Global Solutions to the Compressible Euler Equations with Heat Transport by Convection Around Dyson’s Isothermal Affine Solutions

Abstract

Global solutions to the compressible Euler equations with heat transport by convection in the whole space are shown to exist through perturbations of Dyson’s isothermal affine solutions (Dyson in J Math Mech 18(1):91–101, 1968). This setting presents new difficulties because of the vacuum at infinity behavior of the density. In particular, the perturbation of isothermal motion introduces a Gaussian function into our stability analysis and a novel finite propagation result is proven to handle potentially unbounded terms arising from the presence of the Gaussian. Crucial stabilization-in-time effects of the background motion are mitigated through the use of this finite propagation result however and a careful use of the heat transport formulation in conjunction with new time weight manipulations are used to establish global existence. The heat transport by convection offers unique physical insights into the model and mathematically, we use a controlled spatial perturbation in the analysis of this feature of our system which leads us to exploit source term estimates as part of our techniques.

Appendices

Energy Identities

We give key identities which will be used in our estimates. First from Lemma 4.3 [11] we have the following modified energy identity:

Lemma A.1

Recalling the matrix quantities introduced in (2.63) and (2.64), the following identity holds

$$\begin{aligned} \varLambda _{\ell j}(\nabla _\eta \partial ^\nu \theta )^i_j\varLambda ^{-1}_{im}(\nabla _\eta \partial ^\nu \theta )^m_{\ell ,\tau } = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\tau }\left( \sum _{i,j=1}^3 d_id_j^{-1}\big ([{\mathscr {N}}_\nu ]^j_{i}\big )^2\right) + {\mathcal {T}}_{\nu }. \nonumber \\ \end{aligned}$$

(A.239)

where the error term ${\mathcal {T}}_{\nu }$ is given as

$$\begin{aligned} {\mathcal {T}}_{\nu }= & {} - \frac{1}{2}\sum _{i,j=1}^3\frac{\mathrm{d}}{{\mathrm{d}\tau }}\left( d_id_j^{-1}\right) \big ([{\mathscr {N}}_\nu ]^j_i\big )^2 \nonumber \\&- \text { Tr}\left( Q{\mathscr {N}}_\nu Q^{-1}\left( \partial _\tau P P^\top {\mathscr {N}}_\nu ^\top + {\mathscr {N}}_\nu ^\top P\partial _\tau P^\top \right) \right) . \end{aligned}$$

(A.240)

In the next Lemma, we give some useful results concerning our quantities ${\mathscr {A}}$, ${\mathscr {J}}$ and $\varLambda$.

Lemma A.2

For ${\mathscr {A}}$, ${\mathscr {J}}$ and $\varLambda$, the following identities hold:

$$\begin{aligned} {\mathscr {A}}^k_j{\mathscr {J}}^{-\frac{1}{\alpha }} - \delta ^k_j&= \big ( {\mathscr {A}}^k_j-\delta ^k_j\big ) {\mathscr {J}}^{-\frac{1}{\alpha }} + \delta ^k_j \big ({\mathscr {J}}^{-\frac{1}{\alpha }} -1\big ), \end{aligned}$$

(A.241)

$$\begin{aligned} {\mathscr {A}}^k_j-\delta ^k_j&= -{\mathscr {A}}^k_l [D\theta ]^l_j, \end{aligned}$$

(A.242)

$$\begin{aligned} \varLambda _{ij}&=\varLambda _{ip}\big ({\mathscr {A}}^j_p + {\mathscr {A}}^j_l\theta ^l,_p\big ), \end{aligned}$$

(A.243)

$$\begin{aligned} 1-{\mathscr {J}}^{-\frac{1}{\alpha }}&=\frac{1}{\alpha }{\mathrm{Tr}}[D\theta ] + O(|D\theta |^2), \end{aligned}$$

(A.244)

$$\begin{aligned} {\mathscr {A}}_j^k {\mathscr {J}}^{-\frac{1}{\alpha }} -\delta _j^k&=-{\mathscr {A}}_{\ell }^k[D \theta ]^{\ell }_j {\mathscr {J}}^{-\frac{1}{\alpha }}-\delta _j^k\left( \frac{1}{\alpha }{\mathrm{Tr}}[D \theta ] + O(|D\theta |^2)\right) . \end{aligned}$$

(A.245)

Proof

First (A.241) is straightforward to verify by expanding the right-hand side. For (A.242), first note ${\mathscr {A}}=[D \eta ]^{-1}$ and $\eta =y+\theta$. We then have

$$\begin{aligned} {\mathscr {A}}^k_j-\delta ^k_j= & {} {\mathscr {A}}^k_l \delta ^l_j -{\mathscr {A}}^k_l [D\eta ]^l_j \nonumber \\= & {} {\mathscr {A}}^k_l \big (\delta ^l_j - [D y]^l_j - [D\theta ]^l_j\big ) = -{\mathscr {A}}^k_l [D\theta ]^l_j. \end{aligned}$$

(A.246)

Next, (A.243) is proven in the following calculation where we recall $A=[D \eta ]^{-1}$ and use $\eta =y+\theta$,

$$\begin{aligned} \varLambda _{ij}=\varLambda _{ip}\delta ^j_p = \varLambda _{ip} {\mathscr {A}}^j_l\eta ^l,_p=\varLambda _{ip}\big ({\mathscr {A}}^j_p + {\mathscr {A}}^j_l\theta ^l,_p\big ). \end{aligned}$$

(A.247)

Now (A.244) follows from the calculation

$$\begin{aligned} {\mathscr {J}}=\det [D \eta ] = \det [\mathbf{I }\mathbf{d } + D\theta ] = 1 + \text {Tr}[D \theta ] + O(|D\theta |^2). \end{aligned}$$

(A.248)

Finally (A.245) follows from (A.241)–(A.242) and (A.244). $\quad \square$

Time Based Inequalities

We have the following useful $\tau$ based inequalities, summarized by the Lemma below.

Lemma B.1

Fix an affine motion A(t) from the set ${\mathscr {S}}$ under consideration, namely require

$$\begin{aligned} \det A(t) \sim 1 + t^3, \quad t \ge 0. \end{aligned}$$

(B.249)

Let

$$\begin{aligned} \mu _1:=\lim _{\tau \rightarrow \infty } \frac{\mu _\tau (\tau )}{\mu (\tau )}, \quad \mu _0:=\frac{\sigma }{2}\mu _1. \end{aligned}$$

(B.250)

Then we have the following properties:

$$\begin{aligned}&0<\mu _0=\mu _0(\gamma )\leqq \mu _1, \end{aligned}$$

(B.251)

$$\begin{aligned}&e^{\mu _1\tau } \lesssim \ \mu (\tau ) \lesssim e^{\mu _1\tau }, \ \ \tau \geqq 0, \end{aligned}$$

(B.252)

(B.253)

$$\begin{aligned}&\Vert \varLambda _\tau \Vert \lesssim e^{-\mu _1 \tau }, \quad \Vert \varLambda \Vert + \Vert \varLambda ^{-1} \Vert \leqq C, \end{aligned}$$

(B.254)

$$\begin{aligned}&\sum _{i=1}^3\left( d_i+\frac{1}{d_i}\right) \leqq C \end{aligned}$$

(B.255)

$$\begin{aligned}&\sum _{i=1}^3|\partial _\tau d_i| + \Vert \partial _\tau P\Vert \lesssim e^{-\mu _1\tau } \end{aligned}$$

(B.256)

$$\begin{aligned}&|\mathbf{w}|^2 \lesssim \ \langle \varLambda ^{-1}{} \mathbf{w}, \mathbf{w}\rangle \lesssim |\mathbf{w}|^2 , \ \mathbf{w}\in {\mathbb {R}}^3, \end{aligned}$$

(B.257)

for $C > 0$.

Proof

The result (B.251) is clear from the definition of $\mu _0$. For inequalities (B.252) and (B.254) through (B.257) we first note that by Lemma 1.2, there exist matrices $A_0,A_1,M(t)$ such that

$$\begin{aligned} A(t) = A_0 + t A_1 + M(t), \quad t \ge 0. \end{aligned}$$

(B.258)

where $A_0,A_1$ are time-independent and M(t) satisfies the bounds

$$\begin{aligned} \Vert M(t)\Vert = o_{t \rightarrow \infty }(1+t), \ \ \Vert \partial _t M(t)\Vert \lesssim (1+t)^{3-3\gamma }. \end{aligned}$$

(B.259)

We also note $\det A(t) \sim 1 + t^3$. Then inequalities (B.252) and (B.254) through (B.257) follow from Lemma A.1 [11]. Finally, (B.253) follows from the definition of ${\mathcal {S}}^N$ (2.66) and properties (B.251)–(B.252) above. $\quad \square$

Local Well Posedness

We construct a local solution for our original Euler system (1.1)–(1.4) from generic initial data. First write the Equations (1.1)–(1.3) in terms of $(\rho ,{\mathbf {u}},T)$ as follows where we use the material derivative $\frac{\mathrm{D}}{\mathrm{D}t}=\partial _t + {\mathbf {u}} \cdot \nabla$ and we have multiplied by $\text {diag}(T^2, \rho ^2 T I_3,\alpha \rho ^2)$ where $I_3$ is the $3 \times 3$ identity matrix,

$$\begin{aligned}&T^2 \frac{\mathrm{D} \rho }{\mathrm{D} t} + \rho T^2 \,\text {div}({\mathbf {u}}) = 0 \end{aligned}$$

(C.260)

$$\begin{aligned}&\rho ^2 T \frac{\mathrm{D} {\mathbf {u}}}{\mathrm{D} t} + \rho T^2 \, \nabla \rho + \rho ^2 T \, \nabla T = 0 \end{aligned}$$

(C.261)

$$\begin{aligned}&\alpha \rho ^2 \frac{\mathrm{D} T}{\mathrm{D} t} + \rho ^2 T \, \text {div}({\mathbf {u}}) = 0 \end{aligned}$$

(C.262)

Let $(*)$ denote the symmetric $5 \times 5$ system of Equations (C.260)–(C.262).

We first note a fixed affine solution $(\rho _A,{\mathbf {u}}_A,T_A)$ solves $(*)$ with initial data

$$\begin{aligned} ((\rho _A)_0,({\mathbf {u}}_A)_0,(T_A)_0)=(\rho _A(0,\cdot ),{\mathbf {u}}_A(0,\cdot ),T_A(0,\cdot )). \end{aligned}$$

(C.263)

Next we construct initial data, using generic initial data $(\rho _0,{\mathbf {u}}_0,T_0)$ and our affine initial data $((\rho _A)_0,({\mathbf {u}}_A)_0,(T_A)_0)$, for a modified system which allows to us to avoid the fact that we want to have $\rho _0 \rightarrow 0$ for large x. To this end choose $(\sigma _0^{\rho },\sigma _0^{{\mathbf {u}}},\sigma _0^{T})$ as follows:

$$\begin{aligned} \sigma _0^{\rho }&=\varphi (\rho _0-C_1) + (1-\varphi )\psi ((\rho _A)_0-C_1)+(1-\psi ) \rho _0, \end{aligned}$$

(C.264)

$$\begin{aligned} \sigma _0^{{\mathbf {u}}}&=\varphi {\mathbf {u}}_0 + (1-\varphi ) ({\mathbf {u}}_A)_0, \ \end{aligned}$$

(C.265)

$$\begin{aligned} \sigma _0^{T}&=\varphi T_0 + (1-\varphi ) (T_A)_0, \end{aligned}$$

(C.266)

where $\varphi , \psi \in C_c^\infty ({\mathbf {R}}^3)$ are such that, for fixed $R>0$ and $\eta >0$, $\varphi =1 \text { on } B(0,\frac{R}{2})$, $\varphi = 0 \text { on } {\mathbb {R}}^3 \setminus B(0,R)$, $\psi =1 \text { on } B(0,R+2\eta )$ and $\psi = 0 \text { on } {\mathbb {R}}^3 \setminus B(0,R+3\eta )$, and $C_1 >0$ is such that $\Vert \rho _0 \Vert _{L^\infty ({\mathbb {R}}^3)} \le \frac{C_1}{2}$. Note the existence of $C_1>0$ will be guaranteed through the regularity of $\rho _0$. Then by construction we will have $\inf _{{\mathbb {R}}^3} \{ \sigma _0^{\rho } + C_1 \} > 0$. Now we consider $(\sigma _0^{\rho },\sigma _0^{{\mathbf {u}}},\sigma _0^{T})$ as initial data for the modified system

$$\begin{aligned} T^2 \frac{\mathrm{D} \rho }{\mathrm{D} t} + (\rho +C_1) T^2 \,\text {div}({\mathbf {u}})&= 0 \end{aligned}$$

(C.267)

$$\begin{aligned} (\rho +C_1)^2 T \frac{\mathrm{D} {\mathbf {u}}}{\mathrm{D} t} + (\rho +C_1) T^2 \, \nabla \rho + (\rho +C_1)^2 T \, \nabla T&= 0 \end{aligned}$$

(C.268)

$$\begin{aligned} \alpha (\rho +C_1)^2 \frac{\mathrm{D} T}{\mathrm{D} t} + (\rho +C_1)^2 T \, \text {div}({\mathbf {u}})&= 0 \end{aligned}$$

(C.269)

Let $(\sigma \text {-} *)$ denote the symmetric $5 \times 5$ system of Equations (C.267)–(C.269). Then, with sufficiently regular $(\rho _0,{\mathbf {u}}_0,T_0)$ which will be specified by the Lagrangian formulation, by Theorem II [14] there exists ${\hat{T}} > 0$ such that $(\sigma ^{\rho },\sigma ^{{\mathbf {u}}},\sigma ^{T})$ is a solution to $(\sigma \text {-}*)$ with initial data $(\sigma _0^{\rho },\sigma _0^{{\mathbf {u}}},\sigma _0^{T})$. Now let

$$\begin{aligned} (\rho _B,{\mathbf {u}}_B,T_B)=\big (\sigma ^{\rho }+C_1,\sigma ^{{\mathbf {u}}},\sigma ^{T}\big ). \end{aligned}$$

(C.270)

Since $(\sigma ^{\rho },\sigma ^{{\mathbf {u}}},\sigma ^{T})$ solve $(\sigma \text {-} *)$ we have that $(\rho _B,{\mathbf {u}}_B,T_B)$ solve $(*)$ with initial data $((\rho _B)_0,({\mathbf {u}}_B)_0,(T_B)_0)=(\sigma _0^\rho +C_1,\sigma _0^{{\mathbf {u}}},\sigma _0^T)$.

Next let

$$\begin{aligned} K&=\{(x,t) \, | \, 0 \le t \le T^1, x \in B(0,R+\eta +Mt) \} \nonumber \\ M&=\tfrac{3}{c} \left( \sup _{0 \le t \le T^1-\kappa } \{ \Vert \rho _B (T_B)^2 \Vert _{L^\infty (B(0,R+2\eta ))} + \Vert (\rho _B)^2 T_B {\mathbf {u}}_B \Vert _{L^\infty (B(0,R+2\eta ))}\right. \nonumber \\&\quad + \Vert (\rho _{B})^2 T_{B} \Vert _{L^\infty (B(0,R+2\eta ))} + \Vert \rho _A (T_A)^2 \Vert _{L^\infty (B(0,R+2\eta ))}\nonumber \\&\quad \left. + \Vert (\rho _A)^2 T_A {\mathbf {u}}_A \Vert _{L^\infty (B(0,R+2\eta ))} + \Vert (\rho _A)^2 T_A \Vert _{L^\infty (B(0,R+2\eta ))} \}\right) \nonumber \\ c&>0 \text { is such that } c \le \sup _{0 \le t \le T^*-\kappa } \{ \Vert (T_{B})^2 \Vert _{L^\infty ({\mathbb {R}}^3)} + \Vert (\rho _{B})^2 T_{B} \Vert _{L^\infty ({\mathbb {R}}^3)} + \Vert \alpha (\rho _{B})^2 \Vert _{L^\infty ({\mathbb {R}}^3)} \}, \nonumber \\ T^1&=\min \left( {\hat{T}}-\kappa , \frac{\eta }{2 M} - \kappa \right) , \nonumber \\ \kappa&> 0 \text { is sufficiently small.} \end{aligned}$$

(C.271)

Now we take

$$\begin{aligned} (\rho ,{\mathbf {u}},T)={\left\{ \begin{array}{ll} (\rho _{B},{\mathbf {u}}_{B},T_{B}) &{} \quad \text {in } K, \\ (\rho _A,{\mathbf {u}}_A,T_A) &{}\quad \text {outside } K. \end{array}\right. } \end{aligned}$$

(C.272)

Then $(\rho ,{\mathbf {u}},T)$ is a solution of $(*)$ on ${\mathbb {R}}^3 \times [0,T^1]$ with the following initial data

$$\begin{aligned} \rho _0&=\varphi (\rho _B)_0+ (1-\varphi ) (\rho _A)_0 , \nonumber \\ {\mathbf {u}}_0&=\varphi ({\mathbf {u}}_B)_0 + (1-\varphi ) ({\mathbf {u}}_A)_0, \nonumber \\ T_0&=\varphi (T_B)_0 + (1-\varphi )(T_A)_0, \end{aligned}$$

(C.273)

since $(\rho ,{\mathbf {u}},T)$ is a solution in K and outside K, and applying a classical property of local uniqueness of solutions to $(*)$ to get that $(\rho ,{\mathbf {u}},T)$ is continuous across $\partial K$.

Curl Equations Derivation

Here we give the derivations of the equations satisfied by the modified curl of our velocity and perturbation which will be used for the purpose of our estimates.

Lemma D.1

Let $(\theta , \mathbf{V}):\varOmega \rightarrow {\mathbb {R}}^3\times {\mathbb {R}}^3$ be a unique local solution to (2.55)–(2.56) on $[0,T^*]$ for $T^*>0$ fixed. Then for all $\tau \in [0,T]$ the curl matrices $\text {Curl}_{\varLambda {\mathscr {A}}}{} \mathbf{V}$ and $\text {Curl}_{\varLambda {\mathscr {A}}}{\theta }$ satisfy the equations

$$\begin{aligned}&{\mathrm{Curl}}_{\varLambda {\mathscr {A}}}{} \mathbf{V} = \frac{1}{1+\alpha }\varLambda {\mathscr {A}} y \times {\mathbf {V}} + \frac{\alpha }{(1+\alpha )(1+\beta )} \varLambda {\mathscr {A}} \nabla \beta \times {\mathbf {V}} \nonumber \\&\quad + \frac{\mu (0) {\mathrm{Curl}}_{\varLambda {\mathscr {A}}} (\mathbf{V}(0))}{\mu } - \frac{\mu (0) \varLambda {\mathscr {A}} y \times {\mathbf {V}}(0)}{(1+\alpha ) \mu } - \frac{\alpha \mu (0) \varLambda {\mathscr {A}} \nabla \beta \times {\mathbf {V}}(0)}{(1+\alpha )(1+\beta ) \mu } \nonumber \\&\quad + \frac{1}{\mu }\int _0^{\tau } \mu [\partial _{\tau }, {\mathrm{Curl}}_{\varLambda {\mathscr {A}}}] \mathbf{V} \mathrm{d}\tau ' - \frac{1}{(1+\alpha ) \mu } \int _0^{\tau } \mu [\partial _{\tau },\varLambda {\mathscr {A}} y \times ] {\mathbf {V}} \,\mathrm{d} \tau ' \nonumber \\&\quad - \frac{\alpha }{(1+\alpha )(1+\beta ) \mu } \int _0^{\tau } \mu [\partial _{\tau },\varLambda {\mathscr {A}} \nabla \beta \times ] {\mathbf {V}} \,\mathrm{d} \tau ' \nonumber \\&\quad - \frac{2}{\mu } \int _0^{\tau } \mu \, {\mathrm{Curl}}_{\varLambda {\mathscr {A}}}(\varGamma ^*\mathbf{V}) \mathrm{d}\tau ' + \frac{2}{(1+\alpha ) \mu } \int _0^{\tau } \mu \, \varLambda {\mathscr {A}} y \times (\varGamma ^*{\mathbf {V}}) \,\mathrm{d} \tau ' \nonumber \\&\quad + \frac{2 \alpha }{(1+\alpha )(1+\beta ) \mu } \int _0^{\tau } \mu \, \varLambda {\mathscr {A}} \nabla \beta \times (\varGamma ^*{\mathbf {V}}) \,\mathrm{d} \tau ' \nonumber \\&\quad +\frac{{\overline{C}}}{(1+\alpha ) \mu }\int _0^{\tau } \mu ^{1-\delta -\sigma } \varLambda y \times \varLambda \theta \,\mathrm{d} \tau ' -\frac{{\overline{C}}}{(1+\alpha ) \mu } \int _0^{\tau } \mu ^{1-\delta -\sigma } \varLambda {\mathscr {A}}[D \theta ]y \times \varLambda \eta \,\mathrm{d} \tau ' \nonumber \\&\quad +\frac{\alpha {\overline{C}}}{(1+\alpha )(1+\beta ) \mu }\int _0^{\tau } \mu ^{1-\delta -\sigma } \varLambda {\mathscr {A}} \nabla \beta \times \varLambda \theta \,\mathrm{d} \tau ' \nonumber \\&\quad -\frac{\alpha {\overline{C}}}{(1+\alpha )(1+\beta ) \mu }\int _0^{\tau } \mu ^{1-\delta -\sigma } \varLambda {\mathscr {A}} \nabla \beta \times \varLambda y \,\mathrm{d} \tau '. \end{aligned}$$

(D.274)

and

$$\begin{aligned}&{\mathrm{Curl}}_{\varLambda {\mathscr {A}}}{\theta } = {\mathrm{Curl}}_{\varLambda {\mathscr {A}}}([\theta (0)]) \nonumber \\&\quad +\mu (0) {\mathrm{Curl}}_{\varLambda {\mathscr {A}}} (\mathbf{V}(0)) \int _0^\tau \frac{1}{\mu (\tau ')} \,\mathrm{d} \tau ' - \frac{\mu (0) \varLambda {\mathscr {A}} y \times {\mathbf {V}}(0)}{1+\alpha } \int _0^\tau \frac{1}{\mu (\tau ')} \,\mathrm{d} \tau ' \nonumber \\&\quad - \frac{\alpha \mu (0) \varLambda {\mathscr {A}} \nabla \beta \times {\mathbf {V}}(0)}{(1+\alpha )(1+\beta )}\int _0^{\tau } \frac{1}{\mu (\tau ')} \,\mathrm{d} \tau ' + \int _0^\tau [\partial _{\tau }, {\mathrm{Curl}}_{\varLambda {\mathscr {A}}}] {\theta } \,\mathrm{d}\tau ' \nonumber \\&\quad + \frac{1}{1+\alpha } \int _0^\tau \varLambda {\mathscr {A}} y \times {\mathbf {V}} \mathrm{d}\tau ' + \frac{\alpha }{(1+\alpha )(1+\beta )} \int _0^\tau \varLambda {\mathscr {A}} \nabla \beta \times {\mathbf {V}} \mathrm{d}\tau ' \nonumber \\&\quad +\int _0^\tau \frac{1}{\mu (\tau ')} \int _0^{\tau '} \mu (\tau '') [\partial _{\tau }, {\mathrm{Curl}}_{\varLambda {\mathscr {A}}}] \mathbf{V} \,\mathrm{d}\tau '' \,\mathrm{d} \tau ' \nonumber \\&\quad -\frac{1}{1+\alpha } \int _0^{\tau } \frac{1}{\mu (\tau ')} \int _0^{\tau '} \mu (\tau '') [\partial _{\tau },\varLambda {\mathscr {A}} y \times ] {\mathbf {V}} \,\mathrm{d}\tau '' \,\mathrm{d} \tau ' \nonumber \\&\quad - \frac{\alpha }{(1+\alpha )(1+\beta )} \int _0^{\tau } \frac{1}{\mu (\tau ')} \int _0^{\tau '} \mu (\tau '') [\partial _{\tau },\varLambda {\mathscr {A}} \nabla \beta \times ] {\mathbf {V}} \,\mathrm{d} \tau '' \nonumber \\&\quad - \int _0^\tau \frac{2}{\mu (\tau ')} \int _0^{\tau '} \mu (\tau '') \, {\mathrm{Curl}}_{\varLambda {\mathscr {A}}}(\varGamma ^*\mathbf{V}) \,\mathrm{d}\tau '' \,\mathrm{d} \tau ' \nonumber \\&\quad + \frac{2}{1+\alpha } \int _0^{\tau } \frac{1}{\mu (\tau ')} \int _0^{\tau '} \mu (\tau '') \, \varLambda {\mathscr {A}} x \times (\varGamma ^*{\mathbf {V}}) \,\mathrm{d}\tau '' \,\mathrm{d} \tau ' \nonumber \\&\quad + \frac{2 \alpha }{(1+\alpha )(1+\beta )} \int _0^{\tau } \frac{1}{\mu (\tau ')} \int _0^{\tau '} \mu (\tau '') \, \varLambda {\mathscr {A}} \nabla \beta \times (\varGamma ^*{\mathbf {V}}) \,\mathrm{d}\tau '' \,\mathrm{d} \tau ' \nonumber \\&\quad +\frac{{\overline{C}}}{1+\alpha } \int _0^\tau \frac{1}{\mu (\tau ')} \int _0^{\tau '} \mu (\tau '')^{1-\delta -\sigma } \varLambda x \times \varLambda \theta \,\mathrm{d} \tau '' \,\mathrm{d} \tau ' \nonumber \\&\quad -\frac{{\overline{C}}}{1+\alpha } \int _0^\tau \frac{1}{\mu (\tau ')} \int _0^{\tau '} \mu (\tau '')^{1-\delta -\sigma } \varLambda {\mathscr {A}}[D \theta ]y \times \varLambda \eta \,\mathrm{d} \tau '' \,\mathrm{d} \tau ' \nonumber \\&\quad +\frac{\alpha {\overline{C}}}{(1+\alpha )(1+\beta )} \int _0^\tau \frac{1}{\mu } \int _0^{\tau '} \mu ^{1-\delta -\sigma } \varLambda {\mathscr {A}} \nabla \beta \times \varLambda \theta \,\mathrm{d} \tau '' \,\mathrm{d}\tau ' \nonumber \\&\quad -\frac{\alpha {\overline{C}}}{(1+\alpha )(1+\beta )} \int _0^{\tau } \frac{1}{\mu } \int _0^{\tau '} \mu ^{1-\delta -\sigma } \varLambda {\mathscr {A}} \nabla \beta \times \varLambda y \,\mathrm{d} \tau '' \,\mathrm{d}\tau '. \end{aligned}$$

(D.275)

Proof

Writing (2.55) without the source term outside the nonlinearity, we have

$$\begin{aligned}&\mu ^{\sigma } \partial _{\tau \tau } \theta _i + \mu _{\tau } \mu ^{-1+\sigma } \partial _\tau \theta _i + 2 \mu ^{\sigma } \varGamma ^*_{ij} \partial _{\tau } \theta _j + {\overline{C}} \mu ^{-\delta } \varLambda _{i \ell } \theta _\ell \nonumber \\&\quad + \frac{{\overline{C}} \mu ^{-\delta }}{w} \big (w \varLambda _{ij} \big ( (1+\beta ) {\mathscr {A}}_j^k {\mathscr {J}}^{-\frac{1}{\alpha }} - \delta _j^k\big )\big )_{,k}=0. \end{aligned}$$

(D.276)

Returning back to $\eta$ via $\eta =\theta +y$,

$$\begin{aligned}&\mu ^{\sigma } \partial _{\tau \tau } \eta _i + \mu _{\tau } \mu ^{-1+\sigma } \partial _\tau \eta _i + 2 \mu ^{\sigma } \varGamma ^*_{ij} \partial _{\tau } \eta _j + {\overline{C}} \mu ^{-\delta } \varLambda _{i \ell } \eta _\ell \nonumber \\&\quad + \frac{{\overline{C}} \mu ^{-\delta }}{w} \big (w \varLambda _{ij} (1+\beta ) {\mathscr {A}}_j^k {\mathscr {J}}^{-\frac{1}{\alpha }}\big )_{,k}=0. \end{aligned}$$

(D.277)

Multiply by $w^{\frac{1}{1+\alpha }}(1+\beta )^{-\frac{\alpha }{1+\alpha }}$ to get

$$\begin{aligned}&w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }} \big (\mu ^{\sigma } \partial _{\tau \tau } \eta _i + \mu _{\tau } \mu ^{-1+\sigma } \partial _\tau \eta _i + 2 \mu ^{\sigma } \varGamma ^*_{ij} \partial _{\tau } \eta _j + {\overline{C}} \mu ^{-\delta } \varLambda _{i \ell } \eta _\ell \big ) \nonumber \\&\quad + w^{\frac{1}{1+\alpha }-1} (1+\beta )^{-\frac{\alpha }{1+\alpha }} {\overline{C}} \mu ^{-\delta } \big (w \varLambda _{ij} {\mathscr {A}}^k_j {\mathscr {J}}^{-\frac{1}{\alpha }}\big ),_k=0 \end{aligned}$$

(D.278)

Note that

$$\begin{aligned}&w^{\frac{1}{1+\alpha }-1} (1+\beta )^{-\frac{\alpha }{1+\alpha }} {\overline{C}} \mu ^{-\delta } \big (w \varLambda _{ij} {\mathscr {A}}^k_j {\mathscr {J}}^{-\frac{1}{\alpha }}\big ),_k \nonumber \\&\quad = {\overline{C}} \mu ^{-\delta } (1+\alpha ) \varLambda _{i j} {\mathscr {A}}^k_j \big (w^{\frac{1}{1+\alpha }} (1+\beta )^{\frac{1}{1+\alpha }} {\mathscr {J}}^{-\frac{1}{\alpha }}\big ),_k. \end{aligned}$$

(D.279)

Moving away from coordinates we then have

$$\begin{aligned}&w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }} \big (\partial _{\tau \tau } \theta + \mu _{\tau } \mu ^{-1+\sigma } \theta + 2 \mu ^{\sigma } \varGamma ^* \partial _{\tau } \theta + {\overline{C}} \mu ^{-\delta } \varLambda \eta \big ) \nonumber \\&\quad + {\overline{C}} \mu ^{-\delta } (1+\alpha ) \varLambda {\mathscr {A}}^T \nabla \big (w^{\frac{1}{1+\alpha }} (1+\beta )^{\frac{1}{1+\alpha }} {\mathscr {J}}^{-\frac{1}{\alpha }}\big )=0. \end{aligned}$$

(D.280)

Note that

$$\begin{aligned}&{\overline{C}} \mu ^{-\delta } (1+\alpha ) \varLambda {\mathscr {A}}^T \nabla \big (w^{\frac{1}{1+\alpha }} (1+\beta )^{\frac{1}{1+\alpha }} {\mathscr {J}}^{-\frac{1}{\alpha }}\big )\nonumber \\&\quad ={\overline{C}} \mu ^{-\delta } (1+\alpha ) \varLambda \nabla _{\eta } \big (w^{\frac{1}{1+\alpha }} (1+\beta )^{\frac{1}{1+\alpha }} {\mathscr {J}}^{-\frac{1}{\alpha }}\big ). \end{aligned}$$

(D.281)

Since $\text {Curl}_{\varLambda {\mathscr {A}}} (\varLambda \nabla _\eta f) =0$, apply $\text {Curl}_{\varLambda {\mathscr {A}}}$ to (D.280),

$$\begin{aligned}&\text {Curl}_{\varLambda {\mathscr {A}}} \left( w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }} \big (\mu ^\sigma \partial _{\tau \tau } \theta \nonumber \right. \\&\left. \quad + \mu ^{-1+\sigma } \mu _{\tau } \partial _\tau \theta + 2 \mu ^\sigma \varGamma ^{*} \partial _\tau \theta + {\overline{C}} \mu ^{-\delta } \varLambda \eta ) \right) =0. \end{aligned}$$

(D.282)

Now note that

$$\begin{aligned}&\text {Curl}_{\varLambda {\mathscr {A}}}\big (w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }} {\mathbf {F}}\big )=w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }} \, \text {Curl}_{\varLambda {\mathscr {A}}} {\mathbf {F}} \nonumber \\&\quad + \varLambda {\mathscr {A}} \nabla \big (w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }}\big ) \times {\mathbf {F}}. \end{aligned}$$

(D.283)

Also,

$$\begin{aligned} \big (w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }}\big ),_s&= \frac{1}{1+\alpha } w^{\frac{1}{1+\alpha }-1} w,_s (1+\beta )^{-\frac{\alpha }{1+\alpha }} -\frac{\alpha }{1+\alpha } (1+\beta )^{-\frac{\alpha }{1+\alpha }-1} \beta ,_s w^{\frac{1}{1+\alpha }} \nonumber \\&= -\frac{1}{1+\alpha } w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }} y_s -\frac{\alpha }{1+\alpha } (1+\beta )^{-\frac{1+2\alpha }{1+\alpha }} \beta ,_s w^{\frac{1}{1+\alpha }}, \end{aligned}$$

(D.284)

since $w,_s = - y_s w$. So

$$\begin{aligned}&\varLambda {\mathscr {A}} \nabla \big (w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }}\big ) \times {\mathbf {F}} = - \frac{w^{\frac{1}{1+\alpha }} (1+\beta )^{-\frac{\alpha }{1+\alpha }}}{1+\alpha } \varLambda {\mathscr {A}} y \times {\mathbf {F}} \nonumber \\&\quad - \frac{\alpha (1+\beta )^{-\frac{1+2\alpha }{1+\alpha }} w^{\frac{1}{1+\alpha }}}{1+\alpha } \varLambda {\mathscr {A}} \nabla \beta \times {\mathbf {F}}, \end{aligned}$$

(D.285)

where

$$\begin{aligned} \big [\varLambda {\mathscr {A}} y \times {\mathbf {F}}\big ]^i_j := \varLambda _{jm} {\mathscr {A}}^s_m x_s {\mathbf {F}}^i - \varLambda _{im} {\mathscr {A}}^s_m y_s {\mathbf {F}}^j. \end{aligned}$$

(D.286)

Thus, multiplying (D.282) by $w^{-\frac{1}{1+\alpha }}(1+\beta )^{\frac{\alpha }{1+\alpha }}$ we have and using $\text {Curl}_{\varLambda {\mathscr {A}}} (\varLambda \eta )=0$ we have

$$\begin{aligned}&\mu ^\sigma \text {Curl}_{\varLambda {\mathscr {A}}} (\partial _{\tau \tau } \theta ) + \mu ^{-1+\sigma } \mu _{\tau } \text {Curl}_{\varLambda {\mathscr {A}}}(\partial _\tau \theta ) + 2 \mu ^\sigma \text {Curl}_{\varLambda {\mathscr {A}}}(\varGamma ^{*} \partial _\tau \theta ) \nonumber \\&\quad -\frac{1}{1+\alpha } \varLambda {\mathscr {A}} x \times \left( \mu ^\sigma \partial _{\tau \tau } \theta + \mu ^{-1+\sigma } \mu _{\tau } \partial _\tau \theta + 2 \mu ^\sigma \varGamma ^{*} \partial _\tau \theta + {\overline{C}} \mu ^{-\delta } \varLambda \eta \right) \nonumber \\&\quad -\frac{\alpha }{(1+\alpha )(1+\beta )} \varLambda {\mathscr {A}} \nabla \beta \times \left( \mu ^\sigma \partial _{\tau \tau } \theta + \mu ^{-1+\sigma } \mu _{\tau } \partial _\tau \theta + 2 \mu ^\sigma \varGamma ^{*} \partial _\tau \theta + {\overline{C}} \mu ^{-\delta } \varLambda \eta \right) =0. \end{aligned}$$

(D.287)

Divide by $\mu ^{\sigma }$ to get

$$\begin{aligned}&\text {Curl}_{\varLambda {\mathscr {A}}} (\partial _{\tau \tau } \theta ) + \mu ^{-1} \mu _{\tau } \text {Curl}_{\varLambda {\mathscr {A}}}(\partial _\tau \theta ) + 2 \text {Curl}_{\varLambda {\mathscr {A}}}(\varGamma ^{*} \partial _\tau \theta ) \nonumber \\&\quad -\frac{1}{1+\alpha } \varLambda {\mathscr {A}} x \times \left( \partial _{\tau \tau } \theta + \mu ^{-1} \mu _{\tau } \partial _\tau \theta + 2 \varGamma ^{*} \partial _\tau \theta + {\overline{C}} \mu ^{-\delta -\sigma } \varLambda \eta \right) \nonumber \\&\quad -\frac{\alpha }{(1+\alpha )(1+\beta )} \varLambda {\mathscr {A}} \nabla \beta \times \left( \partial _{\tau \tau } \theta + \mu ^{-1} \mu _{\tau } \partial _\tau \theta + 2 \varGamma ^{*} \partial _\tau \theta + {\overline{C}} \mu ^{-\delta -\sigma } \varLambda \eta \right) =0. \end{aligned}$$

(D.288)

Note that

$$\begin{aligned} \text {Curl}_{\varLambda {\mathscr {A}}} \left( {\mathbf {V}}_ \tau \right) =\partial _\tau \left( \, \text {Curl}_{\varLambda {\mathscr {A}}}\left( \mathbf{V}\right) \right) -[\partial _\tau , \text {Curl}_{\varLambda {\mathscr {A}}}] \left( \mathbf{V}\right) , \end{aligned}$$

(D.289)

where

$$\begin{aligned}{}[\partial _\tau , \text {Curl}_{\varLambda {\mathscr {A}}}] {\mathbf {F}}^i_j := \partial _\tau \left( \varLambda _{jm}{\mathscr {A}}^s_m\right) {\mathbf {F}},_s^i - \partial _\tau \left( \varLambda _{im}{\mathscr {A}}^s_m\right) {\mathbf {F}},_s^j. \end{aligned}$$

(D.290)

Then

$$\begin{aligned}&\partial _\tau \left( \mu \, \text {Curl}_{\varLambda {\mathscr {A}}}\left( \mathbf{V}\right) \right) = \mu [\partial _\tau , \text {Curl}_{\varLambda {\mathscr {A}}}] \left( \mathbf{V}\right) - 2 \mu \, \text {Curl}_{\varLambda {\mathscr {A}}}\left( \varGamma ^*\mathbf{V}\right) \nonumber \\&\quad +\frac{1}{1+\alpha } \varLambda {\mathscr {A}} y \times \left( \mu \partial _{\tau \tau } \theta + \mu _{\tau } \partial _\tau \theta + 2 \mu \varGamma ^{*} \partial _\tau \theta + {\overline{C}} \mu ^{1-\delta -\sigma } \varLambda \eta \right) \nonumber \\&\quad +\frac{\alpha }{(1+\alpha )(1+\beta )} \varLambda {\mathscr {A}} \nabla \beta \times \left( \mu \partial _{\tau \tau } \theta + \mu _{\tau } \partial _\tau \theta + 2 \mu \varGamma ^{*} \partial _\tau \theta + {\overline{C}} \mu ^{1-\delta -\sigma } \varLambda \eta \right) . \end{aligned}$$

(D.291)

Integrate from 0 to $\tau '$, where $\tau ' \in [0,\tau ]$, to that

$$\begin{aligned} \text {Curl}_{\varLambda {\mathscr {A}}}\left( \mathbf{V}\right)&= \frac{\mu (0) \text {Curl}_{\varLambda {\mathscr {A}}} \left( [\mathbf{V}(0)]\right) }{\mu }+\frac{1}{\mu }\int _0^{\tau '} \mu [\partial _{\tau }, \text {Curl}_{\varLambda {\mathscr {A}}}] \left( \mathbf{V}\right) \mathrm{d}\tau '' \nonumber \\&\quad - \frac{2}{\mu } \int _0^{\tau '} \mu \, \text {Curl}_{\varLambda {\mathscr {A}}}\left( \varGamma ^*\mathbf{V}\right) \mathrm{d}\tau '' \nonumber \\&\quad +\frac{1}{(1+\alpha ) \mu } \int _0^{\tau '} \varLambda {\mathscr {A}} x \nonumber \\&\quad \times \left( \mu \partial _{\tau \tau } \theta + \mu _{\tau } \partial _{\tau } \theta + 2 \mu \varGamma ^{*} \partial _{\tau } \theta + {\overline{C}} \mu ^{1-\delta -\sigma } \varLambda \eta \right) \,\mathrm{d} \tau '' \nonumber \\&\quad +\frac{\alpha }{(1+\alpha )(1+\beta )\mu } \int _0^{\tau '} \varLambda {\mathscr {A}} \nabla \beta \nonumber \\&\quad \times \left( \mu \partial _{\tau \tau } \theta + \mu _{\tau } \partial _{\tau } \theta + 2 \mu \varGamma ^{*} \partial _{\tau } \theta + {\overline{C}} \mu ^{1-\delta -\sigma } \varLambda \eta \right) \,\mathrm{d} \tau ''. \end{aligned}$$

(D.292)

Note that

$$\begin{aligned} \mu (\varLambda {\mathscr {A}} x \times \partial _{\tau \tau } \theta )= & {} \partial _{\tau } ( \mu \varLambda {\mathscr {A}} x \times \partial _{\tau } \theta ) - \mu _{\tau } (\varLambda {\mathscr {A}} x \times \partial _{\tau } \theta )\nonumber \\&- \mu [\partial _{\tau },\varLambda {\mathscr {A}} x \times ]\partial _{\tau } \theta , \end{aligned}$$

(D.293)

where

$$\begin{aligned}{}[\partial _{\tau },\varLambda {\mathscr {A}} x \times ] {\mathbf {F}}^i_j := \partial _{\tau } (\varLambda _{jm}{\mathscr {A}}^s_m)x_s {\mathbf {F}}^i - \partial _{\tau }(\varLambda _{im} {\mathscr {A}}^s_m)y_s{\mathbf {F}}^j. \end{aligned}$$

(D.294)

Using a similar result for $\mu (\varLambda {\mathscr {A}} \nabla \beta \times \partial _{\tau \tau } \theta )$, we have

$$\begin{aligned}&\text {Curl}_{\varLambda {\mathscr {A}}}{} \mathbf{V} = \frac{1}{1+\alpha }\varLambda {\mathscr {A}} y \times {\mathbf {V}} + \frac{\alpha }{(1+\alpha )(1+\beta )} \varLambda {\mathscr {A}} \nabla \beta \times {\mathbf {V}} \nonumber \\&\quad + \frac{\mu (0) \text {Curl}_{\varLambda {\mathscr {A}}} (\mathbf{V}(0))}{\mu } - \frac{\mu (0) \varLambda {\mathscr {A}} x \times {\mathbf {V}}(0)}{(1+\alpha ) \mu } - \frac{\alpha \mu (0) \varLambda {\mathscr {A}} \nabla \beta \times {\mathbf {V}}(0)}{(1+\alpha )(1+\beta ) \mu } \nonumber \\&\quad + \frac{1}{\mu }\int _0^{\tau '} \mu [\partial _{\tau }, \text {Curl}_{\varLambda {\mathscr {A}}}] \mathbf{V} \mathrm{d}\tau '' - \frac{1}{(1+\alpha ) \mu } \int _0^{\tau '} \mu [\partial _{\tau },\varLambda {\mathscr {A}} x \times ] {\mathbf {V}} \,\mathrm{d} \tau '' \nonumber \\&\quad - \frac{\alpha }{(1+\alpha )(1+\beta ) \mu } \int _0^{\tau '} \mu [\partial _{\tau },\varLambda {\mathscr {A}} \nabla \beta \times ] {\mathbf {V}} \,\mathrm{d} \tau '' \nonumber \\&\quad - \frac{2}{\mu } \int _0^{\tau '} \mu \, \text {Curl}_{\varLambda {\mathscr {A}}}(\varGamma ^*\mathbf{V}) \mathrm{d}\tau '' + \frac{2}{(1+\alpha ) \mu } \int _0^{\tau '} \mu \, \varLambda {\mathscr {A}} x \times (\varGamma ^*{\mathbf {V}}) \,\mathrm{d} \tau '' \nonumber \\&\quad + \frac{2\alpha }{(1+\alpha )(1+\beta ) \mu } \int _0^{\tau '} \mu \, \varLambda {\mathscr {A}} \nabla \beta \times (\varGamma ^*{\mathbf {V}}) \,\mathrm{d} \tau '' \nonumber \\&\quad +\frac{{\overline{C}}}{(1+\alpha ) \mu }\int _0^{\tau '} \mu ^{1-\delta -\sigma } \varLambda {\mathscr {A}} y \times (\varLambda \eta ) \,\mathrm{d} \tau ''\nonumber \\&\quad +\frac{\alpha {\overline{C}}}{(1+\alpha )(1+\beta ) \mu }\int _0^{\tau '} \mu ^{1-\delta -\sigma } \varLambda {\mathscr {A}} \nabla \beta \times (\varLambda \eta ) \,\mathrm{d} \tau ''. \end{aligned}$$

(D.295)

Now,

$$\begin{aligned}&[\varLambda {\mathscr {A}} x \times (\varLambda \eta )]_j^i = \varLambda _{jm}\big (\delta _m^s - {\mathscr {A}}_\ell ^s [D \theta ]_m^\ell \big ) x_s \varLambda _{ik} \eta ^k - \varLambda _{im}\big (\delta _m^s - {\mathscr {A}}_\ell ^s [D \theta ]_m^\ell \big ) y_s \varLambda _{jk} \eta ^k \nonumber \\&\quad = \varLambda _{js} y_s \varLambda _{ik}\theta ^k - \varLambda _{is} y_s \varLambda _{jk} \theta ^k+\varLambda _{js} y_s\varLambda _{ik}y^k - \varLambda _{is} y_s \varLambda _{jk} y^k \nonumber \\&\qquad -\big (\varLambda _{jm}{\mathscr {A}}_\ell ^s [D \theta ]_m^\ell y_s \varLambda _{ik} \eta ^k - \varLambda _{im} {\mathscr {A}}_\ell ^s [D \theta ]_m^\ell y_s \varLambda _{jk} \eta ^k\big ) \nonumber \\&\quad =\big (\varLambda _{js} y_s \varLambda _{ik}\theta ^k - \varLambda _{is} y_s \varLambda _{jk} \theta ^k\big ) -\big (\varLambda _{jm}{\mathscr {A}}_\ell ^s [D \theta ]_m^\ell x_s \varLambda _{ik} \eta ^k - \varLambda _{im} {\mathscr {A}}_\ell ^s [D \theta ]_m^\ell y_s \varLambda _{jk} \eta ^k\big ) \nonumber \\&\quad :=[\varLambda y \times \varLambda \theta ]_j^i- [\varLambda {\mathscr {A}}[D \theta ]y \times \varLambda \eta ]_j^i. \end{aligned}$$

(D.296)

Second, notice that

$$\begin{aligned}{}[\varLambda {\mathscr {A}} \nabla \beta \times (\varLambda \eta )]_j^i&= \varLambda _{jm}{\mathscr {A}}^s_m \beta _{,s} \varLambda _{ik} \theta ^k - \varLambda _{im}{\mathscr {A}}^s_m \beta _{,s} \varLambda _{jk} \theta ^k\nonumber \\&\quad +\varLambda _{jm}{\mathscr {A}}^s_m \beta _{,s} \varLambda _{ik} y^k - \varLambda _{im}{\mathscr {A}}^s_m \beta _{,s} \varLambda _{jk} y^k \nonumber \\&=[\varLambda {\mathscr {A}} \nabla \beta \times \varLambda \theta ]_j^i + [\varLambda {\mathscr {A}} \nabla \beta \times \varLambda y]_j^i. \end{aligned}$$

(D.297)

Hence we have (D.274). Now for $\text {Curl}_{\varLambda {\mathscr {A}}} \theta$, first note that

$$\begin{aligned} \partial _{\tau } (\text {Curl}_{\varLambda {\mathscr {A}}} \theta ) = \text {Curl}_{\varLambda {\mathscr {A}}} (\partial _{\tau } \theta ) + [\partial _{\tau },\text {Curl}_{\varLambda {\mathscr {A}}}] \theta , \end{aligned}$$

(D.298)

so integrating (D.298) from 0 to $\tau$ via (D.295) we have (D.275). $\quad \square$

Coercivity Estimates

We give a useful result which will allow us to overcome the time weights with negative powers which arise from our equation structure.

Lemma E.1

(Coercivity Estimates) Let $(\theta , \mathbf{V}):\varOmega \rightarrow {\mathbb {R}}^3\times {\mathbb {R}}^3$ be a unique local solution to (2.55)–(2.56) on $[0,T^*]$ for $T^*>0$ fixed with $\mathrm{supp} \,\theta _0 \subseteq B_1({\mathbf {0}})$, $\mathrm{supp}\,{\mathbf {V}}_0 \subseteq B_1({\mathbf {0}})$ and assume $(\theta , \mathbf{V})$ satisfies the a priori assumptions (2.70). Fix $N\ge 4$. Suppose $\beta$ in (2.55) satisfies $\Vert \beta \Vert ^2_{H^{N+1}({\mathbb {R}}^3)} \le \lambda$ and $\mathrm{supp} \, \beta \subseteq B_1({\mathbf {0}})$ where $\lambda > 0$ is fixed. Fix $\nu$ with $0 \le |\nu | \le N-1$. Then for all $\tau \in [0,T^*]$, we have the following inequalities:

$$\begin{aligned} \Vert \partial ^\nu \theta \Vert ^2&\lesssim \sup _{0 \le \tau \le \tau '} \{ \mu ^\sigma \Vert \partial ^\nu {\mathbf {V}}\Vert ^2 \} + \Vert \partial ^\nu \theta (0)\Vert ^2 \end{aligned}$$

(E.299)

$$\begin{aligned} \Vert \nabla _{\eta } \partial ^\nu \theta \Vert ^2&\lesssim \sup _{0 \le \tau \le \tau '} \left\{ \sum _{|\nu '| = |\nu |+1} \mu ^\sigma \Vert \partial ^{\nu '} {\mathbf {V}}\Vert ^2 \right\} + \Vert \nabla _{\eta } \partial ^\nu \theta (0)\Vert ^2 \end{aligned}$$

(E.300)

$$\begin{aligned} \Vert \mathrm{div}_{\eta } \partial ^\nu \theta \Vert ^2&\lesssim \sup _{0 \le \tau \le \tau '} \left\{ \sum _{|\nu '| = |\nu |+1} \mu ^\sigma \Vert \partial ^{\nu '} {\mathbf {V}}\Vert ^2 \right\} + \Vert \mathrm{div}_{\eta } \partial ^\nu \theta (0)\Vert ^2. \end{aligned}$$

(E.301)

Proof of (E.299)

By the fundamental theorem of calculus, and the exponential boundedness of $\mu$ (B.252) and therefore time integrability of negative powers of $\mu$,

$$\begin{aligned} \partial ^\nu \theta = \int _0^\tau \partial ^\nu {\mathbf {V}} \,\mathrm{d} \tau ' + \partial ^\nu \theta (0)&= \int _0^\tau \mu ^{-\tfrac{\sigma }{2}} \mu ^{\tfrac{\sigma }{2}} \partial ^\nu {\mathbf {V}} \,\mathrm{d} \tau ' + \partial ^\nu \theta (0) \nonumber \\&\lesssim \sup _{0 \le \tau \le \tau '} \big \{ \mu ^{\tfrac{\sigma }{2}} \partial ^\nu {\mathbf {V}} \big \} +\partial ^\nu \theta (0). \end{aligned}$$

(E.302)

Therefore, applying Cauchy’s inequality ($ab \lesssim a^2 + b^2,$ $a,b \in {\mathbb {R}}$),

$$\begin{aligned} \Vert \partial ^\nu \theta \Vert ^2 \lesssim \sup _{0 \le \tau \le \tau '} \{ \mu ^\sigma \Vert \partial ^\nu {\mathbf {V}}\Vert ^2 \} + \Vert \partial ^\nu \theta (0)\Vert ^2. \end{aligned}$$

(E.303)

Proof of (E.300)

By a similar coercivity estimate to (E.302)–(E.303),

$$\begin{aligned} \Vert \nabla _{\eta } \partial ^\nu \theta \Vert ^2 \lesssim \sup _{0 \le \tau \le \tau '} \{ \mu ^\sigma \Vert \nabla _{\eta } \partial ^\nu {\mathbf {V}}\Vert ^2 \} + \Vert \nabla _{\eta } \partial ^\nu \theta (0)\Vert ^2. \end{aligned}$$

(E.304)

Now using our a priori bounds (2.70), we have

$$\begin{aligned} \sup _{0 \le \tau \le \tau '} \mu ^\sigma \Vert \nabla _{\eta } \partial ^\nu {\mathbf {V}}\Vert ^2 \lesssim \sup _{0 \le \tau \le \tau '} \left\{ \sum _{|\nu '| = |\nu |+1} \mu ^\sigma \Vert \partial ^{\nu '} {\mathbf {V}}\Vert ^2 \right\} . \end{aligned}$$

(E.305)

Then (E.304)–(E.305) imply (E.300). $\quad \square$

Proof of (E.301)

Finally the proof of (E.301) is similar to the proof of (E.300). $\quad \square$