This section is dedicated to constructing the global weak solutions for the fractional problem (6)-(7)-(8) via the Faedo-Galerkin/penalty method. In particular, the global existence theorem (3.2) for the problem considered will be proved.
4.1 The penalty problem
Let \(\varepsilon> 0\) be a fixed parameter. We construct approximated solutions \((\mathbf{m}^{\varepsilon},\mathbf{u}^{\varepsilon})\) converging, as \(\varepsilon \rightarrow0\), to a solution \((\mathbf{m},\mathbf{u})\) of the problem. Then we consider the following problem:
$$ \textstyle\begin{cases} \dot{\mathbf{m}}^{\varepsilon}+a\Lambda^{2\alpha}\mathbf {m}^{\varepsilon}+\boldsymbol {\ell} (\mathbf{m} ^{\varepsilon},\mathbf{u}^{\varepsilon})+ \frac{|\mathbf {m}^{\varepsilon}|^{2}-1}{\varepsilon}\mathbf{m}^{\varepsilon}=0, \\ \rho\ddot{\mathbf{u}}^{\varepsilon}-\operatorname{div} (\mathcal {S}(\mathbf{u} ^{\varepsilon})+ \frac{1}{2}\mathcal{L}(\mathbf{m}^{\varepsilon}) )=0, \end{cases} $$
(11)
in \(Q=\Omega\times(0,T)\). Note that the last term of the first equation in (11) has been introduced in order to represent the constraint \(|\mathbf{m}| = 1\) in the limit \(\varepsilon\to0\).
System (11) is supplemented with initial and boundary conditions
$$\begin{aligned}& \mathbf{u}^{\varepsilon}(\cdot,0)=\mathbf{u}_{0}, \qquad\dot{\mathbf {u}}^{\varepsilon }(\cdot ,0)=\mathbf{u}_{1},\qquad \mathbf{m}^{\varepsilon}(\cdot,0)=\mathbf{m}_{0},\qquad | \mathbf{m}_{0}|=1 \quad \mbox{a.e. in } \Omega, \end{aligned}$$
(12)
$$\begin{aligned}& \mathbf{u}^{\varepsilon}=0 \quad\mbox{on } \Sigma. \end{aligned}$$
(13)
We construct approximate solutions of (11)-(12)-(13) by using the Galerkin method: let \(\{f_{i}\} _{i\in\mathbb{N}}\) be an orthonormal basis of \(L^{2}(\Omega)\) consisting of all the eigenfunctions for the operator \({\Lambda^{2\alpha}}\) (the existence of such a basis can be proved as in [13], Ch. II)
$$\Lambda^{2\alpha}f_{i}=\alpha_{i} f_{i}, \quad i=1,2,\ldots $$
under periodic boundary conditions, and let \(\{g_{i}\}_{i\in\mathbb{N}}\) be an orthonormal basis of \(L^{2}(\Omega)\) consisting of all the eigenfunctions for the operator −Δ
$$ \textstyle\begin{cases} -\Delta g_{i}=\beta_{i} g_{i}, & i=1,2,\ldots,\\ g_{i}=0 & \mbox{on } \partial\Omega. \end{cases} $$
We consider the following problem in \(Q=\Omega\times(0,T)\):
$$ \textstyle\begin{cases} \dot{\mathbf{m}}^{\varepsilon,N}+a\Lambda^{2\alpha}\mathbf {m}^{\varepsilon ,N}+\boldsymbol {\ell}(\mathbf{m} ^{\varepsilon,N},\mathbf{u}^{\varepsilon,N})+ \frac {|\mathbf{m} ^{\varepsilon,N}|^{2}-1}{\varepsilon}\mathbf{m}^{\varepsilon,N}=0, \\ \rho\ddot{\mathbf{u}}^{\varepsilon,N}-\operatorname{div} (\mathcal {S}(\mathbf{u} ^{\varepsilon,N})+ \frac{1}{2}\mathcal{L}(\mathbf{m} ^{\varepsilon ,N}) )=0, \end{cases} $$
(14)
with initial and boundary conditions
$$\begin{aligned}& \mathbf{u}^{\varepsilon,N}(\cdot,0)=\mathbf{u}^{N}(\cdot,0), \qquad\dot {\mathbf{u} }^{\varepsilon ,N}(\cdot,0)=\dot{\mathbf{u}}^{N}(\cdot,0), \qquad \mathbf{m}^{\varepsilon ,N}(\cdot ,0)=\mathbf{m} ^{N}(\cdot,0)\quad \mbox{in } \Omega, \\& \mathbf{u}^{\varepsilon,N}=0\quad \mbox{on } \Sigma=\partial \Omega \times(0,T) \end{aligned}$$
and
$$\begin{aligned}& \int_{\Omega}\mathbf{u}^{N}(x,0)g_{i}(x)\, \mathrm{d}x= \int_{\Omega }\mathbf{u} _{0}(x)g_{i}(x) \, \mathrm{d}x ,\qquad \int_{\Omega}\dot{\mathbf {u}}^{N}(x,0)g_{i}(x) \,\mathrm{d}x= \int _{\Omega }\mathbf{u}_{1}(x)g_{i}(x) \, \mathrm{d}x , \\& \int_{\Omega}\mathbf{m}^{N}(x,0)f_{i}(x) \, \mathrm{d}x= \int_{\Omega }\mathbf{m} _{0}(x)f_{i}(x) \, \mathrm{d}x. \end{aligned}$$
We are looking for approximate solutions \((\mathbf{m}^{\varepsilon ,N},\mathbf{u} ^{\varepsilon,N})\) to (14) under the form
$$\mathbf{m}^{\varepsilon,N}=\sum_{i=1}^{N} \mathbf {a}_{i}(t)f_{i}(x),\qquad \mathbf{u}^{\varepsilon,N}= \sum_{i=1}^{N} \mathbf{b}_{i}(t)g_{i}(x), $$
where \(\mathbf{a}_{i}\) and \(\mathbf{b}_{i}\) are \(\mathbb{R}^{3}\)-valued vectors.
If we multiply each scalar equation of the first equation of (14) by \(f_{i}\) and the second by \(g_{i}\) and integrate in Ω, we get to a system of integro-differential equations in the unknown \((\mathbf{a} _{i}(t),\mathbf{b}_{i}(t)), i=1, 2,\ldots,N\), that we can write in the form (based on ideas exploited in [2])
$$\textstyle\begin{cases} \mathbf{m}^{\varepsilon,N}=- \int_{0}^{t} (a\Lambda ^{2\alpha }\mathbf{m} ^{\varepsilon,N}(s)+\boldsymbol {\ell}(\mathbf{m}^{\varepsilon ,N}(s),\mathbf{u}^{\varepsilon ,N}(s))+ \frac{|\mathbf{m}^{\varepsilon ,N}(s)|^{2}-1}{\varepsilon}\mathbf{m} ^{\varepsilon,N}(s) )\,\mathrm{d}s+\mathbf{m}^{N}(0),\\ \dot{\mathbf{u}}^{\varepsilon,N}=\frac{1}{\rho} \int _{0}^{t}\operatorname{div} (\mathcal{S}(\mathbf{u}^{\varepsilon,N}(s))+ \frac {1}{2}\mathcal{L}(\mathbf{m}^{\varepsilon,N}(s)) )\,\mathrm{d}s+\dot {\mathbf{u} }^{N}(0), \\ \mathbf{u}^{\varepsilon,N}= \int_{0}^{t}\dot{\mathbf {u}}^{\varepsilon ,N}(s)\,\mathrm{d}s+\mathbf{u}^{N}(0) \end{cases} $$
and if we define \(\mathbf{v}(t):=(\mathbf{v}_{1}(t),\mathbf {v}_{2}(t),\dots,\mathbf{v}_{N}(t))\), where \(\mathbf{v}_{i}(t)=(\mathbf{a}_{i}(t),\mathbf{b}_{i}(t),\dot{\mathbf {b}}_{i}(t)), i=1, 2,\ldots,N\), we can write the last system in the form
$$ \mathbf{v}(t)= \int^{t}_{0}\mathbf{w}\bigl(t,s,\mathbf{v} (s)\bigr) \,\mathrm{d}s+\mathbf{v}(0). $$
Now, for a strictly positive constant α, we take
$$E_{\alpha}:=\bigl\{ \mathbf{x}/ \Vert \mathbf{x} \Vert _{\infty} \leqslant\alpha\bigr\} , $$
where \(\|\mathbf{x}\|_{\infty}= \operatorname{sup}_{0\leqslant t\leqslant\tau} |\mathbf{x}_{i}(t)|\), for \(\tau>0\). We consider the mapping T defined on \(E_{\alpha}\) by
$$\forall\mathbf{x}\in E_{\alpha},\quad T(\mathbf{x})= \int^{t}_{0} \mathbf{w} \bigl(t,s,\mathbf{x}(s) \bigr)\,\mathrm{d}s+\mathbf{v}(0). $$
Then we have, for x in \(E_{\alpha}\),
$$\bigl\Vert \mathbf{w}(\mathbf{x}) \bigr\Vert _{\infty}= \mathop{\operatorname{sup}}_{0\leqslant t\leqslant\tau} \mathop{\operatorname{sup}}_{0\leqslant s\leqslant t} \bigl\vert \mathbf{w}\bigl(t,s,\mathbf{x} (s)\bigr) \bigr\vert \leqslant C\alpha $$
for a positive constant C. So, if we choose \((\alpha, \tau)\) such that \(\|\mathbf{v}(0)\|_{\infty}\leqslant\frac{\alpha}{2}\) and \(\tau \leqslant \frac{1}{2C}\), we can write
$$\begin{aligned} \bigl\Vert T(\mathbf{x}) \bigr\Vert _{\infty} \leqslant & C\alpha \tau+ \bigl\Vert \mathbf{v}(0) \bigr\Vert _{\infty}\\ \leqslant & C\alpha\tau+\frac{\alpha}{2} \\ \leqslant & \alpha, \end{aligned}$$
that implies \(T(\mathbf{x})\in E_{\alpha}\). Moreover, for \(\mathbf {x}^{(1)}\) and \(\mathbf{x} ^{(2)}\) in \(E_{\alpha}\) and for all \(t\in[0,\tau]\),
$$\begin{aligned} \bigl\vert T\bigl(\mathbf{x}^{(1)}\bigr)_{i}-T\bigl( \mathbf{x}^{(2)}\bigr)_{i} \bigr\vert =& \biggl\vert \int^{t}_{0} \bigl( \mathbf{w}^{(1)}_{i}- \mathbf{w} ^{(2)}_{i}\bigr)\,\mathrm{d}s \biggr\vert \\ \leqslant & \tau \bigl\Vert \nabla_{\mathbf{x}} \mathbf{w}(\mathbf{x}) \bigr\Vert _{\infty}\bigl\Vert \mathbf{x}^{(1)}- \mathbf{x}^{(2)} \bigr\Vert _{\infty}. \end{aligned}$$
Since \(\|\nabla_{\mathbf{x}} \mathbf{w}(\mathbf{x})\| _{\infty}\) is bounded in \(E_{\alpha}\) and we can choose τ small enough to have \(\tau\|\nabla_{\mathbf{x}}\mathbf{w}(\mathbf{x})\| _{\infty}<1\), then we obtain
$$\bigl\Vert T\bigl(\mathbf{x}^{(1)}\bigr)-T\bigl(\mathbf{x}^{(2)} \bigr) \bigr\Vert _{\infty}\leqslant C \bigl\Vert \mathbf{x}^{(1)}- \mathbf{x} ^{(2)} \bigr\Vert _{\infty}$$
with \(0< C<1\). Hence T is a contraction mapping of the convex compact set \(E_{\alpha}\) into itself. From the fixed point theorem we deduce the local existence of solutions to the problem that we can extend on \([0,T]\) using a priori estimates. For this, we multiply the first equation of (14) by \(\dot{\mathbf{m}}^{\varepsilon,N}\) and the second by \(\dot{\mathbf{u}}^{\varepsilon,N}\) integrating in Ω, we obtain
$$\textstyle\begin{cases} \int_{\Omega} \vert \dot{\mathbf{m}}^{\varepsilon,N} \vert ^{2} \,\mathrm{d} x+a \int_{\Omega} \Lambda^{2\alpha}\mathbf{m}^{\varepsilon ,N}\cdot \dot {\mathbf{m}}^{\varepsilon,N} \,\mathrm{d}x\\ \quad{}+ \int_{\Omega} \boldsymbol {\ell}(\mathbf {m}^{\varepsilon,N},\mathbf{u} ^{\varepsilon,N})\cdot\dot{\mathbf{m}}^{\varepsilon,N} \,\mathrm{d}x + \frac{1}{4\varepsilon}\frac{\mathrm{d}}{\mathrm {d}t}\int _{\Omega} (|\mathbf{m}^{\varepsilon,N}|^{2}-1)^{2} \,\mathrm{d}x=0, \\ \frac{\rho}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int _{\Omega }|\dot {\mathbf{u}}^{\varepsilon,N}|^{2} \,\mathrm{d}x+ \int_{\Omega} ( \mathcal{S}(\mathbf{u}^{\varepsilon,N})+ \frac{1}{2}\mathcal {L}(\mathbf{m} ^{\varepsilon,N}) )\cdot\nabla\dot{\mathbf{u}}^{\varepsilon ,N} \,\mathrm{d} x= 0. \end{cases} $$
Now we use the following result (the proof can be found in [4]).
Lemma 4.1
If f and g belong to
\(H^{2\alpha}_{per}(\Omega):=\{f\in L^{2}(\Omega)/ \Lambda^{2\alpha}f\in L^{2}(\Omega) \}\), then
$$\int_{\Omega}\Lambda^{2\alpha}f\cdot g \,\mathrm{d}x= \int_{\Omega}\Lambda ^{\alpha }f\cdot\Lambda^{\alpha}g \,\mathrm{d}x. $$
We obtain
$$ \textstyle\begin{cases} \int_{\Omega} \vert \dot{\mathbf{m}}^{\varepsilon,N} \vert ^{2} \,\mathrm{d} x+\frac {a}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega} \vert \Lambda^{\alpha }\mathbf{m} ^{\varepsilon,N} \vert ^{2} \,\mathrm{d}x\\ \quad{}+ \int_{\Omega} \boldsymbol {\ell}(\mathbf {m}^{\varepsilon,N},\mathbf{u} ^{\varepsilon,N})\cdot\dot{\mathbf{m}}^{\varepsilon,N} \,\mathrm{d}x + \frac{1}{4\varepsilon}\frac{\mathrm{d}}{\mathrm {d}t}\int _{\Omega} ( \vert \mathbf{m}^{\varepsilon,N} \vert ^{2}-1)^{2} \,\mathrm{d}x=0, \\ \frac{\rho}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int _{\Omega } \vert \dot {\mathbf{u}}^{\varepsilon,N} \vert ^{2}\, \mathrm{d}x+ \int_{\Omega} ( \mathcal{S}(\mathbf{u}^{\varepsilon,N})+ \frac{1}{2}\mathcal {L}(\mathbf{m} ^{\varepsilon,N}) )\cdot\nabla\dot{\mathbf{u}}^{\varepsilon ,N}\, \mathrm{d} x= 0. \end{cases} $$
(15)
Note that \(\lambda_{ijkl}=\lambda_{jikl}\), hence (using the components of the vector ℓ)
$$\begin{aligned} \int_{\Omega}\boldsymbol {\ell}\bigl(\mathbf{m}^{\varepsilon ,N},\mathbf{u} ^{\varepsilon ,N}\bigr)\cdot\dot{\mathbf{m}}^{\varepsilon,N}\, \mathrm{d}x =& \int _{\Omega}\lambda_{ijkl}m^{\varepsilon,N}_{j} \dot{m}^{\varepsilon ,N}_{i}\epsilon_{kl}\bigl( \mathbf{u}^{\varepsilon,N}\bigr) \,\mathrm{d}x \\ =& \frac{1}{2} \int_{\Omega}\lambda _{ijkl}\bigl(m^{\varepsilon ,N}_{j} \dot{m}^{\varepsilon,N}_{i}+m^{\varepsilon,N}_{i}\dot{m}^{\varepsilon ,N}_{j}\bigr)\epsilon_{kl}\bigl( \mathbf{u}^{\varepsilon,N}\bigr) \,\mathrm{d}x. \end{aligned}$$
From where
$$\begin{aligned} \int_{\Omega}\boldsymbol {\ell}\bigl(\mathbf{m}^{\varepsilon ,N},\mathbf{u} ^{\varepsilon ,N}\bigr)\cdot\dot{\mathbf{m}}^{\varepsilon,N} \,\mathrm{d}x =& \frac {1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega}\lambda _{ijkl}m^{\varepsilon ,N}_{i}m^{\varepsilon,N}_{j} \epsilon_{kl}\bigl(\mathbf{u}^{\varepsilon ,N}\bigr)\,\mathrm{d}x \\ & {}- \frac{1}{2} \int_{\Omega}\lambda _{ijkl}m^{\varepsilon ,N}_{i}m^{\varepsilon,N}_{j} \epsilon_{kl}\bigl(\dot{\mathbf{u}}^{\varepsilon ,N}\bigr)\,\mathrm{d}x, \end{aligned}$$
(16)
and since the tensor \(\mathcal{L}\) is symmetric, we have
$$\begin{aligned} \frac{1}{2} \int_{\Omega}\mathcal{L}\bigl(\mathbf{m}^{\varepsilon ,N}\bigr)\cdot \nabla\dot {\mathbf{u}}^{\varepsilon,N} \,\mathrm{d}x =& \frac{1}{2} \int_{\Omega }\mathcal {L}\bigl(\mathbf{m} ^{\varepsilon,N}\bigr) \cdot\epsilon\bigl(\dot{\mathbf{u}}^{\varepsilon,N}\bigr) \,\mathrm{d} x \\ =& \frac{1}{2} \int_{\Omega}\lambda _{ijkl}m^{\varepsilon,N}_{i}m^{\varepsilon,N}_{j} \epsilon_{kl}\bigl(\dot {\mathbf{u} }^{\varepsilon,N}\bigr)\,\mathrm{d}x. \end{aligned}$$
(17)
Furthermore, (omitting superscripts and) following the idea introduced in [2], we have
$$\begin{aligned} \int_{\Omega}\mathcal{S}\bigl(\mathbf{u}(t)\bigr)\cdot\nabla\dot{ \mathbf {u}}(t) \,\mathrm{d} x =& \int _{\Omega}\mathcal{G}_{ijkl}(0)\epsilon_{ij} \bigl(\mathbf{u}(t)\bigr)\epsilon _{kl}\bigl(\dot{\mathbf{u} }(t)\bigr) \, \mathrm{d}x \\ &{} + \int_{\Omega}\int^{t}_{0}\dot{\mathcal {G}}_{ijkl}(t-s) \epsilon_{ij}\bigl(\mathbf{u}(s)\bigr)\epsilon_{kl}\bigl(\dot {\mathbf{u}}(t)\bigr) \,\mathrm{d} s \,\mathrm{d}x \\ =& \int_{\Omega}\mathcal{G}_{ijkl}(0)\epsilon_{ij} \bigl(\mathbf {u}(t)\bigr)\epsilon _{kl}\bigl(\dot{\mathbf{u}}(t)\bigr) \,\mathrm{d}x \\ &{}+ \int_{\Omega}\int ^{t}_{0}\dot {\mathcal {G}}_{ijkl}(s) \epsilon_{ij}\bigl(\mathbf{u}(t-s)\bigr)\epsilon_{kl}\bigl( \dot {\mathbf{u}}(t)\bigr) \,\mathrm{d} s \,\mathrm{d}x \\ =& - \int_{\Omega}\bigl(\mathcal{G}_{ijkl}(t)- \mathcal{G}_{ijkl}(0) \bigr)\epsilon_{ij}\bigl(\mathbf{u}(t) \bigr)\epsilon_{kl}\bigl(\dot{\mathbf{u}}(t)\bigr) \,\mathrm{d}x \\ &{}+ \int_{\Omega}\mathcal{G}_{ijkl}(t)\epsilon_{ij} \bigl(\mathbf{u} (t)\bigr)\epsilon _{kl}\bigl(\dot{\mathbf{u}}(t)\bigr) \, \mathrm{d}x \\ &{}+ \int_{\Omega}\int^{t}_{0}\dot{\mathcal{G}}_{ijkl}(s) \epsilon _{ij}\bigl(\mathbf{u} (t-s)\bigr)\epsilon_{kl}\bigl( \dot{\mathbf{u}}(t)\bigr) \,\mathrm{d}s \,\mathrm{d}x \\ =& - \int_{\Omega}\int^{t}_{0}\dot{\mathcal{G}}_{ijkl}(s) \epsilon _{ij}\bigl(\mathbf{u} (t)\bigr)\epsilon_{kl}\bigl( \dot{\mathbf{u}}(t)\bigr) \,\mathrm{d}s\, \mathrm{d}x \\ &{}+ \int_{\Omega}\mathcal{G}_{ijkl}(t)\epsilon_{ij} \bigl(\mathbf{u} (t)\bigr)\epsilon _{kl}\bigl(\dot{\mathbf{u}}(t)\bigr) \,\mathrm{d}x \\ &{}+ \int_{\Omega}\int^{t}_{0}\dot{\mathcal{G}}_{ijkl}(s) \epsilon _{ij}\bigl(\mathbf{u} (t-s)\bigr)\epsilon_{kl}\bigl( \dot{\mathbf{u}}(t)\bigr) \,\mathrm{d}s\, \mathrm{d}x. \end{aligned}$$
Then
$$\begin{aligned} \int_{\Omega}\mathcal{S}\bigl(\mathbf{u}(t)\bigr)\cdot\nabla\dot{ \mathbf {u}}(t) \,\mathrm{d} x =& - \int _{\Omega}\int^{t}_{0}\dot{\mathcal{G}}_{ijkl}(s) \bigl(\epsilon _{ij}\bigl(\mathbf{u} (t)\bigr)-\epsilon_{ij} \bigl(\mathbf{u}(t-s)\bigr) \bigr)\epsilon_{kl}\bigl(\dot{\mathbf {u}}(t)\bigr)\, \mathrm{d}s \,\mathrm{d}x \\ &{}+ \int_{\Omega}\mathcal{G}_{ijkl}(t)\epsilon_{ij} \bigl(\mathbf{u} (t)\bigr)\epsilon _{kl}\bigl(\dot{\mathbf{u}}(t)\bigr) \,\mathrm{d}x. \end{aligned}$$
This leads to
$$\begin{aligned} \int_{\Omega}\mathcal{S}\bigl(\mathbf{u}(t)\bigr)\cdot\nabla\dot{ \mathbf {u}}(t) \,\mathrm{d}x =& - \int _{\Omega}\int^{t}_{0}\dot{\mathcal{G}}_{ijkl}(s) \bigl(\epsilon _{ij}\bigl(\mathbf{u} (t)\bigr)-\epsilon_{ij} \bigl(\mathbf{u}(t-s)\bigr) \bigr)\epsilon_{kl}\bigl(\dot{\mathbf {u}}(t)\bigr) \,\mathrm{d}s\, \mathrm{d}x \\ &{}+\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega}\mathcal {G}_{ijkl}(t)\epsilon_{ij} \bigl(\mathbf{u}(t)\bigr)\epsilon_{kl}\bigl(\mathbf {u}(t)\bigr) \, \mathrm{d}x \\ &{}-\frac{1}{2} \int_{\Omega}\dot{\mathcal{G}}_{ijkl}(t)\epsilon _{ij}\bigl(\mathbf{u} (t)\bigr)\epsilon_{kl}\bigl( \mathbf{u}(t)\bigr) \,\mathrm{d}x, \end{aligned}$$
(18)
by using (\(H_{1}\)). On the other hand,
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t} \int^{t}_{0} \int_{\Omega}\dot{\mathcal {G}}_{ijkl}(s) \bigl( \epsilon_{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon _{ij}\bigl( \mathbf{u} (t-s)\bigr) \bigr) \bigl(\epsilon_{kl}\bigl(\mathbf{u}(t) \bigr)-\epsilon_{kl}\bigl(\mathbf {u}(t-s)\bigr) \bigr) \,\mathrm{d} x \,\mathrm{d}s \\& \quad = \int_{\Omega}\dot{\mathcal{G}}_{ijkl}(t) \bigl(\epsilon _{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon_{ij}\bigl( \mathbf{u}(0)\bigr) \bigr) \bigl(\epsilon_{kl}\bigl(\mathbf{u} (t) \bigr)-\epsilon_{kl}\bigl(\mathbf{u}(0)\bigr) \bigr) \,\mathrm{d}x \\& \qquad {}+2 \int_{\Omega}\int^{t}_{0}\dot{\mathcal {G}}_{ijkl}(s) \bigl(\epsilon_{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon _{ij} \bigl(\mathbf{u} (t-s)\bigr) \bigr)\epsilon_{kl}\bigl(\dot{ \mathbf{u}}(t)\bigr) \,\mathrm{d}s \,\mathrm{d}x \\& \qquad {}-2 \int_{\Omega}\int^{t}_{0}\dot{\mathcal {G}}_{ijkl}(s) \bigl(\epsilon_{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon _{ij} \bigl(\mathbf{u} (t-s)\bigr) \bigr)\epsilon_{kl}\bigl(\dot{ \mathbf{u}}(t-s)\bigr) \,\mathrm{d}s\, \mathrm{d}x, \end{aligned}$$
(19)
and we have
$$\begin{aligned}& -2 \int_{\Omega}\int^{t}_{0}\dot{\mathcal{G}}_{ijkl}(s) \bigl(\epsilon _{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon_{ij} \bigl(\mathbf{u}(t-s)\bigr) \bigr)\epsilon _{kl}\bigl(\dot{\mathbf{u} }(t-s)\bigr) \,\mathrm{d}s \,\mathrm{d}x \\& \quad =-2 \int_{\Omega}\int^{t}_{0}\dot{\mathcal {G}}_{ijkl}(s) \bigl(\epsilon_{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon_{ij} \bigl(\mathbf{u}(t-s)\bigr) \bigr)\frac{\partial }{\partial s} \bigl(\epsilon_{kl} \bigl(\mathbf{u}(t)\bigr)-\epsilon _{kl}\bigl(\mathbf{u} (t-s)\bigr) \bigr) \,\mathrm{d}s\, \mathrm{d}x \\& \quad =- \int_{\Omega}\dot{\mathcal{G}}_{ijkl}(t) \bigl(\epsilon _{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon_{ij}\bigl( \mathbf{u}(0)\bigr) \bigr) \bigl(\epsilon_{kl}\bigl(\mathbf{u} (t) \bigr)-\epsilon_{kl}\bigl(\mathbf{u}(0)\bigr) \bigr) \,\mathrm{d}x \\& \qquad {}+ \int_{\Omega}\int^{t}_{0}\ddot{\mathcal {G}}_{ijkl}(s) \bigl(\epsilon_{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon _{ij} \bigl(\mathbf{u} (t-s)\bigr) \bigr) \bigl(\epsilon_{kl}\bigl( \mathbf{u}(t)\bigr)-\epsilon_{kl}\bigl(\mathbf {u}(t-s)\bigr) \bigr) \, \mathrm{d} s \,\mathrm{d}x. \end{aligned}$$
Substituting in (19), we find
$$\begin{aligned}& - \int_{\Omega}\int^{t}_{0}\dot{\mathcal {G}}_{ijkl}(s) \bigl(\epsilon_{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon_{ij} \bigl(\mathbf{u}(t-s)\bigr) \bigr)\epsilon _{kl}\bigl(\dot { \mathbf{u}}(t)\bigr) \,\mathrm{d}s\, \mathrm{d}x \\& \quad =-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int^{t}_{0} \int _{\Omega}\dot{\mathcal{G}}_{ijkl}(s) \bigl( \epsilon_{ij}\bigl(\mathbf {u}(t)\bigr)-\epsilon _{ij}\bigl( \mathbf{u} (t-s)\bigr) \bigr) \bigl(\epsilon_{kl}\bigl(\mathbf{u}(t) \bigr)-\epsilon_{kl}\bigl(\mathbf {u}(t-s)\bigr) \bigr) \,\mathrm{d}x\, \mathrm{d}s \\& \qquad {}+\frac{1}{2} \int^{t}_{0} \int_{\Omega}\ddot{\mathcal {G}}_{ijkl}(s) \bigl( \epsilon_{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon _{ij}\bigl( \mathbf{u} (t-s)\bigr) \bigr) \bigl(\epsilon_{kl}\bigl(\mathbf{u}(t) \bigr)-\epsilon_{kl}\bigl(\mathbf {u}(t-s)\bigr) \bigr) \,\mathrm{d} x \,\mathrm{d}s. \end{aligned}$$
Substituting in (18)
$$\begin{aligned}& \int_{\Omega}\mathcal{S}\bigl(\mathbf{u}(t)\bigr)\cdot\nabla\dot{ \mathbf {u}}(t) \,\mathrm{d}x \\& \quad=-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int^{t}_{0} \int_{\Omega}\dot {\mathcal {G}}_{ijkl}(s) \bigl( \epsilon_{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon _{ij}\bigl( \mathbf{u} (t-s)\bigr) \bigr) \bigl(\epsilon_{kl}\bigl(\mathbf{u}(t) \bigr)-\epsilon_{kl}\bigl(\mathbf {u}(t-s)\bigr) \bigr) \,\mathrm{d} x \,\mathrm{d}s \\& \qquad {}+\frac{1}{2} \int^{t}_{0} \int_{\Omega}\ddot{\mathcal {G}}_{ijkl}(s) \bigl( \epsilon_{ij}\bigl(\mathbf{u}(t)\bigr)-\epsilon_{ij}\bigl( \mathbf{u}(t-s)\bigr) \bigr) \bigl(\epsilon _{kl}\bigl(\mathbf{u}(t) \bigr)-\epsilon_{kl}\bigl(\mathbf{u}(t-s)\bigr) \bigr) \,\mathrm {d}x \, \mathrm{d}s \\& \qquad {}+\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega}\mathcal {G}_{ijkl}(t)\epsilon_{ij} \bigl(\mathbf{u}(t)\bigr)\epsilon_{kl}\bigl(\mathbf {u}(t)\bigr) \, \mathrm{d}x \\& \qquad{}-\frac{1}{2} \int_{\Omega}\dot{\mathcal{G}}_{ijkl}(t)\epsilon _{ij}\bigl(\mathbf{u} (t)\bigr)\epsilon_{kl}\bigl( \mathbf{u}(t)\bigr) \,\mathrm{d}x. \end{aligned}$$
(20)
Substituting (16), (17) and (20) in (15), we obtain after summing
$$\begin{aligned}& \int_{\Omega} \bigl\vert \dot{\mathbf{m}}^{\varepsilon,N} \bigr\vert ^{2} \,\mathrm{d} x+\frac {a}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Omega} \bigl\vert \Lambda^{\alpha }\mathbf{m} ^{\varepsilon,N} \bigr\vert ^{2} \,\mathrm{d}x+\frac{1}{4\varepsilon} \frac {\mathrm{d} }{\mathrm{d} t} \int_{\Omega}\bigl( \bigl\vert \mathbf{m}^{\varepsilon,N} \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x \\& \quad{}+ \frac{\rho}{2}\frac{\mathrm{d}}{\mathrm {d}t} \int _{\Omega} \bigl\vert \dot {\mathbf{u}}^{\varepsilon,N} \bigr\vert ^{2} \,\mathrm{d}x + \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \int _{\Omega}\lambda_{ijkl}m^{\varepsilon,N}_{i}m^{\varepsilon ,N}_{j} \epsilon _{kl}\bigl(\mathbf{u}^{\varepsilon,N}\bigr) \,\mathrm{d}x \\& \quad{}- \frac {1}{2}\frac {\mathrm{d}}{\mathrm{d}t} \int^{t}_{0} \int_{\Omega}\dot{\mathcal {G}}_{ijkl}(s) \bigl( \epsilon_{ij}\bigl(\mathbf{u}^{\varepsilon,N}(t)\bigr)\\& \quad{}- \epsilon_{ij}\bigl(\mathbf{u} ^{\varepsilon ,N}(t-s)\bigr) \bigr) \bigl( \epsilon_{kl}\bigl(\mathbf{u}^{\varepsilon ,N}(t)\bigr)-\epsilon _{kl}\bigl(\mathbf{u}^{\varepsilon,N}(t-s)\bigr) \bigr) \,\mathrm{d}x \, \mathrm {d}s \\& \quad{}+ \frac{1}{2} \int^{t}_{0} \int_{\Omega}\ddot{\mathcal {G}}_{ijkl}(s) \bigl( \epsilon_{ij}\bigl(\mathbf{u}^{\varepsilon ,N}(t)\bigr)-\epsilon _{ij}\bigl(\mathbf{u}^{\varepsilon,N}(t-s)\bigr) \bigr) \bigl(\epsilon _{kl}\bigl(\mathbf{u} ^{\varepsilon ,N}(t)\bigr)-\epsilon_{kl} \bigl(\mathbf{u}^{\varepsilon,N}(t-s)\bigr) \bigr) \,\mathrm {d}x\, \mathrm{d}s \\& \quad{}+ \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int _{\Omega}\mathcal {G}_{ijkl}(t)\epsilon_{ij} \bigl(\mathbf{u}^{\varepsilon,N}(t)\bigr)\epsilon _{kl}\bigl(\mathbf{u} ^{\varepsilon,N}(t)\bigr) \,\mathrm{d}x\\& \quad{}-\frac{1}{2} \int_{\Omega}\dot {\mathcal {G}}_{ijkl}(t) \epsilon_{ij}\bigl(\mathbf{u}^{\varepsilon,N}(t)\bigr)\epsilon _{kl}\bigl(\mathbf{u} ^{\varepsilon,N}(t)\bigr) \,\mathrm{d}x=0. \end{aligned}$$
Now integrating in time
$$\begin{aligned}& \int_{Q} \bigl\vert \dot{\mathbf{m}}^{\varepsilon,N} \bigr\vert ^{2} \,\mathrm {d}x \,\mathrm{d} t+\frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha}\mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+ \frac{1}{4\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x \\& \qquad{}+ \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf {u}}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+ \frac{1}{2} \int_{\Omega }\lambda _{ijkl}m^{\varepsilon,N}_{i}m^{\varepsilon,N}_{j} \epsilon_{kl}\bigl(\mathbf{u} ^{\varepsilon,N}\bigr) (T) \,\mathrm{d}x \\& \qquad{}- \frac{1}{2} \int^{T}_{0} \int_{\Omega}\dot{\mathcal {G}}_{ijkl}(s) \bigl( \epsilon_{ij}\bigl(\mathbf{u}^{\varepsilon ,N}(T)\bigr)-\epsilon _{ij}\bigl(\mathbf{u}^{\varepsilon,N}(T-s)\bigr) \bigr) \bigl(\epsilon _{kl}\bigl(\mathbf{u} ^{\varepsilon ,N}(T)\bigr)\\& \qquad{}-\epsilon_{kl} \bigl(\mathbf{u}^{\varepsilon,N}(T-s)\bigr) \bigr)\, \mathrm {d}s \,\mathrm{d}x \\& \qquad{}+ \frac{1}{2} \int_{Q} \int^{t}_{0}\ddot{\mathcal {G}}_{ijkl}(s) \bigl(\epsilon_{ij}\bigl(\mathbf{u}^{\varepsilon,N}(t)\bigr)- \epsilon_{ij}\bigl(\mathbf{u} ^{\varepsilon ,N}(t-s)\bigr) \bigr) \bigl( \epsilon_{kl}\bigl(\mathbf{u}^{\varepsilon ,N}(t)\bigr)\\& \qquad{}-\epsilon _{kl}\bigl(\mathbf{u}^{\varepsilon,N}(t-s)\bigr) \bigr)\, \mathrm{d}s \, \mathrm {d}x \,\mathrm{d}t \\& \qquad{}+ \frac{1}{2} \int_{\Omega}\mathcal{G}_{ijkl}(T)\epsilon _{ij} \bigl(\mathbf{u}^{\varepsilon,N}(T)\bigr)\epsilon_{kl}\bigl(\mathbf {u}^{\varepsilon ,N}(T)\bigr) \,\mathrm{d}x\\& \qquad{}-\frac{1}{2} \int_{Q}\dot{\mathcal {G}}_{ijkl}(t)\epsilon _{ij}\bigl(\mathbf{u} ^{\varepsilon,N}(t)\bigr)\epsilon_{kl} \bigl(\mathbf{u}^{\varepsilon,N}(t)\bigr) \,\mathrm{d}x \,\mathrm{d}t \\& \quad= \frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha}\mathbf{m} ^{N}(0) \bigr\vert ^{2}\, \mathrm{d}x + \frac{1}{4\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf{m} ^{N}(0) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x + \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf{u}}^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x \\& \qquad{}+ \frac{1}{2} \int_{\Omega}\lambda _{ijkl}m^{N}_{i}m^{N}_{j} \epsilon_{kl}\bigl(\mathbf{u}^{N}\bigr) (0) \,\mathrm {d}x+ \frac{1}{2} \int_{\Omega}\mathcal{G}_{ijkl}(0)\epsilon_{ij} \bigl(\mathbf{u} ^{N}(0)\bigr)\epsilon_{kl}\bigl( \mathbf{u}^{N}(0)\bigr)\, \mathrm{d}x. \end{aligned}$$
Taking into account assumptions (\(H_{3}\)) and (\(H_{4}\))
$$\begin{aligned}& \int_{Q} \bigl\vert \dot{\mathbf{m}}^{\varepsilon,N} \bigr\vert ^{2} \,\mathrm {d}x \,\mathrm{d} t+\frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha}\mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+ \frac{1}{4\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x \\& \qquad{}+ \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf {u}}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+ \frac{1}{2} \int_{\Omega }\lambda _{ijkl}m^{\varepsilon,N}_{i}m^{\varepsilon,N}_{j} \epsilon_{kl}\bigl(\mathbf{u} ^{\varepsilon,N}\bigr) (T)\, \mathrm{d}x \\& \qquad{}+ \frac{1}{2} \int_{\Omega}\mathcal{G}_{ijkl}(T)\epsilon _{ij} \bigl(\mathbf{u}^{\varepsilon,N}(T)\bigr)\epsilon_{kl}\bigl(\mathbf {u}^{\varepsilon ,N}(T)\bigr) \,\mathrm{d}x \\& \quad\leqslant \frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha }\mathbf{m} ^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x + \frac{1}{4\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf{m} ^{N}(0) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x + \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf{u}}^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x \\& \qquad{}+ \frac{1}{2} \int_{\Omega}\lambda _{ijkl}m^{N}_{i}m^{N}_{j} \epsilon_{kl}\bigl(\mathbf{u}^{N}\bigr) (0) \,\mathrm {d}x+ \frac{1}{2} \int_{\Omega}\mathcal{G}_{ijkl}(0)\epsilon_{ij} \bigl(\mathbf{u} ^{N}(0)\bigr)\epsilon_{kl}\bigl( \mathbf{u}^{N}(0)\bigr) \,\mathrm{d}x. \end{aligned}$$
(21)
We call \(\mathcal{A}^{\varepsilon,N}(T)\) the left-hand side of (21) and \(\mathcal{A}^{N}(0)\) its right-hand side.
Now, for a positive parameter λ such that \(\frac {2\lambda}{9}>{\sup}_{ijkl}|\lambda_{ijkl}|\), we have by Young’s inequality, omitting superscripts,
$$\begin{aligned} \bigl\vert \lambda_{ijkl}m_{i}m_{j} \epsilon_{kl}(\mathbf {u}) \bigr\vert \leqslant& \frac {2\lambda}{9} |m_{i}|| m_{j}| \bigl\vert \epsilon_{kl}( \mathbf{u}) \bigr\vert \\ \leqslant & \frac{2\lambda}{9} \biggl(\frac{\lambda}{\beta }|m_{i}|^{2}|m_{j}|^{2} + \frac{\beta}{4\lambda} \bigl\vert \epsilon_{kl}(\mathbf{u}) \bigr\vert ^{2} \biggr), \end{aligned}$$
from where
$$\begin{aligned} \sum_{ijkl} \bigl\vert \lambda_{ijkl}m_{i}m_{j} \epsilon _{kl}(\mathbf{u}) \bigr\vert \leqslant & \frac{2\lambda}{9} \biggl(\frac{9\lambda }{\beta} \sum_{i} \vert m_{i} \vert ^{2} \sum_{j} \vert m_{j} \vert ^{2} + \frac{9\beta }{4\lambda}\sum_{kl} \bigl\vert \epsilon_{kl}(\mathbf{u}) \bigr\vert ^{2} \biggr) \\ =& 2\lambda \biggl(\frac{\lambda}{\beta} \biggl(\sum_{i} \vert m_{i} \vert ^{2} \biggr)^{2} + \frac{\beta}{4\lambda} \sum_{kl} \bigl\vert \epsilon _{kl}(\mathbf{u} ) \bigr\vert ^{2} \biggr) \\ =& \frac{2\lambda^{2}}{\beta} \vert \mathbf{m} \vert ^{4} + \frac{\beta}{2} \sum_{kl} \bigl\vert \epsilon_{kl}(\mathbf{u}) \bigr\vert ^{2}. \end{aligned}$$
Therefore, following the idea introduced in [7], we have
$$\begin{aligned}& \frac{1}{2} \biggl\vert \int_{\Omega}\lambda_{ijkl}m_{i}m_{j} \epsilon _{kl}(\mathbf{u})\,\mathrm{d}x \biggr\vert \\& \quad= \frac{1}{2} \biggl\vert \int_{\Omega} \sum_{ijkl} \lambda_{ijkl}m_{i}m_{j} \epsilon_{kl}(\mathbf{u})\,\mathrm{d}x \biggr\vert \\& \quad\leqslant \frac{1}{2} \int_{\Omega}\sum_{ijkl} \bigl\vert \lambda _{ijkl}m_{i}m_{j}\epsilon_{kl}( \mathbf{u}) \bigr\vert \,\mathrm{d}x \\& \quad\leqslant \frac{\lambda^{2}}{\beta} \int_{\Omega}|\mathbf{m}|^{4} \,\mathrm{d}x + \frac {\beta}{4} \int_{\Omega}\sum_{kl} \bigl\vert \epsilon_{kl}(\mathbf{u} ) \bigr\vert ^{2}\,\mathrm{d} x \\& \quad= \frac{\lambda^{2}}{\beta} \int_{\Omega} \bigl(|\mathbf {m}|^{2}-1+1 \bigr)^{2} \,\mathrm{d}x + \frac{\beta}{4} \int_{\Omega}\sum_{kl} \bigl\vert \epsilon _{kl}(\mathbf{u} ) \bigr\vert ^{2}\,\mathrm{d}x \\& \quad\leqslant \frac{2\lambda^{2}}{\beta} \int_{\Omega} \bigl(|\mathbf{m} |^{2}-1 \bigr)^{2} \,\mathrm{d}x+\frac{2\lambda^{2}}{\beta}\operatorname{vol}(\Omega) + \frac{\beta}{4} \int _{\Omega}\sum_{kl} \bigl\vert \epsilon_{kl}(\mathbf{u}) \bigr\vert ^{2}\,\mathrm {d}x \\& \quad\leqslant \frac{2\lambda^{2}}{\beta} \int_{\Omega} \bigl(|\mathbf{m} |^{2}-1 \bigr)^{2} \,\mathrm{d}x+\frac{2\lambda^{2}}{\beta}\operatorname{vol}(\Omega) + \frac{1}{4} \int _{\Omega }\mathcal{G}_{ijkl}\epsilon_{ij}( \mathbf{u})\epsilon_{kl}(\mathbf {u})\,\mathrm{d}x \end{aligned}$$
by using (\(H_{5}\)). Now, for \(\varepsilon<\frac{\beta}{16\lambda^{2}}\), we have
$$\frac{1}{2} \biggl\vert \int_{\Omega}\lambda_{ijkl}m_{i}m_{j} \epsilon _{kl}(\mathbf{u})\,\mathrm{d}x \biggr\vert \leqslant \frac{1}{8\varepsilon} \int _{\Omega } \bigl(|\mathbf{m} |^{2}-1 \bigr)^{2} \,\mathrm{d}x+\frac{2\lambda^{2}}{\beta}\operatorname{vol}(\Omega) + \frac{1}{4} \int_{\Omega}\mathcal{G}_{ijkl}\epsilon_{ij}( \mathbf{u})\epsilon _{kl}(\mathbf{u} )\,\mathrm{d}x, $$
which implies
$$\begin{aligned} \frac{1}{2} \int_{\Omega}\lambda _{ijkl}m^{N}_{i}m^{N}_{j} \epsilon_{kl}\bigl(\mathbf{u}^{N}\bigr) (0)\,\mathrm{d}x \leq & \frac {1}{8\varepsilon} \int_{\Omega} \bigl( \bigl\vert \mathbf{m}^{N}(0) \bigr\vert ^{2}-1 \bigr)^{2} \,\mathrm{d} x \\ &{}+ \frac{2\lambda^{2}}{\beta}\operatorname{vol}(\Omega) + \frac{1}{4} \int_{\Omega}\mathcal{G}_{ijkl}\epsilon_{ij} \bigl(\mathbf {u}^{N}\bigr)\epsilon _{kl}\bigl(\mathbf{u} ^{N}\bigr) (0)\,\mathrm{d}x, \end{aligned}$$
and
$$\begin{aligned}& -\frac{1}{8\varepsilon} \int_{\Omega} \bigl(\bigl|\mathbf {m}^{\varepsilon ,N}(T)\bigr|^{2}-1 \bigr)^{2} \,\mathrm{d}x-\frac{2\lambda^{2}}{\beta}\operatorname{vol}(\Omega) \\& \qquad{}- \frac{1}{4} \int_{\Omega}\mathcal {G}_{ijkl}\epsilon _{ij} \bigl(\mathbf{u}^{\varepsilon,N}\bigr)\epsilon_{kl}\bigl(\mathbf {u}^{\varepsilon ,N}\bigr) (T)\,\mathrm{d}x \\& \quad\leq \frac{1}{2} \int_{\Omega}\lambda _{ijkl}m^{\varepsilon ,N}_{i}m^{\varepsilon,N}_{j} \epsilon_{kl}\bigl(\mathbf{u}^{\varepsilon ,N}\bigr) (T)\,\mathrm{d}x. \end{aligned}$$
According to the definition of \(\mathcal{A}^{\varepsilon,N}(T)\) and \(\mathcal{A}^{N}(0)\), we can write
$$\begin{aligned}& \int_{Q} \bigl\vert \dot{\mathbf{m}}^{\varepsilon,N} \bigr\vert ^{2} \,\mathrm {d}x \,\mathrm{d} t+\frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha}\mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+ \frac{1}{8\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x \\& \quad{}+ \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf {u}}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+\frac{1}{4} \int_{\Omega}\mathcal {G}_{ijkl}\epsilon _{ij} \bigl(\mathbf{u}^{\varepsilon,N}\bigr)\epsilon_{kl}\bigl(\mathbf {u}^{\varepsilon,N}\bigr) (T)\, \mathrm{d} x-\frac{2\lambda^{2}}{\beta}\operatorname{vol}( \Omega)\leq\mathcal{A}^{\varepsilon,N}(T), \end{aligned}$$
and
$$\begin{aligned} \mathcal{A}^{N}(0) \leq & \frac{a}{2} \int_{\Omega } \bigl\vert \Lambda ^{\alpha} \mathbf{m}^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x + \frac{3}{8\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf{m} ^{N}(0) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x + \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf{u}}^{N}(0) \bigr\vert ^{2}\, \mathrm{d}x \\ &{}+ \frac{3}{4} \int_{\Omega }\mathcal {G}_{ijkl}\epsilon_{ij} \bigl(\mathbf{u}^{N}\bigr)\epsilon_{kl}\bigl(\mathbf {u}^{N}\bigr) (0)\, \mathrm{d}x +\frac{2\lambda^{2}}{\beta}\operatorname{vol}( \Omega). \end{aligned}$$
Since \(\mathcal{A}^{\varepsilon,N}(T)\leqslant\mathcal{A}^{N}(0)\), we have
$$\begin{aligned}& \int_{Q} \bigl\vert \dot{\mathbf{m}}^{\varepsilon,N} \bigr\vert ^{2} \,\mathrm {d}x \,\mathrm{d} t+\frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha}\mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+ \frac{1}{8\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x \\& \qquad{}+ \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf {u}}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+\frac{1}{4} \int_{\Omega}\mathcal {G}_{ijkl}\epsilon _{ij} \bigl(\mathbf{u}^{\varepsilon,N}\bigr)\epsilon_{kl}\bigl(\mathbf {u}^{\varepsilon,N}\bigr) (T) \,\mathrm{d}x \\& \quad\leqslant \frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha }\mathbf{m} ^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x + \frac{3}{8\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf{m} ^{N}(0) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x + \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf{u}}^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x \\& \qquad{}+ \frac{3}{4} \int_{\Omega}\mathcal {G}_{ijkl}\epsilon_{ij} \bigl(\mathbf{u}^{N}\bigr)\epsilon_{kl}\bigl(\mathbf {u}^{N}\bigr) (0) \,\mathrm{d}x +\frac{4\lambda^{2}}{\beta}\operatorname{vol}( \Omega). \end{aligned}$$
Moreover, \(\int_{\Omega}|\nabla\mathbf{u}^{\varepsilon ,N}(T)|^{2} \,\mathrm{d}x\leqslant\int_{\Omega}\sum_{kl}|\epsilon _{kl}(\mathbf{u} ^{\varepsilon,N}(T))|^{2}\,\mathrm{d}x\) and under assumptions (\(H_{5}\)) and (\(H_{6}\)) we have
$$\begin{aligned}& \int_{Q} \bigl\vert \dot{\mathbf{m}}^{\varepsilon,N} \bigr\vert ^{2} \,\mathrm {d}x \,\mathrm{d} t+\frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha}\mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+ \frac{1}{8\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf {m}^{\varepsilon ,N}(T) \bigr\vert ^{2}-1\bigr)^{2}\, \mathrm{d}x \\& \qquad{}+ \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf {u}}^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x+\frac{\beta}{4} \int_{\Omega} \bigl\vert \nabla\mathbf{u} ^{\varepsilon ,N}(T) \bigr\vert ^{2} \,\mathrm{d}x \\& \quad\leqslant \frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha}\mathbf{m} ^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x + \frac{3}{8\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf{m} ^{N}(0) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x + \frac{\rho}{2} \int_{\Omega} \bigl\vert \dot{\mathbf{u}}^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x \\& \qquad{}+ \frac{3\tau}{4} \int_{\Omega} \bigl\vert \nabla\mathbf{u} ^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x +\frac{4\lambda^{2}}{\beta}\operatorname{vol}( \Omega). \end{aligned}$$
(22)
Since \(\mathbf{u}_{0}\in\mathbf{H}^{1}_{0}(\Omega)\), \(\mathbf{u}_{1}\in \mathbf{L}^{2}(\Omega)\) and \(\mathbf{m}_{0}\in \mathbf{H}^{\alpha}(\Omega)\), which is embedded into \(\mathbf {L}^{4}(\Omega)\) for \(1<\alpha<\frac{3}{2}\), the right-hand side is uniformly bounded. Indeed, for constants \(C_{1}\), \(C_{2}\), \(C_{3}\) and \(C_{4}\) independent of N,
$$\begin{aligned} \int_{\Omega}\bigl( \bigl\vert \mathbf{m}^{N}(0) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x =& \int_{\Omega } \bigl\vert \mathbf{m} ^{N}(0) \bigr\vert ^{4} \,\mathrm{d}x-2 \int_{\Omega} \bigl\vert \mathbf{m}^{N}(0) \bigr\vert ^{2} \,\mathrm {d}x+\operatorname{vol}(\Omega) \\ \leqslant& \bigl\Vert \mathbf{m}^{N}(0) \bigr\Vert ^{4}_{\mathbf{L}^{4}(\Omega)}+ \operatorname{vol}(\Omega ) \\ \leqslant & C_{1} \bigl\Vert \mathbf{m}^{N}(0) \bigr\Vert ^{4}_{\mathbf{H}^{\alpha}(\Omega )}+C_{2} \\ \leqslant & C_{3} \end{aligned}$$
and
$$\begin{aligned} \int_{\Omega} \bigl\vert \nabla\mathbf{u}^{N}(0) \bigr\vert ^{2} \,\mathrm{d}x =& \int _{\Omega} \bigl\vert \nabla \mathbf{u} ^{N}(0)- \nabla\mathbf{u}_{0}+\nabla\mathbf{u}_{0} \bigr\vert ^{2} \,\mathrm{d}x \\ \leqslant & 2 \int_{\Omega} \bigl\vert \nabla\mathbf{u}^{N}(0)- \nabla\mathbf {u}_{0} \bigr\vert ^{2} \,\mathrm{d} x+2 \int _{\Omega}|\nabla\mathbf{u}_{0}|^{2} \, \mathrm{d}x \\ \leqslant & 2 \bigl\Vert \mathbf{u}^{N}(0)-\mathbf{u}_{0} \bigr\Vert ^{2}_{\mathbf {H}^{1}_{0}(\Omega)}+2 \Vert \mathbf{u} _{0} \Vert ^{2}_{\mathbf{H} ^{1}_{0}(\Omega)} \\ \leqslant & C_{4}, \end{aligned}$$
thanks to the strong convergences \(\mathbf{m}^{N}(\cdot,0)\rightarrow \mathbf{m}_{0}\) in \(\mathbf{H} ^{\alpha}(\Omega)\) and \(\mathbf{u}^{N}(\cdot,0)\rightarrow\mathbf{u}_{0}\) in \(\mathbf{H} ^{1}_{0}(\Omega)\). For the other term (\(\dot{\mathbf{u}}^{N}(0)\)), the estimate can be carried out in an analogous way using the strong convergence \(\dot{\mathbf{u} }^{N}(\cdot,0)\rightarrow\mathbf{u}_{1}\) in \(\mathbf{L}^{2}(\Omega)\). Moreover, noting that (for a constant C independent of ε and N)
$$\begin{aligned}& \int_{\Omega} \bigl\vert \mathbf{m}^{\varepsilon,N} \bigr\vert ^{2} \,\mathrm{d}x= \int _{\Omega}\bigl( \bigl\vert \mathbf{m} ^{\varepsilon,N} \bigr\vert ^{2}-1+1\bigr) \,\mathrm{d}x \\& \quad \leqslant\frac{1}{2} \int_{\Omega}\bigl( \bigl\vert \mathbf{m}^{\varepsilon ,N} \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x+C. \end{aligned}$$
Therefore, for fixed \(\varepsilon>0\), we have
$$ \begin{aligned} & \bigl(\mathbf{m}^{\varepsilon,N} \bigr)_{N} \mbox{ is bounded in } L^{\infty }\bigl(0,T;\mathbf{H} ^{\alpha}(\Omega)\bigr), \\ & \bigl(\dot{\mathbf{m}}^{\varepsilon,N}\bigr)_{N} \mbox{ is bounded in } L^{2}\bigl(0,T;\mathbf{L} ^{2}(\Omega)\bigr), \\ & \bigl( \bigl\vert \mathbf{m}^{\varepsilon,N} \bigr\vert ^{2}-1 \bigr)_{N} \mbox{ is bounded in } L^{\infty }\bigl(0,T;L^{2}( \Omega)\bigr), \\ & \bigl(\mathbf{u}^{\varepsilon,N}\bigr)_{N} \mbox{ is bounded in } L^{2}\bigl(0,T;\mathbf{H} ^{1}_{0}(\Omega)\bigr), \\ & \bigl(\dot{\mathbf{u}}^{\varepsilon,N}\bigr)_{N} \mbox{ is bounded in } L^{2}\bigl(0,T;\mathbf{L} ^{2}(\Omega)\bigr). \end{aligned} $$
(23)
Note that (23) is due to the Poincaré lemma. Now, from classical compactness results, there exist two subsequences which we still denote by \((\mathbf{m}^{\varepsilon,N})\) and \((\mathbf {u}^{\varepsilon,N})\) such that for fixed \(\varepsilon>0\)
$$ \begin{aligned} & \mathbf{m}^{\varepsilon,N} \rightharpoonup\mathbf{m}^{\varepsilon} \mbox{ weakly in } L^{2} \bigl(0,T;\mathbf{H}^{\alpha}(\Omega)\bigr), \\ & \dot{\mathbf{m} }^{\varepsilon,N}\rightharpoonup\dot{ \mathbf{m}}^{\varepsilon} \mbox{ weakly in } \mathbf{L}^{2}(Q), \\ & \mathbf{m}^{\varepsilon,N}\rightarrow\mathbf{m}^{\varepsilon} \mbox{ strongly in } L^{2}\bigl(0,T,\mathbf{H}^{\beta}(\Omega)\bigr) \mbox{ and a.e. for } 0\leq\beta< \alpha, \\ & \bigl\vert \mathbf{m} ^{\varepsilon,N} \bigr\vert ^{2}-1 \rightharpoonup\zeta \mbox{ weakly in } L^{2}(Q), \\ & \mathbf{u}^{\varepsilon,N}\rightharpoonup\mathbf{u}^{\varepsilon } \mbox{ weakly in } L^{2}\bigl(0,T;\mathbf{H}^{1}_{0}(\Omega) \bigr), \\ &\dot{\mathbf{u}}^{\varepsilon ,N}\rightharpoonup \dot{\mathbf{u}}^{\varepsilon} \mbox{ weakly in } \mathbf{L}^{2}(Q), \\ & \mathbf{u}^{\varepsilon,N}\rightarrow\mathbf{u} ^{\varepsilon} \mbox{ strongly in } \mathbf{L}^{2}(Q). \end{aligned} $$
(24)
Convergence (24) is due to the following lemma (the proof can be found in [14], p.57).
Lemma 4.2
Assume
\(A, B\)
and
C
are three Banach spaces and satisfy
\(A\subset B \subset C\)
where the injections are continuous with compact embedding
\(A\hookrightarrow B\)
and
A, C
are reflexive. Denote
$$D:= \biggl\{ v \Big| v\in L^{p_{0}}(0,T;A), \dot{v}=\frac{dv}{dt}\in L^{p_{1}}(0,T;C) \biggr\} , $$
where
T
is finite and
\(1 < p_{i} < \infty\), \(i=0, 1\). Then
D, equipped with the norm
$$\Vert v\Vert _{ L^{p_{0}}(0,T;A)}+\Vert \dot{v}\Vert _{ L^{p_{1}}(0,T;C)}, $$
is a Banach space and the embedding
\(D \hookrightarrow L^{p_{0}}(0,T;B)\)
is compact.
Another lemma (Lemma 4.3) whose proof can be found in [14], p.12 will ensure that \(\zeta=|\mathbf{m}^{\varepsilon}|^{2}-1\).
Lemma 4.3
Let Θ be a bounded open set of
\(\mathbb{R}^{d}_{x}\times\mathbb {R}_{t}\), \(h_{n}\)
and
h
in
\(L^{q}(\Theta), 1< q<\infty\)
such that
\(\Vert h_{n}\Vert _{L^{q}(\Theta)}\leq C, h_{n}\rightarrow h\)
a.e. in Θ, then
\(h_{n}\rightharpoonup h\)
weakly in
\(L^{q}(\Theta)\).
Now, since \(1<\alpha<\frac{3}{2}\) and from the Sobolev embedding \(H^{\alpha}(Q)\hookrightarrow L^{4}(Q)\), further compactness result follows
$$ m^{\varepsilon,N}_{i} m^{\varepsilon,N}_{j} \rightarrow m_{i}^{\varepsilon }m_{j}^{\varepsilon} \mbox{ strongly in } L^{2}(Q) $$
(25)
and
$$m^{\varepsilon,N}_{i} \phi_{j}\rightarrow m_{i}^{\varepsilon}\phi_{j} \mbox{ strongly in } L^{2}(Q). $$
The above estimates allow us to pass to the limit as N goes to infinity and to get the desired result. Indeed consider the variational formulation of (14)
$$ \textstyle\begin{cases} \int_{Q}\dot{\mathbf{m}}^{\varepsilon,N}\cdot \boldsymbol {\phi} \,\mathrm{d}x\,\mathrm{d}t+a \int_{Q} \Lambda^{\alpha}\mathbf {m}^{\varepsilon,N}\cdot \Lambda ^{\alpha}\boldsymbol {\phi} \,\mathrm{d}x\,\mathrm{d}t\\ \quad{}+ \int_{Q}\lambda_{ijkl}m^{\varepsilon,N}_{j}\epsilon _{kl}(\mathbf{u} ^{\varepsilon,N})\boldsymbol {\phi}_{i} \,\mathrm{d}x\,\mathrm{d}t+\int _{Q}\frac {|\mathbf{m} ^{\varepsilon,N}|^{2}-1}{\varepsilon}\mathbf{m}^{\varepsilon,N}\cdot \boldsymbol {\phi} \,\mathrm{d}x\,\mathrm{d}t=0,\\ -\rho \int_{Q}\dot{\mathbf{u}}^{\varepsilon,N}\cdot \dot{\boldsymbol {\psi}} \,\mathrm{d}x\,\mathrm{d}t+\int_{Q}\mathcal{G}_{ijkl}(0)\epsilon _{ij}(\mathbf{u} ^{\varepsilon,N}(t))\epsilon_{kl}(\boldsymbol {\psi}(t)) \,\mathrm{d} x\,\mathrm{d} t\\ \quad{}+ \int_{Q}\int^{t}_{0}\mathcal{G}_{ijkl}(t-s)\epsilon _{ij}(\mathbf{u}^{\varepsilon,N}(s))\epsilon_{kl}(\boldsymbol {\psi}(t)) \,\mathrm{d} s \,\mathrm{d}x\,\mathrm{d}t\\ \quad{}+ \frac{1}{2} \int_{Q}\lambda _{ijkl}m^{\varepsilon,N}_{i}m^{\varepsilon,N}_{j}\epsilon _{kl}(\boldsymbol {\psi}) \,\mathrm{d}x\,\mathrm{d}t=0 \end{cases} $$
(26)
for any \(\boldsymbol {\phi}\in L^{2}(0,T;\mathbf{H}^{\alpha}(\Omega ))\) and \(\boldsymbol {\psi}\in\mathbf{H}^{1}_{0}(Q)\). Taking \(N\rightarrow \infty\) in (26), we find
$$ \textstyle\begin{cases} \int_{Q}\dot{\mathbf{m}}^{\varepsilon}\cdot \boldsymbol {\phi} \,\mathrm{d}x\,\mathrm{d}t+a \int_{Q} \Lambda^{\alpha}\mathbf {m}^{\varepsilon}\cdot \Lambda ^{\alpha}\boldsymbol {\phi} \,\mathrm{d}x\,\mathrm{d}t\\ \quad{}+ \int_{Q}\lambda_{ijkl}m^{\varepsilon}_{j}\epsilon _{kl}(\mathbf{u} ^{\varepsilon})\boldsymbol {\phi}_{i} \,\mathrm{d}x\,\mathrm{d}t+\int _{Q}\frac {|\mathbf{m} ^{\varepsilon}|^{2}-1}{\varepsilon}\mathbf{m}^{\varepsilon}\cdot \boldsymbol {\phi} \,\mathrm{d}x\,\mathrm{d}t=0,\\ -\rho \int_{Q}\dot{\mathbf{u}}^{\varepsilon}\cdot \dot{\boldsymbol {\psi}} \,\mathrm{d}x\,\mathrm{d}t+\int_{Q}\mathcal{G}_{ijkl}(0)\epsilon _{ij}(\mathbf{u} ^{\varepsilon}(t))\epsilon_{kl}(\boldsymbol {\psi}(t))\, \mathrm {d}x\,\mathrm{d} t\\ \quad{}+ \int_{Q}\int^{t}_{0}\mathcal{G}_{ijkl}(t-s)\epsilon _{ij}(\mathbf{u}^{\varepsilon}(s))\epsilon_{kl}(\boldsymbol {\psi}(t)) \,\mathrm{d} s\, \mathrm{d}x\,\mathrm{d}t\\ \quad{}+ \frac{1}{2} \int_{Q}\lambda _{ijkl}m^{\varepsilon}_{i}m^{\varepsilon}_{j}\epsilon_{kl}(\boldsymbol {\psi }) \,\mathrm{d}x\,\mathrm{d}t=0 \end{cases} $$
(27)
for any \(\boldsymbol {\phi}\in L^{2}(0,T;\mathbf{H}^{\alpha}(\Omega)) \) and \(\boldsymbol {\psi}\in\mathbf{H}^{1}_{0}(Q)\). We proved the following result.
Proposition 4.1
Given
\(\mathbf{m}_{0} \in\mathbf{H}^{\alpha}{(\Omega)}\)
such that
\(|\mathbf{m}_{0}|=1\)
a.e., \(\mathbf{u}_{0} \in\mathbf{H}^{1}_{0}(\Omega)\)
and
\(\mathbf{u}_{1} \in\mathbf{L}^{2}(\Omega)\). Then there exists a solution
\((\mathbf{m}^{\varepsilon},\mathbf{u}^{\varepsilon})\), for any positive
ε
small enough, to problem (11) in the sense of distributions. Moreover, we have the following energy estimate:
$$\begin{aligned}& \int_{Q} \bigl\vert \dot{\mathbf{m}}^{\varepsilon} \bigr\vert ^{2} \,\mathrm{d} x\,\mathrm{d} t+\frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha}\mathbf {m}^{\varepsilon }(T) \bigr\vert ^{2} \,\mathrm{d}x+ \frac{1}{8\varepsilon} \int_{\Omega}\bigl( \bigl\vert \mathbf {m}^{\varepsilon}(T) \bigr\vert ^{2}-1\bigr)^{2} \,\mathrm{d}x \\ & \qquad{}+ \frac{\rho}{2} \int_{\Omega}\bigl|\dot{\mathbf {u}}^{\varepsilon }(T)\bigr|^{2} \, \mathrm{d}x+\frac{\beta}{4} \int_{\Omega}\bigl|\nabla\mathbf {u}^{\varepsilon}(T)\bigr|^{2} \, \mathrm{d}x \\ & \quad\leqslant \frac{a}{2} \int_{\Omega} \bigl\vert \Lambda^{\alpha}\mathbf {m}_{0} \bigr|^{2} \,\mathrm{d}x +\frac{\rho}{2} \int_{\Omega} \vert \mathbf{u}_{1}|^{2} \,\mathrm{d}x + \frac{3\tau}{4} \int_{\Omega} \vert \nabla \mathbf{u}_{0} \vert ^{2} \,\mathrm{d}x +\frac{4\lambda^{2}}{\beta}\operatorname{vol}(\Omega). \end{aligned}$$
(28)
Remark 2
We can deduce (28) by taking the lower semicontinuous limit in (22).
4.2 Convergence of approximate solutions
The limit process as \(\varepsilon\rightarrow0\) makes use also of some convergence results. For this, we will use estimate (28), from which we have
$$\begin{aligned}& \bigl(\mathbf{m}^{\varepsilon}\bigr)_{\varepsilon} \mbox{ is bounded in } L^{\infty}\bigl(0,T;\mathbf{H}^{\alpha}(\Omega)\bigr), \\ & \bigl(\dot{\mathbf{m}}^{\varepsilon}\bigr)_{\varepsilon} \mbox{ is bounded in } L^{2}\bigl(0,T;\mathbf{L}^{2}(\Omega)\bigr), \\ & \bigl( \bigl\vert \mathbf{m}^{\varepsilon} \bigr\vert ^{2}-1 \bigr)_{\varepsilon} \mbox{ is bounded in } L^{\infty}\bigl(0,T;L^{2}( \Omega)\bigr), \\ & \bigl(\mathbf{u}^{\varepsilon}\bigr)_{\varepsilon} \mbox{ is bounded in } L^{2}\bigl(0,T;\mathbf{H}^{1}_{0}(\Omega)\bigr), \\ & \bigl(\dot{\mathbf{u}}^{\varepsilon}\bigr)_{\varepsilon} \mbox{ is bounded in } L^{2}\bigl(0,T;\mathbf{L}^{2}(\Omega)\bigr). \end{aligned}$$
Then there exist two subsequences, which we still denote by \((\mathbf{m} ^{\varepsilon})\) and \((\mathbf{u}^{\varepsilon})\), such that
$$ \begin{aligned} & \mathbf{m}^{\varepsilon }\rightharpoonup\mathbf{m} \mbox{ weakly in } L^{2}\bigl(0,T;\mathbf{H} ^{\alpha}(\Omega) \bigr), \\ & \dot{\mathbf{m} }^{\varepsilon}\rightharpoonup\dot{\mathbf{m}} \mbox{ weakly in } L^{2}\bigl(0,T;\mathbf{L}^{2}(\Omega)\bigr), \\ & \mathbf{m}^{\varepsilon }\rightarrow\mathbf{m} \mbox{ strongly in } L^{2}\bigl(0,T,\mathbf{H} ^{\beta}(\Omega)\bigr) \mbox{ and a.e. for } 0\leqslant\beta < \alpha, \\ & \bigl\vert \mathbf{m} ^{\varepsilon} \bigr\vert ^{2}-1 \rightarrow0 \mbox{ strongly in } L^{2}(Q) \mbox{ and a.e.}, \\ & \mathbf{u}^{\varepsilon}\rightharpoonup\mathbf{u} \mbox{ weakly in } L^{2}\bigl(0,T;\mathbf{H}^{1}_{0}(\Omega)\bigr), \\ & \dot{\mathbf{u}}^{\varepsilon}\rightharpoonup\dot{\mathbf{u}} \mbox{ weakly in } \mathbf{L}^{2}(Q), \\ & \mathbf{u}^{\varepsilon}\rightarrow\mathbf{u} \mbox{ strongly in } \mathbf{L}^{2}(Q). \end{aligned} $$
(29)
It can be shown from convergence (29) that \(|\mathbf {m}|=1\) a.e.
Now, in order to pass to the limit \(\varepsilon\rightarrow0\) in (27), let \(\boldsymbol {\phi}=\mathbf{m}^{\varepsilon}\times\boldsymbol {\varphi}\), where \(\boldsymbol {\varphi} \in\mathbf{C}^{\infty}(\overline{Q})\). As ϕ is in \(L^{2}(0,T;\mathbf{H}^{\alpha}(\Omega))\), there holds
$$ \textstyle\begin{cases} \int_{Q}\dot{\mathbf{m}}^{\varepsilon}\cdot(\mathbf {m}^{\varepsilon }\times \boldsymbol {\varphi}) \,\mathrm{d}x\,\mathrm{d}t+a \int_{Q} \Lambda ^{\alpha }\mathbf{m} ^{\varepsilon}\cdot\Lambda^{\alpha}(\mathbf{m}^{\varepsilon }\times \boldsymbol {\varphi}) \,\mathrm{d}x\,\mathrm{d}t\\ \quad{}+ \int_{Q}\lambda_{ijkl}m^{\varepsilon}_{j}\epsilon _{kl}(\mathbf{u} ^{\varepsilon})(\mathbf{m}^{\varepsilon}\times\boldsymbol {\varphi})_{i} \,\mathrm{d} x\,\mathrm{d}t=0,\\ -\rho \int_{Q}\dot{\mathbf{u}}^{\varepsilon}\cdot \dot{\boldsymbol {\psi}} \,\mathrm{d}x\,\mathrm{d}t+\int_{Q}\mathcal{G}_{ijkl}(0)\epsilon _{ij}(\mathbf{u} ^{\varepsilon}(t))\epsilon_{kl}(\boldsymbol {\psi}(t)) \,\mathrm {d}x\,\mathrm{d} t\\ \quad{}+ \int_{Q}\int^{t}_{0}\mathcal{G}_{ijkl}(t-s)\epsilon _{ij}(\mathbf{u}^{\varepsilon}(s))\epsilon_{kl}(\boldsymbol {\psi}(t)) \,\mathrm{d} s \,\mathrm{d}x\,\mathrm{d}t\\ \quad{}+ \frac{1}{2} \int_{Q}\lambda _{ijkl}m^{\varepsilon}_{i}m^{\varepsilon}_{j}\epsilon_{kl}(\boldsymbol {\psi}) \,\mathrm{d}x\,\mathrm{d}t=0. \end{cases} $$
(30)
In (30) we can easily pass to the limit \(\varepsilon \rightarrow 0\) (with the exception of the terms where there is the fractional Laplacian) thanks to recent convergences that we have set and a result like the one in (25).
Now we consider the convergence of the second term of the first equation. This is by no means obvious since we encounter the fractional order derivatives; for this reason, the classical methods are not applied anymore. However, commutator estimates (Lemma 4.4 (see [15–17] for a proof)) provide us with proper tools, to which the success in the following owes a lot.
Lemma 4.4
Commutator estimates
Suppose that
\(s>0\)
and
\(p\in(1,+\infty)\). If
\(f, g \in\mathcal{S}\) (the Schwartz class), then
$$ \bigl\Vert \Lambda^{s}(fg)-f\Lambda^{s}g \bigr\Vert _{L^{p}}\leq C \bigl( \Vert \nabla f \Vert _{L^{p_{1}}} \Vert g \Vert _{\dot{W}^{s-1,p_{2}}}+ \Vert f \Vert _{\dot{W}^{s,p_{3}}}\|g \|_{L^{p_{4}}} \bigr) $$
(31)
and
$$ \bigl\Vert \Lambda^{s}(fg) \bigr\Vert _{L^{p}}\leq C \bigl( \Vert f \Vert _{L^{p_{1}}}\|g \|_{\dot {W}^{s,p_{2}}}+\|f\|_{\dot{W}^{s,p_{3}}}\|g\|_{L^{p_{4}}} \bigr) $$
(32)
with
\(p_{2}, p_{3} \in(1,+\infty)\)
such that
\(\frac{1}{p}=\frac {1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}\).
We start firstly by showing that \(\Lambda^{\alpha}(\mathbf {m}^{\varepsilon }\times\boldsymbol {\varphi})\in\mathbf{L}^{2}(Q)\) (then we comment that the second term in (9) makes sense), indeed applying the multiplicative estimates (32) in Lemma 4.4 to \(\mathbf{m} ^{\varepsilon}\) and φ (for \(s=\alpha, p=2, p_{1}=\frac{6}{3-2\alpha},p_{2}=\frac{3}{\alpha}, p_{3}=2\) and \(p_{4}=\infty\)), we find for a constant C independent of ε
$$\begin{aligned} \bigl\Vert \Lambda^{\alpha}\bigl(\mathbf{m}^{\varepsilon} \times\boldsymbol {\varphi}\bigr) \bigr\Vert _{\mathbf{L} ^{2}(\Omega)} \leqslant & C \bigl( \bigl\Vert \mathbf{m}^{\varepsilon} \bigr\Vert _{\mathbf{L} ^{p_{1}}(\Omega)}\|\boldsymbol { \varphi}\|_{\dot{\mathbf{W}}^{\alpha,p_{2}}(\Omega)}+ \bigl\Vert \mathbf{m} ^{\varepsilon} \bigr\Vert _{\dot{\mathbf{H}}^{\alpha}(\Omega)} \Vert \boldsymbol {\varphi} \Vert _{\mathbf{L}^{\infty}(\Omega)} \bigr) \\ =& C \bigl( \bigl\Vert \mathbf{m}^{\varepsilon} \bigr\Vert _{\mathbf{L}^{p_{1}}(\Omega)} \bigl\Vert \Lambda ^{\alpha}\boldsymbol {\varphi} \bigr\Vert _{\mathbf{L}^{p_{2}}(\Omega)}+ \bigl\Vert \Lambda^{\alpha}\mathbf{m} ^{\varepsilon} \bigr\Vert _{\mathbf{L}^{2}(\Omega)} \Vert \boldsymbol {\varphi} \Vert _{\mathbf{L} ^{\infty }(\Omega)} \bigr). \end{aligned}$$
Here is another lemma (see [3] for a detailed proof and for more details on fractional calculus).
Lemma 4.5
Suppose that
\(p>q>1\)
and
\(\frac{1}{p}+ \frac{s}{d}= \frac{1}{q}\). Assume that
\(\Lambda^{s}f\in L^{q}\), then
\(f\in L^{p}\)
and there is a constant
\(C>0\)
such that
$$\Vert f \Vert _{ L^{p}}\leq C \bigl\Vert \Lambda^{s}f \bigr\Vert _{ L^{q}}. $$
In Lemma 4.5, we take \(f=\mathbf{m}^{\varepsilon}, q=2, s=\alpha , p=p_{1}\), then \(\|\mathbf{m}^{\varepsilon}\|_{ L^{p_{1}}(\Omega)}\leq C_{1}\|\Lambda ^{\alpha }\mathbf{m}^{\varepsilon}\|_{ L^{2}(\Omega)}\). Therefore
$$\begin{aligned} \bigl\Vert \Lambda^{\alpha}\bigl(\mathbf{m}^{\varepsilon} \times\boldsymbol {\varphi}\bigr) \bigr\Vert _{\mathbf{L} ^{2}(\Omega)} \leqslant & C \bigl(C_{1} \bigl\Vert \Lambda^{\alpha}\mathbf {m}^{\varepsilon}\bigr\Vert _{ \mathbf{L} ^{2}(\Omega)} \bigl\Vert \Lambda^{\alpha}\boldsymbol {\varphi} \bigr\Vert _{\mathbf{L} ^{p_{2}}(\Omega)}+ \bigl\Vert \Lambda^{\alpha}\mathbf{m}^{\varepsilon} \bigr\Vert _{\mathbf{L}^{2}(\Omega)} \Vert \boldsymbol {\varphi } \Vert _{\mathbf{L}^{\infty}(\Omega)} \bigr) \\ \leqslant & C \bigl\Vert \Lambda^{\alpha}\mathbf{m}^{\varepsilon}\bigr\Vert _{ \mathbf {L}^{2}(\Omega )} \bigl(C_{1} \bigl\Vert \Lambda^{\alpha}\boldsymbol {\varphi} \bigr\Vert _{\mathbf {L}^{p_{2}}(\Omega)}+ \Vert \boldsymbol {\varphi} \Vert _{\mathbf{L}^{\infty}(\Omega)} \bigr) \\ \leqslant & C_{2}, \end{aligned}$$
where the constants \(C_{1}\), \(C_{2}\) and C are independent of ε.
Now, to ensure the convergence for the nonlinear nonlocal term, we introduce the commutator (see [12])
$$\Gamma_{\boldsymbol {\varphi}}(\mathbf{m}):=\Lambda ^{\alpha }(\mathbf{m} \times \boldsymbol {\varphi})-\boldsymbol {\varphi}\times\Lambda ^{\alpha }\mathbf{m} $$
and we begin by showing that \(\Gamma_{\boldsymbol {\varphi}}(\mathbf {m})\in \mathbf{L} ^{2}(Q)\). Indeed, applying (31) for \(p_{1}=\infty, p_{2}=2, p_{3}=\frac {3}{\beta}\) and \(p_{4}=\frac{6}{3-2\beta}\) with \(\beta=\alpha-1\) (note that for the choice of \(p_{4}\) we have \(\dot{\mathbf{H}}^{\beta }(\Omega )\hookrightarrow\mathbf{L}^{p_{4}}(\Omega)\)), we find
$$\begin{aligned} \bigl\Vert \Gamma_{\boldsymbol {\varphi}}(\mathbf{m}) \bigr\Vert _{\mathbf{L}^{2}(\Omega )} \leq & C_{1} \bigl( \Vert \nabla\boldsymbol {\varphi} \Vert _{\mathbf{L}^{\infty}(\Omega)} \Vert \mathbf{m} \Vert _{\dot {\mathbf{H} }^{\beta}(\Omega)} + \Vert \boldsymbol {\varphi} \Vert _{\dot{\mathbf{W}}^{\alpha,p_{3}}(\Omega )} \Vert \mathbf{m} \Vert _{\mathbf{L} ^{p_{4}}(\Omega)} \bigr) \\ \leqslant & C_{1} \bigl( \Vert \nabla\boldsymbol {\varphi} \Vert _{\mathbf {L}^{\infty }(\Omega )} \Vert \mathbf{m} \Vert _{\dot{\mathbf{H}}^{\beta}(\Omega)} +C_{2}\| \boldsymbol {\varphi}\|_{\dot{\mathbf{W}}^{\alpha,p_{3}}(\Omega )}\| \mathbf{m}\| _{\dot{\mathbf{H}}^{\beta}(\Omega)} \bigr) \\ \leqslant & C \Vert \mathbf{m} \Vert _{\mathbf{H}^{\beta}(\Omega)} \bigl(\| \nabla\boldsymbol { \varphi}\|_{\mathbf{L}^{\infty}(\Omega)} +\|\boldsymbol {\varphi}\|_{\dot{\mathbf{W}}^{\alpha,p_{3}}(\Omega )} \bigr) \\ \leqslant & C'\|\mathbf{m}\|_{\mathbf{H}^{\beta}(\Omega)}, \end{aligned}$$
where \(C_{1}\), \(C_{2}\), C and \(C'\) are constants. Then
$$ \bigl\Vert \Gamma_{\boldsymbol {\varphi}}(\mathbf{m}) \bigr\Vert _{\mathbf {L}^{2}(Q)} \leqslant C\|\mathbf{m} \| _{L^{2}(0,T;\mathbf{H}^{\beta}(\Omega))}. $$
Once again, similarly
$$ \bigl\Vert \Gamma_{\boldsymbol {\varphi}}\bigl(\mathbf{m}^{\varepsilon}-\mathbf {m} \bigr) \bigr\Vert _{\mathbf{L} ^{2}(Q)}\leqslant C \bigl\Vert \mathbf{m}^{\varepsilon}- \mathbf{m} \bigr\Vert _{L^{2}(0,T;\mathbf{H}^{\beta }(\Omega))}. $$
In what follows, we focus on the convergence of the following term:
$$ \mathfrak{I}_{\varepsilon}:= \int_{Q} \Lambda^{\alpha}\mathbf{m} ^{\varepsilon }\cdot \Lambda^{\alpha}\bigl(\mathbf{m}^{\varepsilon}\times\boldsymbol {\varphi}\bigr) \,\mathrm{d}x\,\mathrm{d}t. $$
Let \(\mathfrak{I}:= \int_{Q} \Lambda^{\alpha}\mathbf {m}\cdot \Lambda ^{\alpha}(\mathbf{m}\times\boldsymbol {\varphi}) \,\mathrm {d}x\,\mathrm{d}t\), since \(\Lambda^{\alpha}\mathbf{m}\cdot(\Lambda^{\alpha}\mathbf{m}\times \boldsymbol {\varphi})=0\), we have
$$\mathfrak{I}_{\varepsilon}= \int_{Q} \Lambda ^{\alpha}\mathbf{m}^{\varepsilon}\cdot \Gamma_{\boldsymbol {\varphi }}\bigl(\mathbf{m} ^{\varepsilon}\bigr) \,\mathrm{d}x\, \mathrm{d}t \quad \mbox{and}\quad \mathfrak{I}= \int_{Q} \Lambda^{\alpha}\mathbf{m} \cdot\Gamma _{\boldsymbol {\varphi}}(\mathbf{m}) \,\mathrm{d}x\,\mathrm{d}t, $$
and note that these two integrals are well defined since \(\Gamma _{\boldsymbol {\varphi}}(\mathbf{m}^{\varepsilon})\) and \(\Gamma _{\boldsymbol {\varphi}}(\mathbf{m})\) are in \(\mathbf{L}^{2}(Q)\). Now we will show that \(\mathfrak{I}_{\varepsilon}\rightarrow\mathfrak{I}\) as \(\varepsilon\rightarrow0\).
We have
$$\begin{aligned} \vert \mathfrak{I}_{\varepsilon}-\mathfrak{I} \vert =& \biggl\vert \int_{Q} \Lambda^{\alpha }\mathbf{m} ^{\varepsilon}\cdot \Gamma_{\boldsymbol {\varphi}}\bigl(\mathbf {m}^{\varepsilon }\bigr) \,\mathrm{d}x\, \mathrm{d}t- \int_{Q} \Lambda^{\alpha}\mathbf{m}\cdot \Gamma_{\boldsymbol {\varphi}}(\mathbf{m}) \,\mathrm{d}x\,\mathrm{d}t \biggr\vert \\ =& \biggl\vert \int_{Q} \Lambda^{\alpha}\mathbf{m}^{\varepsilon}\cdot \Gamma _{\boldsymbol {\varphi}}\bigl(\mathbf{m}^{\varepsilon}-\mathbf{m}\bigr) \, \mathrm{d}x\,\mathrm {d}t+ \int_{Q} \Lambda ^{\alpha }\bigl(\mathbf{m}^{\varepsilon}- \mathbf{m}\bigr)\cdot\Gamma_{\boldsymbol {\varphi}}(\mathbf{m} ) \,\mathrm{d} x\, \mathrm{d}t \biggr\vert \\ \leqslant & \int_{Q} \bigl\vert \Lambda^{\alpha} \mathbf{m}^{\varepsilon}\cdot \Gamma _{\boldsymbol {\varphi}}\bigl(\mathbf{m}^{\varepsilon}- \mathbf{m}\bigr) \bigr\vert \,\mathrm{d}x\,\mathrm{d} t+ \biggl\vert \int_{Q} \Lambda^{\alpha}\bigl(\mathbf{m}^{\varepsilon}- \mathbf{m}\bigr)\cdot\Gamma _{\boldsymbol {\varphi }}(\mathbf{m})\, \mathrm{d}x\, \mathrm{d}t \biggr\vert \\ \leqslant & C \bigl\Vert \Gamma_{\boldsymbol {\varphi}}\bigl(\mathbf{m}^{\varepsilon }- \mathbf{m} \bigr) \bigr\Vert _{\mathbf{L}^{2}(Q)}+ \biggl\vert \int_{Q} \Lambda^{\alpha}\bigl(\mathbf {m}^{\varepsilon}-\mathbf{m}\bigr)\cdot \Gamma _{\boldsymbol {\varphi}}(\mathbf{m}) \, \mathrm{d}x\,\mathrm{d}t \biggr\vert \\ \leqslant & C' \bigl\Vert \mathbf{m}^{\varepsilon}-\mathbf{m} \bigr\Vert _{L^{2}(0,T;\mathbf{H}^{\beta}(\Omega ))}+ \biggl\vert \int_{Q} \Lambda^{\alpha}\bigl(\mathbf{m}^{\varepsilon}- \mathbf {m}\bigr)\cdot\Gamma _{\boldsymbol {\varphi}}(\mathbf{m}) \,\mathrm{d}x\, \mathrm{d}t \biggr\vert \\ \rightarrow & 0 \end{aligned}$$
by the strong convergence for \(\mathbf{m}^{\varepsilon}\) to m in \(L^{2}(0,T;\mathbf{H}^{\beta}(\Omega))\) and the weak convergence in \(L^{2}(0,T;\mathbf{H} ^{\alpha}(\Omega))\).
Therefore
$$\textstyle\begin{cases} \int_{Q}\dot{\mathbf{m}}\cdot(\mathbf{m}\times \boldsymbol {\varphi }) \,\mathrm{d}x\,\mathrm{d}t+a \int_{Q} \Lambda^{\alpha}\mathbf {m}\cdot\Lambda^{\alpha }(\mathbf{m} \times\boldsymbol {\varphi}) \,\mathrm{d}x\,\mathrm{d}t\\ \quad{}+ \int_{Q}\lambda_{ijkl}m_{j}\epsilon_{kl}(\mathbf {u})(\mathbf{m} \times \boldsymbol {\varphi})_{i} \,\mathrm{d}x\,\mathrm{d}t=0,\\ -\rho \int_{Q}\dot{\mathbf{u}}\cdot\dot{\boldsymbol {\psi}} \,\mathrm{d} x\,\mathrm{d}t+\int_{Q}\mathcal{G}_{ijkl}(0)\epsilon_{ij}(\mathbf {u}(t))\epsilon _{kl}(\boldsymbol {\psi}(t)) \,\mathrm{d}x\,\mathrm{d}t\\ \quad{}+ \int _{Q}\int^{t}_{0}\mathcal{G}_{ijkl}(t-s)\epsilon_{ij}(\mathbf {u}(s))\epsilon _{kl}(\boldsymbol {\psi}(t)) \,\mathrm{d}s\, \mathrm{d}x\,\mathrm {d}t+ \frac {1}{2} \int_{Q}\lambda_{ijkl}m_{i}m_{j}\epsilon_{kl}(\boldsymbol {\psi}) \,\mathrm{d}x\,\mathrm{d}t=0. \end{cases} $$
This being true for every \(\boldsymbol {\varphi} \in\mathbf {C}^{\infty}(\overline{Q})\), \(\boldsymbol {\psi} \in\mathbf{H}^{1}_{0}(Q)\). Note that from (28) one can easily get (10). Hence \((\mathbf{m},\mathbf{u})\) is a solution of problem (6)-(7)-(8) in the sense of Definition 3.1. The proof of Theorem 3.2 is complete.