a > 0 we infer,

$$ \begin{array}{@{}rcl@{}} &&{{\int}_{0}^{t}} \left| \left\langle P_{n} \sigma(\mathbf{u}_{n}(s)) \theta(s), \mathbf{u}_{n}(s) \right\rangle \right| ds \leq {{\int}_{0}^{t}} |\sigma(\mathbf{u}_{n}(s))|_{\mathrm{L}_{2}} |\theta(s)|_{\mathrm{H}} |\mathbf{u}_{n}(s)|_{\mathrm{H}} ds\\ &&\leq \sqrt{K} {{\int}_{0}^{t}} \sqrt{1+ \left| \mathbf{u}_{n}(s) \right|^{2}_{\mathrm{H}}} \ |\theta(s)|_{\mathrm{H}} |\mathbf{u}_{n}(s)|_{\mathrm{H}} ds \leq \sqrt{K} {{\int}_{0}^{t}} \left( 1+ |\mathbf{u}_{n}(s)|_{\mathrm{H}}^{2} \right) |\theta(s)|_{\mathrm{H}_{0}}\\ &&\leq \sqrt{K} {{\int}_{0}^{t}} \left( 1+|\mathbf{u}_{n}(s)|_{\mathrm{H}}^{2} \right) \left( 1+ |\theta(s)|^{2}_{\mathrm{H}_{0}}\right) ds\\ &&= {{\int}_{0}^{t}} \sqrt{K} ds + \sqrt{K} {{\int}_{0}^{t}} |\theta(s)|^{2}_{\mathrm{H}_{0}} ds + \sqrt{K} {{\int}_{0}^{t}} |\mathbf{u}_{n}(s)|_{\mathrm{H}}^{2} \left( 1+ |\theta(s)|^{2}_{\mathrm{H}_{0}}\right) ds. \end{array} $$

(A.6)

Similar procedure will give,

$$ \begin{array}{@{}rcl@{}} &&{{\int}_{0}^{t}} \left| \left\langle \tilde{P}_{n} G(\mathbf{d}_{n}(s)) \rho(s), f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s) \right\rangle \right| ds\\ &&\leq {{\int}_{0}^{t}} |G(\mathbf{d}_{n}(s))|_{L^{2}} |\rho(s)|_{\mathbb{R}} |f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s)|_{L^{2}} ds\\ &&\leq \frac{C(\mathbf{h})}{2} {{\int}_{0}^{t}} |\mathbf{d}_{n}(s)|^{2}_{L^{2}} |\rho(s)|^{2}_{\mathbb{R}} ds + \frac{1}{2} {{\int}_{0}^{t}} |f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s)|^{2}_{L^{2}} ds. \end{array} $$

(A.7)

Using (A.6) and (A.7) in (A.5) we obtain,

$$ \begin{array}{@{}rcl@{}} &&| \mathbf{u}_{n}(t)|^{2} + {\Psi}(\mathbf{d}_{n}(t)) + {{\int}_{0}^{t}} \left( | \nabla \mathbf{u}_{n}(s) |^{2} + \frac{1}{2} |f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s)|^{2} \right) ds\\ &&\leq | \mathbf{u}_{0n}|^{2} + {\Psi}(\mathbf{d}_{0n}) + \tilde{C}(K, T) + \sqrt{K} {{\int}_{0}^{t}} |\mathbf{u}_{n}(s)|_{\mathrm{H}}^{2} \left( 1+ |\theta(s)|^{2}_{\mathrm{H}_{0}}\right) dss\\ &&\quad+ \frac{C(\mathbf{h})}{2} \sup_{s \in [0, T]} |\mathbf{d}_{n}(s)|^{2}_{L^{2}} {{\int}_{0}^{t}} |\rho(s)|^{2}_{\mathbb{R}} ds. \end{array} $$

(A.8)

Now using Proposition A.2 (for p = 2) in (A.8) we finally get,

$$ \begin{array}{@{}rcl@{}} \left| \mathbf{u}_{n}(t) \right|_{\mathrm{H}}^{2} &\leq& | \mathbf{u}_{0n}|^{2} + {\Psi}(\mathbf{d}_{0n}) + C(K, T) + C(\mathbf{h}, T, |\rho|_{L^{2}(0, T; \mathbb{R})})\\ &&+ \sqrt{K} {{\int}_{0}^{t}} \left|\mathbf{u}_{n}(s) \right|_{\mathrm{H}}^{2} \left( 1+ \left|\theta(s) \right|^{2}_{\mathrm{H}_{0}}\right) ds. \end{array} $$

(A.9)

Using the Gronwall lemma we infer from (A.9),

$$ \begin{array}{@{}rcl@{}} \sup_{s \in [0, T]} | \mathbf{u}_{n}(s)|_{\mathrm{H}}^{2} &\leq& \left[ | \mathbf{u}_{0n}|^{2} + {\Psi}(\mathbf{d}_{0n}) + C(K, \mathbf{h}, |\rho|_{L^{2}(0, T; \mathbb{R})}, T) \right] \\&&\times \exp \left\{ \sqrt{K} \left( 1+|\theta|^{2}_{L^{2}(0, T; \mathrm{H}_{0})} \right) \right\}. \end{array} $$

(A.10)

Since $(\theta , \rho ) \in L^{2}(0, T; \mathrm {H}_{0} \times \mathbb {R}),$ finally using (A.10) we obtain,

$$ \begin{array}{@{}rcl@{}} \sup_{n \geq 1} \left[ \sup_{s \in [0, T]} | \mathbf{u}_{n}(s)|_{\mathrm{H}}^{2} + {\Psi}(\mathbf{d}_{n}(s)) + {{\int}_{0}^{T}} \left( \|\mathbf{u}_{n}(s) \|^{2} + \frac{1}{2} |f_{n}(\mathbf{d}_{n}(s)) + \mathcal{A} \mathbf{d}_{n}(s)|_{L^{2}}^{2} \right) ds \right] \leq \bar{C}, \end{array} $$

where the positive constant

$$\bar{C} := \left[ | \mathbf{u}_{0n}|^{2} + {\Psi}(\mathbf{d}_{0n}) + C(K, \mathbf{h}, |\rho|_{L^{2}(0, T; \mathbb{R})}, T) \right] \exp \left\{C(K, |\theta|_{L^{2}(0, T; \mathrm{H}_{0})}, T) \right\}.$$

□

Proposition A.4

Let $\beta \in (0, \frac {1}{2})$. Then there exist positive constants C = C_β and $\tilde {C}=\tilde {C}_{\beta }$such that

$$ \begin{array}{@{}rcl@{}} \sup_{n \geq 1} \left| \mathbf{u}_{n} \right|_{\mathrm{W}^{\beta, 2}([0, T]; \mathrm{V}^{\prime})}^{2} \leq C \qquad \text{and} \qquad \sup_{n \geq 1} \left| \mathbf{d}_{n} \right|_{\mathrm{W}^{\beta, 2}([0, T]; (D(\mathcal{A}))')}^{2} \leq \tilde{C}. \end{array} $$

Proof

We write the approximated system as:

$$ \begin{array}{@{}rcl@{}} \mathbf{u}_{n}(t) &=& \mathbf{u}_{0n} - {{\int}_{0}^{t}} \mathscr{A}\mathbf{u}_{n}(s) ds - {{\int}_{0}^{t}} B_{n}(\mathbf{u}_{n}(s)) ds - {{\int}_{0}^{t}} M_{n}(\mathbf{d}_{n}(s)) ds +{{\int}_{0}^{t}} P_{n} \sigma(\mathbf{u}_{n}(s)) \theta(s) ds, \\ &:=& {I^{1}_{n}} + {I^{2}_{n}}(t) + {I^{3}_{n}}(t) + {I^{4}_{n}}(t) + {I^{5}_{n}}(t). \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \mathbf{d}_{n}(t) &=& \mathbf{d}_{0n} - {{\int}_{0}^{t}} \mathcal{A} \mathbf{d}_{n}(s) ds - {{\int}_{0}^{t}} \tilde{B}_{n}(\mathbf{u}_{n}(s), \mathbf{d}_{n}(s)) ds - {{\int}_{0}^{t}} f_{n}(\mathbf{d}_{n}(s)) ds \\ &&+ {{\int}_{0}^{t}} \tilde{P}_{n} G(\mathbf{d}_{n}(s)) \rho(s) ds := {J^{1}_{n}} +{J^{2}_{n}}(t) + {J^{3}_{n}}(t) + {J^{4}_{n}}(t) + {J^{5}_{n}}(t). \end{array} $$

(A.11)

Using the same arguments as in Theorem 3.1 of Flandoli and Gatarek [17], we obtain

$$ \begin{array}{@{}rcl@{}} \left| {I^{1}_{n}} \right|^{2}_{\mathrm{H}} \leq k_{1}, \quad \left| {I^{2}_{n}} \right|_{\mathrm{W}^{\beta, 2}([0, T]; \mathrm{V}^{\prime})}^{2} \leq k_{2}, \quad \left| {I^{3}_{n}} \right|_{\mathrm{W}^{\beta, 2}([0, T]; \mathrm{V}^{\prime})}^{2} \leq k_{3}. \end{array} $$

For t > s, using the Cauchy-Schwarz and Young’s inequality and the property of σ, we get

$$ \begin{array}{@{}rcl@{}} &&\left| {I^{5}_{n}}(t) - {I^{5}_{n}}(s) \right|^{2}_{\mathrm{H}} = \bigg| {{\int}_{s}^{t}} P_{n} \sigma(\mathbf{u}_{n}(l)) \theta(l) dl \bigg|^{2}_{\mathrm{H}} \leq \left( {{\int}_{s}^{t}} | P_{n} \sigma(\mathbf{u}_{n}(l)) \theta(l) |_{\mathrm{H}} dl \right)^{2} \\ &&\!\leq\! \left( {{\int}_{s}^{t}} \sqrt{K} \sqrt{1+|\mathbf{u}_{n}(l)|_{\mathrm{H}}^{2}} |\theta(l) |_{\mathrm{H}} dl \right)^{2} \leq c \left( 1+ \sup_{l \in [0, T]} |\mathbf{u}_{n}(l)|_{\mathrm{H}}^{2} \right) {{\int}_{s}^{t}} K dl {{\int}_{s}^{t}} |\theta(l) |^{2}_{\mathrm{H}_{0}} dl.\\ \end{array} $$

(A.12)

Taking s = 0, then integrating from 0 to T we obtain,

$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{T}} \left| {I^{5}_{n}}(l) \right|^{2}_{\mathrm{H}} dl \leq C(T) \left( 1+ \sup_{l \in [0, T]} |\mathbf{u}_{n}(l)|_{\mathrm{H}}^{2} \right) {{\int}_{0}^{T}} K dl {{\int}_{0}^{T}} |\theta(l) |^{2}_{\mathrm{H}_{0}} dl, \end{array} $$

(A.13)

and

$$ \begin{array}{@{}rcl@{}} &&{}{{\int}_{0}^{T}}{{\int}_{0}^{T}} \frac{\left| {I^{5}_{n}}(t) - {I^{5}_{n}}(s) \right|^{2}_{\mathrm{H}}}{|t-s|^{1+2\beta}} dt ds \leq c \left( 1+ \sup_{l \in [0, T]} |\mathbf{u}_{n}(l)|_{\mathrm{H}}^{2} \right) {{\int}_{0}^{T}} K dl {{\int}_{0}^{T}}{{\int}_{0}^{T}} {{\int}_{s}^{t}}\\ && \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad ~~~ \frac{| \theta(l)|^{2}_{\mathrm{H}_{0}}}{|t-s|^{1+2\beta}} dl dt ds.\end{array} $$

(A.14)

Using Fubini’s theorem for $\beta \in (0, \frac {1}{2})$,

$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{T}}{{\int}_{0}^{T}} {{\int}_{s}^{t}} \frac{| \theta(l)|^{2}_{\mathrm{H}}}{|t-s|^{1+2\beta}} dl dt ds \leq \tilde{c} {{\int}_{0}^{T}} | \theta(l)|^{2}_{\mathrm{H}_{0}} dl \leq C. \end{array} $$

(A.15)

Using Proposition A.3, inequalities (A.12), (A.13), (A.14) and (A.15) we get,

$$ \begin{array}{@{}rcl@{}} \left| {I^{5}_{n}} \right|_{\mathrm{W}^{\beta, 2}([0, T]; \mathrm{H})}^{2} \leq k_{5}. \end{array} $$

Now we calculate for t > s,

$$ \begin{array}{@{}rcl@{}} &&\left| {I^{4}_{n}}(t) - {I^{4}_{n}}(s) \right|^{2}_{\mathrm{V}^{\prime}} = \bigg| {{\int}_{s}^{t}} M_{n}(\mathbf{d}_{n}(l)) dl \bigg|^{2}_{\mathrm{V}^{\prime}} \leq \left( {{\int}_{s}^{t}} | M_{n}(\mathbf{d}_{n}(l)) |_{\mathrm{V}^{\prime}} dl \right)^{2} \\ &&\leq \left( {{\int}_{s}^{t}} |\nabla \mathbf{d}_{n}(l)) |_{L^{2}} | \mathcal{A} \mathbf{d}_{n}(l)) |_{L^{2}} dl \right)^{2} \leq C_{T} \sup_{l \in [0, T]} |\nabla \mathbf{d}_{n}(l)) |_{L^{2}}^{2} {{\int}_{s}^{t}} | \mathcal{A} \mathbf{d}_{n}(l)) |_{L^{2}}^{2} dl. \end{array} $$

Now using the similar procedure as done to get inequalities (A.12), (A.13), (A.14), (A.15) with the help of Proposition A.2 and Proposition A.3 we get,

$$ \begin{array}{@{}rcl@{}} \left| {I^{4}_{n}} \right|_{\mathrm{W}^{\beta, 2}([0, T]; \mathrm{V}^{\prime})}^{2} \leq k_{4}. \end{array} $$

Now consider (A.11). From Theorem 3.1 of Flandoli and Gatarek [17] we get,

$$\left| {J^{1}_{n}} \right|^{2}_{H^{1}} \leq C_{1}, \qquad \left| {J^{2}_{n}} \right|_{\mathrm{W}^{1, 2}([0, T]; (D(\mathcal{A}))')}^{2} \leq C_{2}.$$

Again for t > s, from (2.6) we have,

$$ \begin{array}{@{}rcl@{}} &&\left| {J^{3}_{n}}(t) - {J^{3}_{n}}(s) \right|^{2}_{(D(\mathcal{A}))^{\prime}} = \left| {{\int}_{s}^{t}} \tilde{B}_{n}(\mathbf{u}_{n}(l), \mathbf{d}_{n}(l)) dl \right|^{2}_{(D(\mathcal{A}))^{\prime}} \leq \left( {{\int}_{s}^{t}} | \tilde{B}_{n}(\mathbf{u}_{n}(l), \mathbf{d}_{n}(l)) |_{L^{2}} dl \right)^{2} \\ &&\leq c \left( {{\int}_{s}^{t}} \left\{ |\mathbf{u}_{n}(l)| | \nabla \mathbf{u}_{n}(l)| + | \nabla \mathbf{d}_{n}(l)| | \mathcal{A} \mathbf{d}_{n}(l)| \right\} dl \right)^{2} \\ &&\leq C_{T} {{\int}_{s}^{t}} \left\{ |\mathbf{u}_{n}(l)|^{2} | \nabla \mathbf{u}_{n}(l)|^{2} + | \nabla \mathbf{d}_{n}(l)|^{2} | \mathcal{A} \mathbf{d}_{n}(l)|^{2} \right\} dl \\ &&\leq C_{T} \left[ \sup_{l \in [0, T]} |\mathbf{u}_{n}(l)|_{\mathrm{H}}^{2} {{\int}_{s}^{t}} \| \mathbf{u}_{n}(l)\|^{2} dl + \sup_{l \in [0, T]} | \nabla \mathbf{d}_{n}(l) |_{L^{2}}^{2} {{\int}_{s}^{t}} | \mathcal{A} \mathbf{d}_{n}(l) |_{L^{2}}^{2} dl \right]. \end{array} $$

Now using Proposition A.2 and Proposition A.3, following the same technique as (A.12)-(A.15), for $\beta \in (0, \frac {1}{2})$ we obtain

$$ \begin{array}{@{}rcl@{}} \left| {J^{3}_{n}} \right|_{\mathrm{W}^{\beta, 2}([0, T]; (D(\mathcal{A}))')}^{2} \leq C_{3}. \end{array} $$

Following Lemma 6.1 (equation (6.26)) in [9] and similar procedure as (A.12)-(A.15) will give for $\beta \in (0, \frac {1}{2})$,

$$ \begin{array}{@{}rcl@{}} \left| {J^{4}_{n}} \right|_{\mathrm{W}^{\beta, 2}([0, T]; (D(\mathcal{A}))')}^{2} \leq C_{4}. \end{array} $$

Now for ${J^{5}_{n}},$ since $\rho \in S^{\alpha }_{2},$ using the property of G, Proposition A.2 and similar technique as (A.12)-(A.15) we conclude for $\beta \in (0, \frac {1}{2})$,

$$ \begin{array}{@{}rcl@{}} \left| {J^{5}_{n}} \right|_{\mathrm{W}^{\beta, 2}([0, T]; L^{2})}^{2} \leq C_{5}. \end{array} $$

Since H↪V^′ and $L^{2} \hookrightarrow (D(\mathcal {A}))',$ from above estimates for $\beta \in (0, \frac {1}{2})$ we get

$$ \begin{array}{@{}rcl@{}} \sup_{n \geq 1} \left| \mathbf{u}_{n} \right|_{\mathrm{W}^{\beta, 2}([0, T]; \mathrm{V}^{\prime})}^{2} \leq C_{\beta} \qquad \text{and} \qquad \sup_{n \geq 1} \left| \mathbf{d}_{n} \right|_{\mathrm{W}^{\beta, 2}([0, T]; (D(\mathcal{A}))')}^{2} \leq \tilde{C}_{\beta} \end{array} $$

□

Finally we are ready to prove Theorem 5.5.

1.2 A.2. Proof of Theorem 5.5

Proof

Existence:

Let us choose and fix $\beta \in (0, {\frac 12})$. From Proposition A.2, A.3 and A.4 we deduce that, there exist positive constants C₁, C₂, C₃(β), C₄(β) such that (7.3), (7.4) and (7.5) hold. From the above estimates and Lemma 2.3, we infer that there exists a subsequence $(\mathbf {u}_{m^{\prime }}, \mathbf {d}_{m^{\prime }})$ and an element

$$ (\mathbf{u}, \mathbf{d}) \in L^{2}([0, T]; \mathrm{V}) \cap L^{\infty}([0, T]; \mathrm{H}) \times L^{2}([0, T]; D(\mathcal{A})) \cap L^{\infty}([0, T]; H^{1}) $$

such that as m^′→∞ we have,

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} &\mathbf{u}_{m^{\prime}} \to \mathbf{u} \quad \text{in} \quad L^{2}([0, T]; \mathrm{V}) \quad \text{weakly}, \\ &\mathbf{u}_{m^{\prime}} \to \mathbf{u} \quad \text{in} \quad L^{\infty}([0, T]; \mathrm{H}) \quad \text{weak-star}, \\ &\mathbf{d}_{m^{\prime}} \to \mathbf{d} \quad \text{in} \quad L^{2}([0, T]; D(\mathcal{A})) \quad \text{weakly}, \\ &\mathbf{d}_{m^{\prime}} \to \mathbf{d} \quad \text{in} \quad L^{\infty}([0, T]; H^{1}) \quad \text{weak-star}. \\ &\mathbf{u}_{m^{\prime}} \to \mathbf{u} \quad \text{in} \quad L^{2}([0, T]; \mathrm{H}) \quad \text{strongly}, \\ & \mathbf{d}_{m^{\prime}} \to \mathbf{d} \quad \text{in} \quad L^{2}([0, T]; H^{1}) \quad \text{strongly}. \end{array}\right. \end{array} $$

Finally, we show (u, d) is the unique solution to (5.8)-(5.9). For this we will argue similarly as in the proof of Theorem 3.1 in Temam [25], Section 3.

The similar technique to prove (7.11) (in the proof of Condition 1), gives us the convergence as m^′→∞, for individual terms involving control parameters. For all other linear and nonlinear terms, we follow the calculations of our earlier work (see [9]).

Using similar arguments as in the proof of Theorem 3.1 in Temam [25], Section 3, Chapter III, we infer that (u, d) is the desired solution.

Eqs. 5.10, 5.11 can be proved similarly as Eqs. 7.3 and 7.4. Now using the similar arguments as in the proof of Theorem 3.2 in Temam [25], Section 3, Chapter III, we also have

$$ \begin{array}{@{}rcl@{}} \frac{d\mathbf{u}}{dt} \in L^{2}([0, T]; \mathrm{V}^{\prime}) + L^{1}([0, T]; \mathrm{H}) \quad \text{and} \quad \frac{d\mathbf{d}}{dt} \in L^{2}([0, T]; (D(\mathcal{A}))') + L^{1}([0, T]; H^{1}). \end{array} $$

Due to Lemma 1.2, (1.84) and (1.85) in Temam [25], Chapter III, we infer that

$$ \mathbf{u} \in \mathbb{C}([0, T]; \mathrm{H}) \quad \text{and} \quad \mathbf{d} \in \mathbb{C}([0, T]; H^{1}). $$

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Uniqueness

The uniqueness will follow the same technique required to get (7.21) and (7.22) in the proof of Condition 1.