On a thermodynamically consistent model for magnetoviscoelastic fluids in 3D

Abstract

We introduce a system of equations that models a non-isothermal magnetoviscoelastic fluid. We show that the model is thermodynamically consistent, and that the critical points of the entropy functional with prescribed energy correspond exactly with the equilibria of the system. The system is investigated in the framework of quasilinear parabolic systems and shown to be locally well-posed in an $L_p$-setting. Furthermore, we prove that constant equilibria are normally stable. In particular, we show that solutions that start close to a constant equilibrium exist globally and converge exponentially fast to a (possibly different) constant equilibrium. Finally, we establish that the negative entropy serves as a strict Lyapunov functional and we then show that every solution that is eventually bounded in the topology of the natural state space exists globally and converges to the set of equilibria.

Appendices

Appendix A: Properties of fractional Sobolev spaces with temporal weights

For $r\in (0, 1)$, fractional Sobolev spaces with temporal weight can also be defined by means of interpolation. It then holds that

$$\begin{aligned} W^r_{p,\mu }(J_T;X) {\mathop {=}\limits ^{\cdot }} \left( L_{p,\mu }(J_T;X), W^1_{p,\mu }(J_T;X) \right) _{r,p}, \end{aligned}$$

where the symbol ${\mathop {=}\limits ^{\cdot }}$ means equivalent norms; see [26, Proposition 1.1.13], or [28, equation (2.6)]. The corresponding norm is called the interpolation norm of $W^r_{p,\mu }(J_T;X)$. It is pointed out in [26, Remark 1.1.15] that the equivalence constant between the intrinsic norm (1.5) and the interpolation norm of $W^r_{p,\mu }(J_T;X)$ blows up as $T~\rightarrow ~0^+$. Using interpolation norms can cause difficulties in obtaining uniform estimates for nonlinear terms on short time intervals (0, T) that are independent of T. This difficulty can often be circumvented by using intrinsic norms.

In this section, we establish some useful results for fractional Sobolev spaces with temporal weights by exclusively using intrinsic norms. These results are also interesting in their own right. Analogous results have been obtained in [26], see also [28], by using interpolation norms. For instance, it is shown in [26, Lemma 1.1.15] that there exists an extension operator $\mathcal {E}_T: {}_{0}{W}^r_{p,\mu }(J_T;X) \rightarrow {}_{0}{W}^r_{p,\mu }(\mathbb {R}_+;X)$ whose norm is independent of T, where both spaces are equipped with the corresponding interpolation norms. The merit of Proposition A.4 lies in the fact that we can completely rely on intrinsic norms. This greatly facilitates deriving estimates for nonlinear boundary terms.

The results obtained in this section employed in obtaining estimates for nonlinear mappings, but are also of independent interest. We recall that

$$\begin{aligned} {}_{0}{W}^r_{p,\overline{\mu }}(J_T; X):=\{u\in W^r_{p,\overline{\mu }}(J_T; X): \, \gamma _0 u=0 \}, \end{aligned}$$

where $r,\overline{\mu }\in (1/p,1]$ with $r+\overline{\mu }>1+1/p$, and X is a Banach space; see [28, Proposition 2.10].

Lemma A.1

Let X be a Banach space. Given $p\in (1,\infty )$ and $r,\mu \in (1/p,1]$ such that $r+\mu >1+1/p$, there exists a constant $C>0$, which is independent of $T\in (0,\infty ]$, such that

$$\begin{aligned} \left( \int \nolimits _0^T t^{(1-\mu -r)p} \Vert u(t)\Vert _X^p \, \textrm{d}t \right) ^{1/p} \le C \Vert u\Vert _{W^r_{p,\mu }(J_T;X)} \end{aligned}$$

for all $u\in {}_{0}{W}^r_{p,\mu }(J_T;X)$, where ${}_{0}{W}^r_{p,\mu }(J_T;X)$ is equipped with the intrinsic norm.

Proof

The case $r \in \{0,1\}$ follows from the definition of $L_{p,\mu }(J_T;X)$ and [26, Lemma 1.1.2(b)]. When $r\in (0,1)$,

$$\begin{aligned}&\left( \int \nolimits _0^T t^{(1-\mu -r)p} \Vert u (t)\Vert _X^p \, \textrm{d}t \right) ^{1/p} \nonumber \\&\quad = \left( \int \nolimits _0^T \left( t^{ -\mu -r } \int \nolimits _0^t \Vert u (t)\Vert _X \, \textrm{d}s \right) ^p \, \textrm{d}t \right) ^{1/p}\nonumber \\&\quad \le \left( \int \nolimits _0^T \left( t^{ -\mu -r } \int \nolimits _0^t \Vert u (t)-u(s)\Vert _X \, \textrm{d}s \right) ^p \, \textrm{d}t \right) ^{1/p}\nonumber \\&\qquad + \left( \int \nolimits _0^T \left( t^{ -\mu -r } \int \nolimits _0^t \Vert u (s)\Vert _X \, \textrm{d}s \right) ^p \, \textrm{d}t \right) ^{1/p}. \end{aligned}$$

(A.1)

We will use Hölder’s inequality to estimate the first term in (A.1) as follows:

$$\begin{aligned}&\left( \int \nolimits _0^T \left( t^{ -\mu -r } \int \nolimits _0^t \Vert u (t)-u(s)\Vert _X \, \textrm{d}s \right) ^p \, \textrm{d}t \right) ^{1/p}\\&\quad \le \left( \int \nolimits _0^T t^{ (-\mu -r)p } \left( \int \nolimits _0^t s^{(1-\mu )p} \Vert u (t)-u(s)\Vert _X^p \, \textrm{d}s \right) \left( \int \nolimits _0^t s^{(\mu -1)p'} \, \textrm{d}s \right) ^{p/p'} \, \textrm{d}t \right) ^{1/p}\\&\quad \le C \left( \int \nolimits _0^T t^{ -1-r p } \left( \int \nolimits _0^t s^{(1-\mu )p} \Vert u (t)-u(s)\Vert _X^p \, \textrm{d}s \right) \, \textrm{d}t \right) ^{1/p}\\&\quad \le C [u]_{W^r_{p,\mu }(J_T;X)}. \end{aligned}$$

In the last line, we used that $1/t < 1/( t-s)$ for $s\in (0,t)$. Observe that it follows from the condition $\mu \in (1/p,1]$ that $(\mu -1)p'>-1$. To estimate the second term in (A.1), we will apply Hardy’s inequality, c.f. [32, Lemma 3.4.5], to obtain

$$\begin{aligned} \left( \int \nolimits _0^T \left( t^{ -\mu -r } \int \nolimits _0^t \Vert u (s)\Vert _X \, \textrm{d}s \right) ^p \, \textrm{d}t \right) ^{1/p} \le \frac{1}{(\mu +r-1/p)} \left( \int \nolimits _0^T t^{(1-\mu -r)p} \Vert u(t)\Vert _X^p \, \textrm{d}t \right) ^{1/p}. \end{aligned}$$

Hence, we have shown that

$$\begin{aligned}&\left( \int \nolimits _0^T t^{(1-\mu -r)p} \Vert u (t)\Vert _X^p \, \textrm{d}t \right) ^{1/p} \le C \Vert u\Vert _{W^r_{p,\mu }(J_T;X)} + \frac{1}{(\mu +r-1/p)}\\&\qquad \left( \int \nolimits _0^T t^{(1-\mu -r)p} \Vert u(t)\Vert _X^p \, \textrm{d}t \right) ^{1/p}. \end{aligned}$$

In view of the condition $r+\mu >1+1/p$, the asserted estimate then follows. $\square$

Lemma A.2

Suppose $\mu \in [0,1]$. Then, we have

$$\begin{aligned} \left( t^{1-\mu } - s^{1-\mu } \right) ^p{} & {} \le t^{-\mu p} (t-s)^p\quad \text {and}\quad |t^{\mu -1} - s^{\mu -1}|^p\le s^{(\mu -1)p} t^{-p} (t-s)^p,\\{} & {} \quad 0<s<t < \infty . \end{aligned}$$

Proof

The assertions are clear for $\mu \in \{0,1\}$. In case $\mu \in (0,1)$, we obtain

$$\begin{aligned} (t^{1-\mu } - s^{1-\mu })^p = t^{(1-\mu )p} \left( 1- \left( s/t\right) ^{1-\mu }\right) ^p \le t^{(1-\mu )p} \left( 1- (s/t)\right) ^p = t^{-\mu p} (t-s)^p. \end{aligned}$$

This estimate, in turn, yields

$$\begin{aligned} |t^{\mu -1}- s^{\mu -1}|^p = s^{(\mu -1)p} t^{(\mu -1)p} (t^{1-\mu }- s^{1-\mu })^p \le s^{(\mu -1)p} t^{-p} (t-s)^p. \end{aligned}$$

$\square$

For $u\in L_{1,\textrm{loc}}(J_T;X)$, we define $(\Phi _\mu u)(t):= t^{1-\mu } u(t)$, see [31]. It is then clear that

$$\begin{aligned} \Phi _\mu : L_{p,\mu }(J_T; X) \rightarrow L_p(J_T;X)\quad \text {is an isometric isomorphism}, \end{aligned}$$

(A.2)

and its inverse $\Phi ^{-1}_\mu$ is given by $(\Phi _\mu ^{-1} v)(t)= t^{\mu -1}v(t)$. The next result shows that $\Phi _\mu$ induces an isomorphism for the Sobolev spaces ${_0}W^r_{\mu , p}(J_T;X)$.

Lemma A.3

Let X be a Banach space. Suppose that $r,\mu \in (1/p,1]$ and $r+\mu > 1+1/p$. Then, it holds that

$$\begin{aligned} \Phi _\mu \in {\mathcal {L}}is (_{0}W^r_{p,\mu }(J_T;X), {_0}W^r_p(J_T;X)). \end{aligned}$$

Moreover, there exists a constant C which is independent of $T\in (0,\infty ]$ such that

$$\begin{aligned} \Vert \Phi _\mu u\Vert _{W^r_{p,\mu }(J_T;X)}\le C \Vert u \Vert _{W^r_{p}(J_T;X)},\qquad \Vert \Phi ^{-1}_\mu v\Vert _{W^r_{p}(J_T;X)}\le C \Vert v \Vert _{W^r_{p,\mu }(J_T;X)}, \nonumber \\ \end{aligned}$$

(A.3)

where the spaces are equipped with their respective intrinsic norms.

Proof

The first part of the assertion has been established in [28, Lemma 2.3], where the spaces are equipped with their respective interpolation norms.

We will now establish the uniform estimates in (A.3) for intrinsic norms. The case $r=1$ follows readily from Lemma  A.1. For the reader’s convenience, we include a proof (see also [26, Lemma 1.1.3]). Suppose $u\in W^1_{p,\mu }(J_T;X)$. Then, we obtain

$$\begin{aligned} \Vert (\Phi _\mu u)^\prime \Vert _{L_{p}(J_T;X)}{} & {} \le \left( \int \nolimits _0^T \Vert t^{1-\mu } u^\prime (t)\Vert ^p_X\, \textrm{d}t \right) ^{1/p} + (1-\mu ) \left( \int \nolimits _0^T t^{-\mu p} \Vert u(t) \Vert ^p_X\, \textrm{d}t \right) ^{1/p}\\ {}{} & {} \le C \Vert u\Vert _{W^1_{p,\mu } (J_T;X)}, \end{aligned}$$

where we used Lemma A.1 with $r=1$. Suppose now that $v\in {_0}W^1_{p,\mu }(J_T; X)$. Then, we obtain

$$\begin{aligned} \Vert (\Phi ^{-1}_\mu v)^\prime \Vert _{L_{p,\mu }(J_T;X)}{} & {} \le \Big (\int \nolimits _0^T \Vert v^\prime (t)\Vert ^p_X\,\textrm{d}t \Big )^{1/p} + |\mu -1| \left( \int \nolimits _0^T t^{-1} \Vert v \Vert ^p_X\,\textrm{d}t\right) ^{1/p}\\ {}{} & {} \le \Vert v \Vert _{W^1_p(J_T;X)}, \end{aligned}$$

where we employed, once more, Lemma A.1 with $r=\mu =1.$ These estimates together with (A.2) imply the assertion.

We will now consider the case $r<1$ and $r+\mu > 1+1/p$. Suppose $u\in {_0}W^r_{p,\mu }(J_T;X)$. Then, we obtain

$$\begin{aligned}{}[\Phi _\mu u ]_{W^r_{p}(J_T;X)}&= \left( \int \nolimits _0^T \int \nolimits _0^t \frac{\Vert (\Phi _\mu u)(t)- (\Phi _\mu u)(s)\Vert _X^p}{(t-s)^{1+rp}}\, \textrm{d}s \, d t \right) ^{1/p} \nonumber \\&\le \left( \int \nolimits _0^T \int \nolimits _0^t s^{p(1-\mu )} \frac{\Vert u(t) - u(s) \Vert _X^p}{(t-s)^{1+rp}}\, \textrm{d}s \, d t \right) ^{1/p} \nonumber \\&\quad + \left( \int \nolimits _0^T \int \nolimits _0^t \frac{ (t^{1-\mu } - s^{1-\mu })^p}{(t-s)^{1+rp}} \Vert u (t) \Vert ^p_X \, \textrm{d}s \, d t \right) ^{1/p} \nonumber \\&\le [u ]_{W^r_{p,\mu }(J_T;X)} + \left( \int \nolimits _0^T \int \nolimits _0^t t^{-\mu p} (t-s)^{(1-r)p-1} \Vert u (t)\Vert ^p_X \, \textrm{d}s \, d t \right) ^{1/p} \nonumber \\&\le [u ]_{W^r_{p,\mu }(J_T;X)} +c(r,p) \left( \int \nolimits _0^T t^{(1-\mu -r )p} \Vert u (t)\Vert ^p_X \, d t \right) ^{1/p} \end{aligned}$$

(A.4)

$$\begin{aligned}&\le C \Vert u \Vert _{W^r_{p,\mu }(J_T;X)}. \end{aligned}$$

(A.5)

We used Lemma A.2 in (A.4) and Lemma A.1 in (A.5).

Suppose that $v\in {_0}W^r_{p}(J_T;X)$. Then, we obtain

$$\begin{aligned}{}[(\Phi _\mu )^{-1} v ]_{W^r_{p,\mu }(J_T;X)}&= \left( \int \nolimits _0^T \int \nolimits _0^t s^{(1-\mu )p} \frac{\Vert (\Phi ^{-1}_\mu v)(t)- (\Phi ^{-1}_\mu v)(s)\Vert _X^p}{(t-s)^{1+rp}}\, \textrm{d}s \, d t \right) ^{1/p} \nonumber \\&\le \left( \int \nolimits _0^T \int \nolimits _0^t \frac{\Vert v(t)- v(s)\Vert _X^p}{(t-s)^{1+rp}}\, \textrm{d}s \, d t \right) ^{1/p} \nonumber \\&\quad + \left( \int \nolimits _0^T \int \nolimits _0^t s^{(1-\mu )p} \frac{|t^{\mu -1}- s^{\mu -1}|^p}{ (t-s)^{1+rp}} \Vert v(t) \Vert ^p_X\, \textrm{d}s \, d t \right) ^{1/p} \nonumber \\&\le [u ]_{W^r_{p}(J_T;X)} +c(r,p) \left( \int \nolimits _0^T t^{-rp} \Vert u (t)\Vert ^p_X \, d t \right) ^{1/p} \end{aligned}$$

(A.6)

$$\begin{aligned}&\le C \Vert u \Vert _{W^r_{p}(J_T;X)}. \end{aligned}$$

(A.7)

Here we used, once more, Lemma A.2 in (A.6) and Lemma A.1 in (A.7). $\square$

Proposition A.4

Let X be a Banach space. Suppose $r,\mu \in (1/p,1]$ and $r+\mu >1+1/p$. Then, there exists an extension operator:

$$\begin{aligned} \mathcal {E}_{J_T}: {}_{0}{W}^r_{p,\mu }(J_T;X) \rightarrow {}_{0}{W}^r_{p,\mu }(\mathbb {R}_+;X) \end{aligned}$$

such that its norm is independent of $T\in (0, \infty ]$, where the spaces are equipped with their intrinsic norms.

Proof

We define the extension operator by

$$\begin{aligned} \mathcal {E}_{J_T} u (t) := {\left\{ \begin{array}{ll} u(t) \quad &{}\text {for } 0<t\le T \\ \left( \frac{2T-t}{t}\right) ^{1-\mu } u(2T-t) &{} \text {for } T< t \le 2T \\ 0 &{}\text {for } 2T<t. \end{array}\right. } \end{aligned}$$

The statement follows from Lemma A.3, [30, Proposition 6.1], and the commutativity of the diagram

$$\begin{aligned} \begin{aligned}&{_0}W^r_{p,\mu }(J_T;X)\quad \overset{\Phi _\mu }{\longrightarrow }{} & {} {_0}W^r_{p}(J_T;X) \\&\quad \downarrow \mathcal {E}_{J_T}{} & {} \quad \ \downarrow \mathcal {E}_T \\&{_0}W^r_{p,\mu }(\mathbb {R}_+;X)\quad \overset{{\Phi ^{-1}_\mu }}{\longleftarrow }{} & {} \ {_0}W^r_p(\mathbb {R}_+;X), \end{aligned} \end{aligned}$$

where the extension operator $\mathcal {E}_T$ on the right side is defined in [30, Proposition 6.1]. $\square$

The following result is used in Sect. 6 in order to show stability of (constant) equilibria.

Lemma A.4

Let $T>0$, $r\in (0,1)$, $\omega \in \mathbb {R}$, and $\mu \in (1/p,1]$. We then set

$$\begin{aligned} B_T =\{(s,t)\in (0,T)^2: 0<s<t\}\quad \text {and} \quad B_T^1=\{(s,t)\in (0,T)^2: 0<t-s<1\}. \end{aligned}$$

Suppose that X is a Banach space and $u\in W^r_{p,\mu }(J_T;X)$. Then,

$$\begin{aligned}{}[ e_\omega u ]_{W^r_{p,\mu }(J_T; X)}&\le C \Vert e_\omega u \Vert _{L_{p,\mu } (J_T;X)} + \left( \iint _{B_T^1} \frac{\Vert s^{1-\mu } e^{\omega s} (u(t)-u(s)) \Vert _X^p}{(t-s)^{1+r p}} \, \textrm{d}s \, \textrm{d}t \right) ^{1/p} \\&\le C \Vert e_\omega u \Vert _{W^r_{p,\mu }(J_T; X)} , \end{aligned}$$

where the constant $C=C(p,r, \omega )$ is independent of T and $e_\omega : L_{1,loc}(\mathbb {R}_+) \rightarrow L_{1,loc}(\mathbb {R}_+): u \mapsto e^{\omega t} u$.

Proof

Using (1.5), we estimate as in [22, Lemma 11] and obtain

$$\begin{aligned}&[ e_\omega u ]_{W^r_{p,\mu }(J_T; X)} \\&\quad \le \left( \iint _{B_T\setminus B_T^1} s^{p(1-\mu )} \frac{\Vert e^{\omega t} u(t)- e^{\omega s}u(s)\Vert _X^p}{(t-s)^{1+r p}}\, \textrm{d}s \, d t \right) ^{1/p} \\&\qquad + \left( \iint _{B_T^1} s^{p(1-\mu )} \frac{\Vert e^{\omega t} u(t)- e^{\omega s} u(s)\Vert _X^p}{(t-s)^{1+r p}}\, \textrm{d}s \, d t \right) ^{1/p} \\&\quad \le \left( \int \nolimits _0^T \int \nolimits _0^{t-1} \frac{\Vert t^{1-\mu }\, e^{\omega t} u(t) \Vert _X^p}{(t-s)^{1+r p}} \, \textrm{d}s \, d t \right) ^{1/p} + \left( \int \nolimits _0^T \int \nolimits _{s+1}^T \frac{ \Vert s^{1-\mu }\, e^{\omega s} u(s) \Vert _X^p }{(t-s)^{1+r p}} \, \textrm{d}t \, d s \right) ^{1/p} \\&\qquad + \left( \iint _{B_T^1} s^{p(1-\mu )} e^{\omega t p} \Vert u(t)\Vert _X^p \frac{|e^{-\omega (t-s)}-1|^p}{(t-s)^{1+r p}}\, \textrm{d}s \, d t \right) ^{1/p}\\&\qquad + \left( \iint _{B_T^1} s^{p(1-\mu )} e^{\omega s p} \frac{\Vert u(t)- u(s)\Vert _X^p}{(t-s)^{1+r p}}\, \textrm{d}s \, d t \right) ^{1/p} \\&\quad \le \Vert e_\omega u\Vert _{L_{p,\mu } (J_T;X)} \left[ 2 \left( \int \nolimits _1^\infty \frac{d\tau }{\tau ^{1+r p}}\right) ^{1/p} + c(\omega ) \left( \int \nolimits _0^1 \frac{d\tau }{\tau ^{1+(r -1)p}}\right) ^{1/p}\right] \\&\qquad + \left( \iint _{B_T^1} s^{p(1-\mu )} e^{\omega s p} \frac{\Vert u(t)- u(s)\Vert _X^p}{(t-s)^{1+r p}}\, \textrm{d}s \, d t \right) ^{1/p} \\&\quad \le C \Vert e_\omega u\Vert _{L_{p,\mu } (J_T;X)} + \left( \iint _{B_T^1} s^{p(1-\mu )} e^{\omega s p} \frac{\Vert u(t)- u(s)\Vert _X^p}{(t-s)^{1+r p}}\, \textrm{d}s \, d t \right) ^{1/p} . \end{aligned}$$

In the derivation above, we have used that $s<t$ for $(s,t)\in B_T$. $\square$

Our next result deals with multiplication properties in weighted Sobolev spaces.

Lemma A.5

Let $T_*>0$ be given.

(i):

There exists a constant $C>0$, which is independent of $T\in (0,T_*]$, such that

$$\begin{aligned} \Vert u v \Vert _{\mathbb {F}_\mu (J_T)} \le C \Vert u \Vert _{\mathbb {F}_\mu (J_T)} \Vert v \Vert _{\mathbb {F}_\mu (J_T)},\quad \text {for all } u,v\in {}_{0}{\mathbb {F}}_\mu (J_T). \end{aligned}$$

(ii):

There exists a constant $C>0$, which is independent of $T\in (0,T_*]$, such that

$$\begin{aligned} \Vert u v \Vert _{\mathbb {F}_\mu (J_T)} \le C \Vert u \Vert _{\mathbb {F}_\mu (J_T)} \Vert v \Vert _{\mathbb {F}_{1,\mu }(J_T)},\quad \text {for all } (u,v)\in \mathbb {F}_\mu (J_T) \times {}_{0}{\mathbb {F}}_{1,\mu }(J_T), \end{aligned}$$

where $\mathbb {F}_{1,\mu }(J_T)$ is defined as

$$\begin{aligned} \mathbb {F}_{1,\mu }(J_T):= W^{1-1/2p}_{p,\mu }(J_T;L_p(\partial \Omega ))\cap C([0,T];W^{2\mu -3/p}_p(\partial \Omega )). \end{aligned}$$

(A.8)

Proof

(i) The assertion follows from the fact that $\mathbb {F}_\mu (J_T)$ is a Banach algebra and Proposition A.4. See also [26, Lemma 1.3.23].

(ii) To explain the occurrence of the space $C([0,T];W^{2\mu -3/p}_p(\partial \Omega ))$ in (A.8), we note that

$$\begin{aligned} W^{1-1/2p}_{p,\mu }(J_T; L_p(\partial \Omega ))\cap L_{p,\mu }(J_T; W^{2-1/p}_p(\partial \Omega )) \hookrightarrow C([0,T];W^{2\mu -3/p}_p(\partial \Omega )) \end{aligned}$$

see [28, equation (4.10)]. It is an easy task to check that

$$\begin{aligned} \Vert u v \Vert _{L_{p,\mu }(J_T; W^{1-1/p}_p(\partial \Omega ))}\le C \Vert v\Vert _{L_\infty (J_T;W^{1-1/p}_p(\partial \Omega ))} \Vert u \Vert _{L_{p,\mu }(J_T; W^{1-1/p}_p(\partial \Omega ))} \end{aligned}$$

for some $C>0$ independent of $T\in (0,T_*]$. In addition, one has

$$\begin{aligned}&\Vert u v \Vert _{W^{1/2-1/2p}_{p,\mu }(J_T; L_p(\partial \Omega ))} \nonumber \\&\le \Vert v\Vert _{C([0,T]\times \overline{\Omega }) } \Vert u \Vert _{L_{p,\mu }(J_T; L_p(\partial \Omega ))} + [u v]_{W^{1/2-1/2p}_{p,\mu }(J_T; L_p(\partial \Omega ))} . \end{aligned}$$

(A.9)

We can derive from Proposition A.4 and equation (1.4) in [29, Theorem 1.1] (by choosing $p_i=q_i=p$ for $i=0,1$, $\gamma _0=(1-\mu )p$, and $\gamma _1=0$) that

$$\begin{aligned} {}_{0}{W}^{1-1/2p}_{p,\mu }(J_T; L_p(\partial \Omega )) \hookrightarrow {}_{0}{W}^{s_1}_p(J_T; L_p(\partial \Omega )) \end{aligned}$$

for $s_1=\mu -1/p$ with embedding constant independent of $T\in (0,T_*]$. Due to assumption (3.2), we have $s_1>1/2+2/p$. This implies

$$\begin{aligned} {}_{0}{\mathbb {F}}_{1,\mu }(J_T) \hookrightarrow C^\sigma ([0,T]; L_p(\partial \Omega )) \end{aligned}$$

for some $\sigma >1/2+1/p$ with embedding constant independent of T. Therefore, the second term on the right-hand side of (A.9) can be estimated as follows:

$$\begin{aligned}&[u v]_{W^{1/2-1/2p}_{p,\mu }(J_T; L_p(\partial \Omega ))}^p \\&\quad \le \int \nolimits _0^T \int \nolimits _0^t s^{p(1-\mu )} \frac{\Vert u(t) v(t) - u(s) v(s) \Vert _{L_p(\partial \Omega )}^p}{ (t-s)^{1/2+p/2} } \, \textrm{d}s \textrm{d}t \\&\quad \le \Vert v\Vert _{C([0,T]\times \overline{\Omega }) }^p [ u]_{W^{1/2-1/2p}_{p,\mu }(J_T; L_p(\partial \Omega ))}^p \\&\qquad + \int \nolimits _0^T \int \nolimits _0^t s^{p(1-\mu )} \Vert u(s)\Vert _{C( \overline{\Omega }) }^p \frac{\Vert v(t) - v(s) \Vert _{L_p(\partial \Omega )}^p}{ (t-s)^{1/2+p/2} } \, \textrm{d}s \textrm{d}t \\&\quad \le \Vert v\Vert _{C([0,T]\times \overline{\Omega }) }^p [ u]_{W^{1/2-1/2p}_{p,\mu }(J_T; L_p(\partial \Omega ))}^p \\&\qquad + \Vert v\Vert _{C^\sigma ([0,T]; L_p(\partial \Omega ))}^p \int \nolimits _0^T \int \nolimits _0^t s^{p(1-\mu )} \Vert u(s)\Vert _{C( \overline{\Omega }) }^p (t-s)^{\sigma p-(1/2+p/2)} \, \textrm{d}s \textrm{d}t \\&\quad \le \Vert v\Vert _{C([0,T]\times \overline{\Omega }) }^p [ u]_{W^{1/2-1/2p}_{p,\mu }(J_T; L_p(\partial \Omega ))}^p + C_1 \Vert v\Vert _{C^\sigma ([0,T]; L_p(\partial \Omega ))}^p \Vert u\Vert _{L_{p,\mu }(J_T;C( \overline{\Omega })) }^p \end{aligned}$$

for some constant $C_1=C_1(T)>0$ that is uniform in $T\in (0,T_*]$. This implies

$$\begin{aligned}{}[u v]_{W^{1/2-1/2p}_{p,\mu }(J_T; L_p(\partial \Omega ))}\le C \Vert v\Vert _{_{0}\mathbb {F}_{1,\mu }(J_T)} \Vert v\Vert _{\mathbb {F}_\mu (J_T)}, \end{aligned}$$

where $C=C(T)$ is uniform in $T\in (0,T_*]$. $\square$

Appendix B: Properties of nonlinear maps

In this section, we establish some mapping properties for the nonlinear operators in (1.1). Our first step is to study the Nemyskii operators induced by the functions in (2.1).

Lemma B.1

Suppose $\varphi \in C^5(\mathbb {R})$ and $X\in \{W^{2\mu -2/p}_p(\Omega ), W^{2\mu +1-2/p}_p(\Omega ), \mathbb {E}_{2,\mu }^k(J_T), \mathbb {F}_\mu (J_T)\}$, where

$$\begin{aligned} \mathbb {E}_{2,\mu }^k(J_T):= W^1_{p,\mu }(J_T;W^k_p( \Omega ))\cap L_{p,\mu }(J_T; W^{k+2}_p(\Omega )), \quad k=0,1. \end{aligned}$$

Then, the Nemyskii operator induced by $\varphi$, still denoted by $\varphi$, satisfies

$$\begin{aligned} \varphi \in C^1(X). \end{aligned}$$

Moreover, given $T_*>0$

$$\begin{aligned} \Vert \varphi (u) \Vert _{\mathbb {F}_\mu (J_T)} \le C \left( \Vert \varphi '(u)\Vert _\infty \Vert u\Vert _{\mathbb {F}_\mu (J_T)} + \Vert \varphi (u)\Vert _\infty \right) ,\quad u\in \mathbb {F}_\mu (J_T). \end{aligned}$$

(B.1)

The constant $C>0$ is uniform with respect to $T\in (0,T_*]$.

Proof

The mapping property $\varphi \in C^1(\mathbb {E}_{2,\mu }^k(J_T))$ can be proved via direct computations and the fact that $\varphi \in C^5(\mathbb {R})$. It follows from Lemma 3.1(b) that there exists a bounded right inverse $\gamma ^c_0$ for the initial trace operator

$$\begin{aligned} \gamma _0: \mathbb {E}_{2,\mu }^0(J_T) \rightarrow W^{2\mu -2/p}_p(\Omega ). \end{aligned}$$

The $C^1$-continuity of $\varphi$ in $W^{2\mu -2/p}_p(\Omega )$ then follows from the relationship

$$\begin{aligned} \varphi (u)= \gamma _0 \varphi (\gamma ^c_0(u)), \quad u\in W^{2\mu -2/p}_p(\Omega ). \end{aligned}$$

The case $X=W^{2\mu +1-2/p}_p(\Omega )$ follows from a similar argument. The assertion in (B.1) has been proved in [26, Lemma 4.2.3(a)]. A close look at its proof shows that the constant in [26, Lemma 4.2.3(a)] is uniform with respect to $T\in (0,T_*]$. The $C^1$-continuity of $\varphi$ in $\mathbb {F}_\mu (J_T)$ can be derived from (B.1) by a mean value theorem argument and the fact that $\mathbb {F}_\mu (J_T)$ is a Banach algebra. $\square$

Next, we will establish some relevant mapping properties of the operators in (3.7). For the analysis below, note that by Proposition A.4 and [28, Theorems 4.2 and 4.5], there exists a constant $C>0$ such that

$$\begin{aligned} \Vert \textrm{tr}_{\partial \Omega } v\Vert _{\mathbb {F}_{1,\mu }(J_T)} \le C \Vert v\Vert _{\mathbb {E}_{2,\mu }^0(J_T) }, \quad v\in \mathbb {E}_{2,\mu }^0(J_T), \end{aligned}$$

(B.2)

where the embedding constant is independent of T if $v\in {}_{0}{\mathbb {E}}_{2,\mu }^0(J_T)$.

Suppose that $\phi _1\in C^5(\mathbb {R}^{16} )$, $\phi _2\in C^5(\mathbb {R}^{48} )$ and $\phi _3\in C^5(\mathbb {R}^{9} )$. In order to derive an estimate for

$$\begin{aligned} \Vert \mathcal {A}(z_1+z_2) - \mathcal {A}(z_1) -\mathcal {A}'(z_1)z_2\Vert _{\mathbb {E}_{0,\mu }(J_T)} \end{aligned}$$

for proper functions $z_1,z_2\in \mathbb {E}_{1,\mu }(J_T)$, we will consider five types of mappings, given by

$$\begin{aligned} \begin{aligned} G_1 (z)&= \phi _1( z) \phi _2(\partial z), \\ G_2 (z)&= \phi _1 (z) \partial _{ij} z, \\ G_3 (z)&= \phi _1 (z) \phi _3(\partial m), \\ G_4 (z)&= \phi _1 (z) \phi _3(\partial m) \partial _{ij} m, \\ G_5 (z)&= \phi _1(z) |\Delta m |^2, \end{aligned} \end{aligned}$$

(B.3)

where for any function $z=(u,F,\theta ,m)\in C^1(\Omega ,\mathbb {R}^{16})$, we define $\partial z\in C(\Omega , \mathbb {R}^{48})$ by $\partial z=( \partial _1 z, \partial _2 z, \partial _3 z),$ and $\partial m\in C(\Omega , \mathbb {R}^9)$ by $\partial m= ( \partial _1 m, \partial _2 m, \partial _3 m).$

All terms in $\mathcal {A}(z)$ can be estimated by using one of the functions $G_i$. For instance,

• terms like $\mu '(\theta )\partial _i \theta \partial _i u$ can be estimated by using $G_1$ with $\phi _1(z)=\mu '(\theta )$ and $\phi _2(\partial z)=\partial _i \theta \partial _i u$;

• terms like $\mu (\theta ) \partial _{ii} u$ can be estimated by using $G_2$ with $\phi _1(z)=\mu (\theta )$;

• the term $K(z):\nabla ^2 \theta =K_{ij}(z)\partial _{ij}\theta$ can be estimated by using $G_2$ with $\phi _1(z)= K_{ij}(z)$;

• terms like $\alpha (\theta ) |m|^2 |\nabla m|^2$ can be estimated by $G_3$ with $\phi _1(z)=\alpha (\theta ) |m|^2$ and $\phi _3(\partial m)=|\nabla m|^2$;

• the scalar components of $(\alpha (\theta ) I_3 - \beta (\theta )M(m))\Delta m$ can be estimated by using $G_4$ with $\phi _3\equiv 1$ and $\phi _1$ properly chosen;

• the term $\alpha (\theta ) |\nabla m|^2 \;m\cdot \Delta m$, appearing in the $\theta$-equation, can be estimated by using $G_4$ with $\phi _3(\partial m)= |\nabla m|^2$ and $\phi _1$ properly chosen;

• lastly, the term $\alpha (\theta ) |\Delta m|^2$ can be estimated by using $G_5$ with $\phi _1(z)=\alpha (\theta )$.

Lemma B.2

Let the functions $G_i, 1\le i\le 5$ be given by (B.3). Then,

$$\begin{aligned} G_1, G_2, G_5 \in C^1(\mathbb {E}_{1,\mu } (J_T), \mathbb {E}_{0,\mu }^0 (J_T)), \quad G_3, G_4 \in C^1(\mathbb {E}_{1,\mu } (J_T), \mathbb {E}_{0,\mu }^1 (J_T)), \end{aligned}$$

where $\mathbb {E}_{0,\mu }^k (J_T)=L_{p,\mu }(J_T; W^k_p(\Omega ))$, $k=0,1$. Furthermore, given $T_0, R_0 >0$, then for any $T\in (0,T_0]$, $R\in (0,R_0]$ and any $z_1=(u_1,F_1,\theta _1,m_1) \in \mathbb {E}_{1,\mu } (J_T)$ and $z_2=(u_2,F_2,\theta _2,m_2)\in {}_{0}{\mathbb {E}}_{1,\mu } (J_T)$ satisfying

$$\begin{aligned} \Vert z_1\Vert _{\mathbb {B}_\mu (J_T)},\; \Vert z_1\Vert _{\mathbb {E}_{1,\mu } (J_T)},\; \Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} \le R, \end{aligned}$$

the following estimate holds

$$\begin{aligned} \begin{aligned} \Vert G_i (z_1+z_2) - G_i(z_1) - G_i'(z_1)z_2 \Vert _{\mathbb {E}_{0,\mu }^0(J_T)}&\le \Phi (\Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} ) \Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} ,\\ \qquad i=1,2,5,\\ \Vert G_i (z_1+z_2) - G_i(z_1) - G_i'(z_1)z_2 \Vert _{\mathbb {E}_{0,\mu }^1(J_T)}&\le \Phi (\Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} ) \Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} ,\\ \qquad i=3,4, \end{aligned} \end{aligned}$$

(B.4)

where $G_i'$ is the Frechét derivative of $G_i$.

Proof

The continuous differentiability of $G_i$ follows by direct computations. We will only establish the estimates in (B.4). Easy computations lead to

$$\begin{aligned}&G_1 (z_1+z_2) - G_1(z_1) - G_1'(z_1)z_2 \\&\quad = \left( \phi _1(z_1 +z_2) - \phi _1(z_1 ) - \phi _1'(z_1 )z_2)\right) \phi _2(\partial z_1 ) \\&\qquad + \phi _1(z_1 +z_2) \left( \phi _2(\partial z_1 + \partial z_2) - \phi _2(\partial z_1 ) -\phi _2'(\partial z_1)\partial z_2 \right) \\&\qquad + \left( \phi _1(z_1 +z_2) - \phi _1(z_1 ) \right) \phi _2'(\partial z_1)\partial z_2. \end{aligned}$$

Then the mean value theorem, Lemma 3.1(a), (3.3) implies

$$\begin{aligned}&\left\| \left( \phi _1(z_1 +z_2) - \phi _1(z_1 ) - \phi _1'(z_1 )z_2)\right) \phi _2(\partial z_1 )\right\| _{\mathbb {E}_{0,\mu }^0(J_T)} \\&\quad \le \Vert \phi _2(\partial z_1 ) \Vert _\infty \; \Vert z_2\Vert _\infty \; \int \nolimits _0^1 \left\| \phi _1'(z_1 +\sigma z_2) - \phi _1'(z_1) \right\| _{\mathbb {E}_{0,\mu }^0(J_T)}\, d\sigma \\&\quad \le \Vert \phi _2(\partial z_1 ) \Vert _\infty \; \Vert z_2\Vert _\infty \; \int \nolimits _{[0,1]\times [0,1]} \left\| \phi _1''(z_1 +\tau \sigma z_2)\right\| _\infty \; \left\| z_2 \right\| _{\mathbb {E}_{0,\mu }^0(J_T)}\, d\sigma \, d\tau \\&\quad \le \Phi (\Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} ) \Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)}. \end{aligned}$$

In the above, $\phi '(z)$ denotes the Frechét derivative of the Nemyskii operator induced by $\phi$ and we have used the fact that $\phi '(z) = \sum _{j=1}^{16} \partial _{j} \phi (z) \otimes e_j$, where $\partial _{j} \phi$ is the partial derivative of $\phi$. We will take advantage of this observation in the sequel. Note that the function $\Phi$ above is uniform with respect to $T\in (0,T_0]$ in view of Lemma 3.1(a). Estimating in the same way, we have

$$\begin{aligned}&\Vert \phi _1(z_1 +z_2) \left( \phi _2(\partial z_1 + \partial z_2) - \phi _2(\partial z_1 ) -\phi _2'(\partial z_1)\partial z_2 \right) \Vert _{\mathbb {E}_{0,\mu }^0(J_T)}\\&\quad \le \Phi (\Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} ) \Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} . \end{aligned}$$

The remaining terms can be estimated again by using the mean value theorem as follows:

$$\begin{aligned}&\Vert \left( \phi _1(z_1 +z_2) - \phi _1(z_1 ) \right) \phi _2'(\partial z_1)\partial z_2 \Vert _{\mathbb {E}_{0,\mu }^0(J_T)} \\&\quad \le \Vert \phi _2'(\partial z_1)\Vert _\infty \; \Vert \partial z_2\Vert _\infty \; \int \nolimits _0^1 \left( \Vert \phi _1'(z_1 + \sigma z_2)\Vert _\infty \; \Vert z_2 \Vert _{\mathbb {E}_{0,\mu }^0(J_T)} \right) \, d\sigma \\&\quad \le \Phi (\Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} ) \Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} . \end{aligned}$$

The estimate for $G_3$ and $G_5$ can be obtained in the same manner in view of the additional regularity of m.

The estimate for $G_2$ will be slightly different in the sense that we need to evaluate $\partial _{ij} z_k$, $k=1,2$, by using the $\mathbb {E}_{0,\mu }^0(J_T)$-norm. First, notice that

$$\begin{aligned}&G_2 (z_1+z_2) - G_2(z_1) - G_2'(z_1)z_2, \\&\quad = \left( \phi _1(z_1 +z_2) - \phi _1(z_1 ) - \phi _1'(z_1 )z_2 \right) \partial _{ij} z_1 + \left( \phi _1(z_1 +z_2) - \phi _1(z_1 ) \right) \partial _{ij} z_2 . \end{aligned}$$

Then,

$$\begin{aligned}&\Vert \left( \phi _1(z_1 +z_2) - \phi _1(z_1 ) - \phi _1'(z_1 )z_2 \right) \partial _{ij} z_1\Vert _{\mathbb {E}_{0,\mu }^0(J_T)} \\&\quad \le \Vert \partial _{ij} z_1\Vert _{\mathbb {E}_{0,\mu }^0(J_T)} \; \Vert z_2\Vert _\infty \; \int \nolimits _0^1 \left\| \phi _1'(z_1 +\sigma z_2) - \phi _1'(z_1) \right\| _\infty \, d\sigma \\&\quad \le \Phi (\Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} ) \Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} . \end{aligned}$$

Similarly,

$$\begin{aligned} \Vert \left( \phi _1(z_1 +z_2) - \phi _1(z_1 ) \right) \partial _{ij} z_2\Vert _{ \mathbb {E}_{0,\mu }^0(J_T) } \le \Phi (\Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} ) \Vert z_2\Vert _{\mathbb {E}_{1,\mu } (J_T)} . \end{aligned}$$

The estimate for $G_4$ can be derived in a similar way by utilizing the additional regularity of m and the facts that

$$\begin{aligned} \Vert G_6(z_1 + z_2) -G_6(z_1) -G_6'(z_1)z_2 \Vert _{C([0,T];C^1(\overline{\Omega }))}&\le \Phi (\Vert z_2\Vert _{C([0,T];C^1(\overline{\Omega }))})\\&\quad \Vert z_2\Vert _{C([0,T];C^1(\overline{\Omega }))} , \\ \Vert G_6(z_1 + z_2) -G_6(z_1) \Vert _{C([0,T];C^1(\overline{\Omega }))}&\le M \Vert z_2\Vert _{C([0,T];C^1(\overline{\Omega }))} , \end{aligned}$$

where $G_6(z)=\phi _1(z) \phi _3(\partial m)$. $\square$

We are now ready to establish the differentiability of $(\textsf{A},\textsf{B},\textsf{F})$ as operators defined on $\mathbb {E}_{1,\mu }(J_T)$.

Proposition B.3

Assume (2.1) and (3.2). Then,

$$\begin{aligned} \begin{aligned} \mathcal {A}&\in C^1(\mathbb {E}_{1,\mu }(J_T), \mathbb {E}_{0,\mu }(J_T)), \quad{} & {} \mathcal {A}^\prime (z_*)z= \textsf{A}(z_*)z + [\textsf{A}^\prime (z_*)z] z_*, \\ \textsf{F}&\in C^1(\mathbb {E}_{1,\mu }(J_T),\mathbb {E}_{0,\mu }(J_T))), \quad{} & {} \\ \mathcal {B}&\in C^1(\mathbb {E}_{1,\mu }(J_T), \mathbb {F}_\mu (J_T)), \quad{} & {} \mathcal {B}^\prime (z_*)z= \textsf{B}(z_*)z + [\textsf{B}^\prime (z_*)z] z_*, \end{aligned} \end{aligned}$$

for $z_*, z\in \mathbb {E}_{1,\mu }(J_T)$, where the mappings $(\mathcal {A}, \mathcal {B})$ were introduced in (4.12). Moreover, given $T_0, R_0 >0$, then for any $T\in (0,T_0]$, $R\in (0,R_0]$ and any $z_*\in \mathbb {E}_{1,\mu }(J_T)$, $z\in {}_{0}{\mathbb {E}}_{1,\mu }(J_T)$ satisfying

$$\begin{aligned} \Vert \textrm{tr}_{\partial \Omega } z_*\Vert _{\mathbb {F}_\mu (J_T)},\; \Vert \textrm{tr}_{\partial \Omega } \nabla z_*\Vert _{\mathbb {F}_\mu (J_T)}, \; \Vert z_* \Vert _{ \mathbb {B}_\mu (J_T) },\; \Vert z_*\Vert _{\mathbb {E}_{1,\mu }(J_T)},\; \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}\le R, \end{aligned}$$

the following estimates hold:

$$\begin{aligned} \begin{aligned} \Vert \mathcal {A}(z_* + z) - \mathcal {A}(z_*) -\mathcal {A}'(z_*)z \Vert _{\mathbb {E}_{0,\mu }(J_T)}&\le \Phi (\Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)} ) \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}, \\ \Vert \textsf{F}(z_* + z) - \textsf{F}(z_*) - \textsf{F}'(z_*) z \Vert _{\mathbb {E}_{0,\mu }(J_T)}&\le \Phi (\Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)} ) \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}, \\ \Vert \mathcal {B}(z_* + z) - \mathcal {B}(z_*) - \mathcal {B}'(z_*)z \Vert _{\mathbb {F}_\mu (J_T)}&\le \Phi (\Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)} ) \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}. \end{aligned} \end{aligned}$$

(B.5)

If, in addition, ${\overline{z}}\in \mathbb {E}_{1,\mu }(J_T)$ with $z_*(0)={\overline{z}}(0)$ satisfies

$$\begin{aligned} \Vert \textrm{tr}_{\partial \Omega } {\overline{z}}\Vert _{\mathbb {F}_\mu (J_T)}, \Vert \textrm{tr}_{\partial \Omega } \nabla {\overline{z}}\Vert _{\mathbb {F}_\mu (J_T)} , \Vert {\overline{z}}\Vert _{\mathbb {E}_{1,\mu }(J_T)}, \Vert {\overline{z}}\Vert _{\mathbb {B}_\mu (J_T)} \le R, \end{aligned}$$

then the following estimates hold

$$\begin{aligned} \begin{aligned} \Vert \mathcal {A}'(z_*) z - \mathcal {A}'({\overline{z}}) z \Vert _{\mathbb {E}_{0,\mu }(J_T)}&\le \Phi (\Vert z_* - {\overline{z}}\Vert _{\mathbb {E}_{1,\mu }(J_T)} ) \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}, \\ \Vert \textsf{F}'(z_*) z - \textsf{F}'({\overline{z}}) z \Vert _{\mathbb {E}_{0,\mu }(J_T)}&\le \Phi (\Vert z_* - {\overline{z}}\Vert _{\mathbb {E}_{1,\mu }(J_T)} ) \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}, \\ \Vert \mathcal {B}'(z_*) z - \mathcal {B}'({\overline{z}}) z \Vert _{\mathbb {F}_\mu (J_T)}&\le \Phi (\Vert z_*-{\overline{z}}\Vert _{\mathbb {E}_{1,\mu }(J_T)} ) \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}. \end{aligned} \end{aligned}$$

(B.6)

Proof

The continuous differentiability of $\mathcal {A}$ and $\textsf{F}$ and the first two estimates in (B.5) are immediate consequences of Lemma B.2. The continuous differentiability of $\mathcal {B}$ is a direct consequence of Lemma B.1 and the fact that $\mathbb {F}_\mu (J_T)$ is a Banach algebra. To establish the last estimate in (B.5), we set $z=(z_j)_{j=1}^{16} =(u,F,\theta ,m )$ and $z_*=(u_*,F_*,\theta _*,m_*)$. Then, we can apply (B.1), (B.2), Lemma A.5(i) and (ii), Proposition A.4, and [28, Theorem 4.5] to obtain (where we suppress $\textrm{tr}_{\partial \Omega }$ in the following computations)

$$\begin{aligned}&\Vert \mathcal {B}(z_*+z) - \mathcal {B}(z_*) - \mathcal {B}'(z_*)z \Vert _{\mathbb {F}_\mu (J_T)} \\&\quad \le \left\| \left[ \left( \int \nolimits _0^1 \left( K'(z_*+\sigma z ) - K'(z_*) \right) \, d\sigma \right) z \right] \nabla \theta _* \right\| _{\mathbb {F}_\mu (J_T)}\\&\qquad + \left\| \left[ \left( \int \nolimits _0^1 \left( K'(z_*+\sigma z ) \right) \, d\sigma \right) z \right] \nabla \theta \right\| _{\mathbb {F}_\mu (J_T)} \\&\quad \le C \left\| \int \nolimits _0^1 \left( K'(z_*+\sigma z ) - K'(z_*) \right) \, d\sigma \right\| _{\mathbb {F}_\mu (J_T)} \Vert \nabla \theta _* \Vert _{\mathbb {F}_\mu (J_T)} \; \Vert z\Vert _{\mathbb {F}_{1,\mu }(J_T)} \\&\qquad + C \left\| \int \nolimits _0^1 K'(z_*+\sigma z ) \, d\sigma \right\| _{\mathbb {F}_\mu (J_T)} \left\| \nabla \theta \right\| _{\mathbb {F}_\mu (J_T)}\; \Vert z\Vert _{\mathbb {F}_{1,\mu }(J_T)}\\&\quad \le \Phi ( \Vert z\Vert _{\mathbb {E}_{1,\mu } (J_T)} ) \Vert z\Vert _{\mathbb {E}_{1,\mu } (J_T)} . \end{aligned}$$

This establishes the last estimate in (B.5).

Concerning the estimates in (B.6), we will only establish the last one. The remaining two follow from a similar argument.

$$\begin{aligned} \Vert \mathcal {B}'(z_*) z - \mathcal {B}'({\overline{z}}) z \Vert _{\mathbb {F}_\mu (J_T)}&= \Vert \textsf{B}(z_* )z - \textsf{B}({\overline{z}}) z - [\textsf{B}'(z_*)z] z_* + [\textsf{B}'({\overline{z}})z] {\overline{z}} \Vert _{\mathbb {F}_\mu (J_T)} \\&\le \Vert \textsf{B}(z_* )z - \textsf{B}({\overline{z}}) z \Vert _{\mathbb {F}_\mu (J_T)} + \Vert [\textsf{B}'(z_*)z] z_* - [\textsf{B}'({\overline{z}})z] {\overline{z}} \Vert _{\mathbb {F}_\mu (J_T)}. \end{aligned}$$

Let ${\overline{z}}=({\overline{u}},{\overline{F}},\overline{\theta },{\overline{m}})$. Then, the first term on the right-hand side can be estimated by using (B.1), (B.2), Lemma A.5(i) and (ii), Proposition A.4, and [28, Theorems 4.2 and 4.5] as follows:

$$\begin{aligned} \Vert \textsf{B}(z_* )z - \textsf{B}({\overline{z}}) z \Vert _{\mathbb {F}_\mu (J_T)}&\le C \Vert K(z_*)- K({\overline{z}} ) \Vert _{\mathbb {F}_\mu (J_T) } \Vert \nabla \theta \Vert _{\mathbb {F}_\mu (J_T)} \\&\le C \int \nolimits _0^1 \Vert K'( \sigma z_* + (1-\sigma ) {\overline{z}} ) \Vert _{\mathbb {F}_\mu (J_T)}\, d\sigma \Vert z_* \\&\quad -{\overline{z}}\Vert _{\mathbb {F}_{1,\mu }(J_T)} \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}\\&\le \Phi (\Vert z_*-{\overline{z}}\Vert _{\mathbb {E}_{1,\mu }(J_T)} ) \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}. \end{aligned}$$

The estimate of the second term can be obtained analogously:

$$\begin{aligned}&\Vert [\textsf{B}'(z_*)z] z_* - [\textsf{B}'({\overline{z}})z] {\overline{z}} \Vert _{\mathbb {F}_\mu (J_T)} \\&\quad \le \Vert [ K '(z_*) z] \nabla (\theta _* -\overline{\theta }) \Vert _{\mathbb {F}_\mu (J_T)} + \Vert [ (K '(z_*) - K '({\overline{z}}) ) z ]\nabla \overline{\theta } \Vert _{\mathbb {F}_\mu (J_T)} \\&\quad \le C \left( \Vert K' (z_*) z\Vert _{\mathbb {F}_\mu (J_T)} \Vert \nabla (\theta _* -\overline{\theta }) \Vert _{\mathbb {F}_\mu (J_T)} + \sum _{j=1}^{16} \Vert \left( \partial _{j} K (z_*) - \partial _{j} K ({\overline{z}}) \right) \Vert _{\mathbb {F}_\mu (J_T)} \Vert z_j \nabla \overline{\theta } \Vert _{\mathbb {F}_\mu (J_T)} \right) \\&\quad \le C \left( \Vert K' (z_*)\Vert _{\mathbb {F}_\mu (J_T)} + \Phi (R) \Vert \nabla \overline{\theta }\Vert _{\mathbb {F}_\mu (J_T)} \right) \Vert z\Vert _{\mathbb {F}_{1,\mu }(J_T)} \Vert z_*-{\overline{z}}\Vert _{\mathbb {E}_{1,\mu }(J_T)}\\&\quad \le \Phi (\Vert z_*-{\overline{z}}\Vert _{\mathbb {E}_{1,\mu }(J_T)} ) \Vert z\Vert _{\mathbb {E}_{1,\mu }(J_T)}. \end{aligned}$$

$\square$

To study the continuous dependence of solutions to (3.7) on the initial data, see Theorem 5.1(b), we need the following result.

Lemma B.3

Let $\mathcal {B}$ be as in (4.12). Then, we have

• (a) $\mathcal {B}\in C^1(X_{\gamma , \mu }, Y_{\gamma ,\mu })\quad \text {and} \quad \mathcal {B}^\prime (z_0)z = \textsf{B}(z_0)z + [\textsf{B}^\prime (z_0)z]z_0,\quad z_0,z\in X_{\gamma ,\mu }.$

• (b) For each $z_0\in X_{\gamma ,\mu }$, $\mathcal {B}^\prime (z_0)\in \mathcal {L}( X_{\gamma ,\mu }, Y_{\gamma ,\mu } )$ has a bounded right inverse $\mathcal {R}(z_0)\in \mathcal {L}( Y_{\gamma ,\mu }, X_{\gamma ,\mu })$.

Proof

• (a) By Lemma 3.1, the trace operator $\gamma _0\in \mathcal {L}(\mathbb {E}_{1,\mu }(J_T),X_{\gamma ,\mu } )$ has a right inverse $\gamma _0^c \in \mathcal {L}(X_{\gamma ,\mu }, \mathbb {E}_{1,\mu }(J_T) )$. It is then easy to see that $\mathcal {B}(z)= \widetilde{\gamma _0}\mathcal {B}(\gamma ^c_0 (z))$, where $\widetilde{\gamma _0}$ denotes the initial time trace operator for functions defined on $\mathbb {F}_\mu (J_T)$. The assertions then follow from Proposition B.3.

• (b) The existence of $\mathcal {R}(z_0)$ is proved in [26, Proposition 2.5.1]. $\square$