Assume (7.1.4) with $q>\frac{11}{5}$, $\mathbf{f}\in L^{q^{\prime }}(Q)$ and ${\mathbf{v}}_0\in L^2(G)$. Then there is a solution $\mathbf{v}\in L^{\infty }(0,T;L^2(G))\cap L^q(0,T;W_{0,\mathrm{div}}^{1,q}(G))$ to

$\begin{array}{cc}\hfill {\int }_Q\mathbf{S}(\mathit{\epsilon }(\mathbf{v})):\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t& ={\int }_Q\mathbf{f}\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q\mathbf{v}\otimes \mathbf{v}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_Q\mathbf{v}\cdot {\partial }_t\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_G{\mathbf{v}}_0\cdot \mathit{\phi }(0)\mathrm{d}x\hfill \end{array}$

for all $\mathit{\phi }\in C_{0,\mathrm{div}}^{\infty }([0,T)\times G)$.

We mainly follow the ideas of [111], chapter 5. We separate space and time and approximate the corresponding Sobolev space by a finite dimensional subspace. From [111] we infer the existence of a sequence $({\lambda }_k)\subset \mathbb{R}$ and a sequence of functions $({\mathbf{w}}_k)\subset W_{0,\mathrm{div}}^{l,2}(G)$, $l\in \mathbb{N}$, such that

i) ${\mathbf{w}}_k$ is an eigenvector to the eigenvalue ${\lambda }_k$ of the Stokes-operator in the sense that

${\langle {\mathbf{w}}_k,\mathit{\phi }\rangle }_{W_0^{l,2}}={\lambda }_k{\int }_G{\mathbf{w}}_k\cdot \mathit{\phi }\mathrm{d}x\text{for all}\mathit{\phi }\in W_{0,\mathrm{div}}^{l,2}(G),$

ii) ${\int }_G{\mathbf{w}}_k{\mathbf{w}}_m\mathrm{d}x={\delta }_{km}$ for all $k,m\in \mathbb{N}$,

iii) $1⩽{\lambda }_1⩽{\lambda }_2⩽...$ and ${\lambda }_k\to \infty$,

iv) ${\langle \frac{{\mathbf{w}}_k}{\sqrt{{\lambda }_k}},\frac{{\mathbf{w}}_m}{\sqrt{{\lambda }_m}}\rangle }_{W_0^{l,2}}={\delta }_{km}$ for all $k,m\in \mathbb{N}$,

v) $({\mathbf{w}}_k)$ is a basis of $W_{0,\mathrm{div}}^{l,2}(G)$.

We choose $l>1+\frac{d}{2}$ so that $W_0^{l,2}(G)\hookrightarrow W^{1,\infty }(G)$. We are looking for an approximate solution ${\mathbf{v}}_N$ of the form

${\mathbf{v}}_N=\sum _{k=1}^N{c}_N^k{\mathbf{w}}_k$

where ${\mathbf{C}}_N={(c_N^k)}_{k=1}^N:(0,T)\to {\mathbb{R}}^N$. We will construct ${\mathbf{C}}_N$ so that ${\mathbf{v}}_N$ is a solution to

$\begin{array}{cc}\hfill {\int }_G\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_N)):\mathit{\epsilon }({\mathbf{w}}_k)\mathrm{d}x& =-{\int }_G{\partial }_t{\mathbf{v}}_N\cdot {\mathbf{w}}_k\mathrm{d}x+{\int }_G{\mathbf{v}}_N\otimes {\mathbf{v}}_N:\mathit{\epsilon }({\mathbf{w}}_k)\mathrm{d}x\hfill \\ \hfill & +{\int }_G\mathbf{f}\cdot {\mathbf{w}}_k\mathrm{d}x,k=1,...,N\hfill \\ \hfill {\mathbf{v}}_N(0,\cdot )& ={\mathcal{P}}_N{\mathbf{v}}_0.\hfill \end{array}$

Here ${\mathcal{P}}_N$ is the $L^2(G)$-orthogonal projection into ${\mathcal{X}}_N:=\mathrm{span}\{{\mathbf{w}}_1,...,{\mathbf{w}}_N\}$, i.e.

${\mathcal{P}}_N(\mathbf{u}):=\sum _{k=1}^N({\int }_G{\mathbf{w}}_k\cdot \mathbf{u}\mathrm{d}x){\mathbf{w}}_k.$

On account of the properties of $({\mathbf{w}}_k)$ equation (7.1.6) is equivalent to

$\begin{array}{cc}\hfill \frac{\mathrm{d}{c}_N^k}{\mathrm{d}t}& =-{\int }_G\mathbf{S}({\mathbf{C}}_N\cdot \mathit{\epsilon }({\mathbf{w}}_N)):\mathit{\epsilon }({\mathbf{w}}_k)\mathrm{d}x+\sum _{l,j}{c}_N^l{c}_N^j{\int }_G{\mathbf{w}}_l\otimes {\mathbf{w}}_j:\mathrm{\nabla }{\mathbf{w}}_k\mathrm{d}x\hfill \\ \hfill & +{\int }_G\mathbf{f}\cdot {\mathbf{w}}_k\mathrm{d}x,k=1,...,N\hfill \\ \hfill c_N^k(0)& ={\int }_G{\mathbf{w}}_k\cdot {\mathbf{v}}_0\mathrm{d}x,k=1,...,N.\hfill \end{array}$

Since the right-hand-side is not globally Lipschitz continuous in ${\mathbf{C}}_N$ the Picard–Lindelöff Theorem only gives a local solution which does not suffice for our purpose. The following lemma helps (see [143], chapter 30).

So we need to show boundedness of ${\mathbf{C}}_N$ in t (assuming its existence). Therefore we multiply the k-th equation of (7.1.6) by $c_N^k$ and sum with respect to k. Using ${\int }_G{\mathbf{v}}_N\otimes {\mathbf{v}}_N:\mathrm{\nabla }{\mathbf{v}}_N\mathrm{d}x$ we obtain

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_G|{\mathbf{v}}_N{|}^2\mathrm{d}x+{\int }_G|\mathrm{\nabla }{\mathbf{v}}_N{|}^q\mathrm{d}x={\int }_G\mathbf{f}\cdot {\mathbf{v}}_N\mathrm{d}x.$

Integration over $[0,s]$ with $0<s⩽T$ implies for all $\kappa >0$

$\begin{array}{cc}\hfill & \frac{1}{2}{\int }_G|{\mathbf{v}}_N(s,\cdot ){|}^2\mathrm{d}x+\lambda {\int }_{Q_s}(|\mathrm{\nabla }{\mathbf{v}}_N{|}^q-1)\mathrm{d}x⩽{\int }_{Q_s}\mathbf{f}\cdot {\mathbf{v}}^N\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ⩽c(\kappa ){\int }_{Q_s}|\mathbf{f}{|}^{q^{\prime }}\mathrm{d}x\mathrm{d}t+\kappa {\int }_0^s{\int }_G|{\mathbf{v}}_N{|}^q\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ⩽c(\kappa ){\int }_{Q_s}|\mathbf{f}{|}^{q^{\prime }}\mathrm{d}x\mathrm{d}t+\kappa {\int }_0^s{\int }_G|\mathrm{\nabla }{\mathbf{v}}_N{|}^q\mathrm{d}x\mathrm{d}t,\hfill \end{array}$

where we used the inequalities of Korn, Young and Poincaré as well as (7.1.4). Choosing κ small enough leads to

$\underset{t\in (0,T)}{\sup }{\int }_G|{\mathbf{v}}_N(t,\cdot ){|}^2\mathrm{d}x+{\int }_Q|\mathrm{\nabla }{\mathbf{v}}_N{|}^q\mathrm{d}x\mathrm{d}t⩽c{\int }_Q(|\mathbf{f}{|}^{q^{\prime }}+1)\mathrm{d}x\mathrm{d}t$

which also implies (7.1.8). Lemma 7.1.1 shows the existence of a solution ${\mathbf{v}}_N$ to (7.1.6). Moreover we established with (7.1.9) a useful a priori estimate. Passing to a subsequence implies

${\mathbf{v}}_N\rightharpoonup :\mathbf{v}\text{in}{L}^q(0,T;W_{0,div}^{1,q}(G)),$

${\mathbf{v}}_N{\rightharpoonup }^{⁎}\mathbf{v}\text{in}{L}^{\infty }(0,T;L^2(G)).$

In order to pass to the limit in the convective term we need compactness of ${\mathbf{v}}_N$. We obtain from (7.1.6)

$\begin{array}{cc}\hfill & {\int }_G{\partial }_t{\mathbf{v}}_N\cdot \mathit{\phi }\mathrm{d}x={\int }_G{\partial }_t{\mathbf{v}}_N\cdot {\mathcal{P}}_N^l\mathit{\phi }\mathrm{d}x\hfill \\ \hfill & =-{\int }_G\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}^N)):\mathit{\epsilon }({\mathcal{P}}_N^l\mathit{\phi })\mathrm{d}x+{\int }_G{\mathbf{v}}_N\otimes {\mathbf{v}}_N:\mathit{\epsilon }({\mathcal{P}}_N^l\mathit{\phi })\mathrm{d}x+{\int }_G\mathbf{F}:\mathrm{\nabla }{\mathcal{P}}_N^l\mathit{\phi }\mathrm{d}x\hfill \\ \hfill & =:{\int }_G{\mathbf{H}}_N:\mathrm{\nabla }{\mathcal{P}}_N^l\mathit{\phi }\mathrm{d}x\hfill \end{array}$

for all $\mathit{\phi }\in W_{0,div}^{l,2}(G)$ (setting $\mathbf{F}:=\mathrm{\nabla }{\mathrm{\Delta }}^{-1}\mathbf{f}$). Here ${\mathcal{P}}_N^l$ denotes the orthogonal projection into ${\mathcal{X}}_N$ with respect to the $W_0^{l,2}(G)$ inner product. We have uniformly in N

${\mathbf{H}}_N\in L^{q_0}(Q),q_0=:\min \{\frac{5}{6}q,q^{\prime }\}>1,$

as a consequence of (7.1.9), (7.1.4) and $\mathbf{F}\in L^{q^{\prime }}(Q)$ (which follows from $\mathbf{f}\in L^{q^{\prime }}(Q)$). On account of (7.1.12) and Sobolev's embedding (recall the choice of l) we obtain

$\begin{array}{cc}\hfill & {\Vert {\partial }_t{\mathbf{v}}_N\Vert }_{L^{q_0}(0,T;W_{div}^{-l,2}(G))}={\Vert {\partial }_t{\mathbf{v}}_N\Vert }_{L^{q_0^{\prime }}{(0,T;W_{0,div}^{l,2}(G))}^{\prime }}\hfill \\ \hfill & =\underset{{\Vert \mathit{\phi }\Vert }_{L^{q_0^{\prime }}(0,T;W_{0,div}^{l,2}(G))}⩽1}{\sup }{\int }_0^T{\int }_G{\partial }_t{\mathbf{v}}_N\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & =\underset{{\Vert \mathit{\phi }\Vert }_{L^{q_0^{\prime }}(0,T;W_{0,div}^{l,2}(G))}⩽1}{\sup }{\int }_0^T{\int }_G{\mathbf{H}}_N:\mathrm{\nabla }{P}_N^l\mathit{\phi }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ⩽\underset{{\Vert \mathit{\phi }\Vert }_{L^{q_0^{\prime }}(0,T;W_{0,div}^{l,2}(G))}⩽1}{\sup }{({\int }_Q|{\mathbf{H}}_N{|}^{q_0}\mathrm{d}x\mathrm{d}t)}^{\frac{1}{q_0}}{({\int }_Q|\mathrm{\nabla }{P}_l^N\mathit{\phi }{|}^{q_0^{\prime }}\mathrm{d}x\mathrm{d}t)}^{\frac{1}{q_0^{\prime }}}\hfill \end{array}$

and finally

$\begin{array}{cc}\hfill \Vert {\partial }_t{\mathbf{v}}_N\Vert & {}_{L^{q_0}(0,T;W_{div}^{-l,2}(G))}\hfill \\ \hfill & ⩽c\underset{{\Vert \mathit{\phi }\Vert }_{L^{q_0^{\prime }}(0,T;W_{0,div}^{l,2}(G))}⩽1}{\sup }{\Vert \mathrm{\nabla }{\mathcal{P}}_N^l\mathit{\phi }\Vert }_{L^{q_0}(0,T;L^{\infty }(G))}\hfill \\ \hfill & ⩽c\underset{{\Vert \mathit{\phi }\Vert }_{L^{q_0^{\prime }}(0,T;W_{0,div}^{l,2}(G))}⩽1}{\sup }{\Vert {\mathcal{P}}_N^l\mathit{\phi }\Vert }_{L^{q_0}(0,T;W_{0,div}^{l,2}(G))}⩽c.\hfill \end{array}$

Combining (7.1.9) and (7.1.13) with the Aubin–Lions compactness Theorem (see Theorem 5.1.23) shows ${\mathbf{v}}_N\to \mathbf{v}$ in $L^2(0,T;L_{\mathrm{div}}^2(G))$ (recall that $q>\frac{11}{5}$). This and a parabolic interpolation imply

${\mathbf{v}}_N\otimes {\mathbf{v}}_N\to \mathbf{v}\otimes \mathbf{v}\text{in}{L}^s(Q)$

for all $s<\frac{5}{6}p$. Due to (7.1.9) and (7.1.4) we know that $\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_N))$ is bounded in $L^{q^{\prime }}(G)$ thus

$\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_N))\rightharpoonup \tilde{\mathbf{S}}\text{in}{L}^{q^{\prime }}(Q).$

Passing to the limit in (7.1.6) leads to

$\begin{array}{cc}\hfill {\int }_Q\tilde{\mathbf{S}}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t& ={\int }_Q\mathbf{f}\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q\mathbf{v}\otimes \mathbf{v}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_Q\mathbf{v}\cdot {\partial }_t\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_G{\mathbf{v}}_0\cdot \mathit{\phi }(0)\mathrm{d}x\hfill \end{array}$

for all $\mathit{\phi }\in C_{0,\mathrm{div}}^{\infty }([0,T)\times G)$. Note that the class of test-functions which factorize in space and time is dense, see Lemma 5.1.2. In (7.1.16) we also used (for $k\in \mathbb{N}$ and $g\in C_0^{\infty }[0,T)$)

${\int }_Q{\partial }_t{\mathbf{v}}_N\cdot g{\mathbf{w}}_k\mathrm{d}x\mathrm{d}t=-{\int }_G{\mathcal{P}}_N{\mathbf{v}}_0\cdot {\mathbf{w}}_k\mathrm{d}xg(0)-{\int }_0^T{\int }_G{\mathbf{v}}_N\cdot {\mathbf{w}}_k{\partial }_{t}g\mathrm{d}x\mathrm{d}t$

and ${\mathcal{P}}_N{\mathbf{v}}_0\to {\mathbf{v}}_0$ in $L^2(G)$.

Finally we need to show that

$\tilde{\mathbf{S}}=\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))$

holds. We first investigate the time derivative of v. On account of $q>\frac{11}{5}$ the mapping

$\mathit{\phi }\mapsto {\int }_Q\mathbf{v}\otimes \mathbf{v}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t$

belongs to $L^{q^{\prime }}(0,T;W_{div}^{-1,q^{\prime }}(G))$. The same is true for

$\mathit{\phi }\mapsto {\int }_Q\tilde{\mathbf{S}}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t,\mathit{\phi }\mapsto {\int }_Q\mathbf{f}\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t.$

Thus we have ${\partial }_t\mathbf{v}\in L^{q^{\prime }}(0,T;W_{div}^{-1,q^{\prime }}(G))$ by (7.1.16) and

$\begin{array}{cc}\hfill {\int }_Q\tilde{\mathbf{S}}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t& ={\int }_Q\mathbf{f}\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q\mathbf{v}\otimes \mathbf{v}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_0^T\langle {\partial }_t\mathbf{v},\mathbf{v}\rangle \mathrm{d}t\hfill \end{array}$

for all $\mathit{\phi }\in L^q(0,T;W_{0,div}^{1,q}(G))$. Especially v is an admissible test function. We claim that

$\mathbf{v}\in C_w([0,T];L^2(G)).$

Hence we have $\mathbf{v}(0)={\mathbf{v}}_0$ and $\mathbf{v}(t)$ is uniquely determined for every $t\in [0,T]$. Sobolev's Theorem for Bochner spaces leads to

$\mathbf{v}\in C([0,T];W_{div}^{-1,q^{\prime }}(G)).$

Let $(t_n)\subset [0,T]$ for which $t_n\to t_0$ and $\mathbf{v}(t_n,\cdot )\in L^2(G)$ for all $n\in \mathbb{N}$. Then $\mathbf{v}(t_n,\cdot )$ is bounded in $L^2(G)$ by (7.1.9) thus

$\mathbf{v}(t_n,\cdot )\rightharpoonup :\mathbf{w}\text{in}{L}^2(G).$

We have as a consequence of (7.1.20)

$\mathbf{v}(t_n,\cdot )\to \mathbf{v}(t_0,\cdot )\text{in}{W}_0^{-1,q^{\prime }}(G).$

This leads to $\mathbf{w}=\mathbf{v}(t_0,\cdot )$ and (7.1.21) implies (7.1.19).

We apply monotone operator theory to show (7.1.17). On account of

$\begin{array}{cc}\hfill & {\int }_Q(\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_N))-\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))):\mathit{\epsilon }({\mathbf{v}}_N-\mathbf{v})\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ={\int }_Q(\tilde{\mathbf{S}}-\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_N))):\mathit{\epsilon }(\mathbf{v})\mathrm{d}x\mathrm{d}t-{\int }_Q\mathbf{S}(\mathit{\epsilon }(\mathbf{v})):\mathit{\epsilon }({\mathbf{v}}_N-\mathbf{v})\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_Q(\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_N)):\mathit{\epsilon }({\mathbf{v}}_N)\mathrm{d}x\mathrm{d}t-{\int }_Q\tilde{\mathbf{S}}:\mathit{\epsilon }(\mathbf{v})\mathrm{d}x\mathrm{d}t\hfill \end{array}$

we obtain from (7.1.10) and (7.1.15)

$\begin{array}{cc}\hfill & {\int }_Q(\tilde{\mathbf{S}}-\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_N))):\mathit{\epsilon }(\mathbf{v})\mathrm{d}x\mathrm{d}t⟶0,N\to \infty ,\hfill \\ \hfill & {\int }_Q\mathbf{S}(\mathit{\epsilon }(\mathbf{v})):\mathit{\epsilon }({\mathbf{v}}_N-\mathbf{v})\mathrm{d}x\mathrm{d}t⟶0,N\to \infty .\hfill \end{array}$

Due to (7.1.6) and (7.1.18) we have

$\begin{array}{cc}\hfill & {\int }_Q(\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_N)):\mathit{\epsilon }({\mathbf{v}}_N)\mathrm{d}x\mathrm{d}t-{\int }_Q\tilde{\mathbf{S}}:\mathit{\epsilon }(\mathbf{v})\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ={\int }_Q\mathbf{f}\cdot ({\mathbf{v}}_N-\mathbf{v})\mathrm{d}x\mathrm{d}t+{\int }_Q({\mathbf{v}}_N\otimes {\mathbf{v}}_N:\mathit{\epsilon }({\mathbf{v}}_N)-\mathbf{v}\otimes \mathbf{v}:\mathit{\epsilon }(\mathbf{v}))\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_Q{\partial }_t{\mathbf{v}}_N\cdot {\mathbf{v}}_N\mathrm{d}x\mathrm{d}t+{\int }_0^T\langle {\partial }_t\mathbf{v},\mathbf{v}\rangle \mathrm{d}t\hfill \\ \hfill & =:(I)+(II)+(III).\hfill \end{array}$

We deduce from (7.1.10), (7.1.14) and (7.1.15) that

$\underset{N\to \infty }{\lim }(I)=\underset{N\to \infty }{\lim }(II)=0.$

For the integral involving the convective term we used the assumption $q>\frac{11}{5}$. Finally, we obtain

$\begin{array}{cc}\hfill (III)& =\frac{1}{2}{\int }_G|{\mathbf{v}}_N(0){|}^2\mathrm{d}x-\frac{1}{2}{\int }_G|\mathbf{v}(0){|}^2\mathrm{d}x+\frac{1}{2}{\int }_G|\mathbf{v}(T){|}^2\mathrm{d}x\hfill \\ \hfill & -\frac{1}{2}{\int }_G|{\mathbf{v}}_N(T){|}^2\mathrm{d}x\hfill \\ \hfill & =-\frac{1}{2}{\int }_G|{\mathbf{v}}_N(T)-\mathbf{v}(T){|}^2\mathrm{d}x-{\int }_G{\mathbf{v}}_N(T)\cdot \mathbf{v}(T)\mathrm{d}x+{\int }_G|\mathbf{v}(T){|}^2\mathrm{d}x\hfill \\ \hfill & +\frac{1}{2}{\int }_G|{\mathcal{P}}_N{\mathbf{v}}_0{|}^2\mathrm{d}x-\frac{1}{2}{\int }_G|{\mathbf{v}}_0{|}^2\mathrm{d}x.\hfill \end{array}$

We infer from (7.1.10) and the continuity of ${\mathcal{P}}_N$ that ${\lim \sup }_{N\to \infty }(III)⩽0$ (here we took into account (7.1.19) and used ${\mathbf{v}}_N(T)\rightharpoonup \mathbf{v}(T)$ in $L^2(G)$ as a consequence of (7.1.9) and passing to a subsequence) thus

${\int }_Q(\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_N))-\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))):\mathit{\epsilon }({\mathbf{v}}_N-\mathbf{v})\mathrm{d}x\mathrm{d}t⟶0,N\to \infty .$

Monotonicity of S (which follows from (7.1.4)) implies (7.1.17).  □

Under the assumptions of Theorem 7.0.27 there is a function $\tilde{\pi }\in C_w^0([0,T];L_0^{q^{\prime }}(G))$ for which

$\begin{array}{cc}\hfill & {\int }_Q\mathbf{S}(\mathit{\epsilon }(\mathbf{v})):\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t={\int }_Q\mathbf{f}\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q\mathbf{v}\otimes \mathbf{v}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_Q\mathbf{v}\cdot {\partial }_t\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_G{\mathbf{v}}_0\mathit{\phi }(0)\mathrm{d}x+{\int }_Q\tilde{\pi }{\partial }_t\mathrm{div}\mathit{\phi }\mathrm{d}x\mathrm{d}t\hfill \end{array}$

for all $\mathit{\phi }\in C_0^{\infty }([0,T)\times G)$.

We follow [140], Thm. 2.6. Let $\mathit{\phi }(t,x)=g(t)\mathit{\psi }(x)$ with $g\in C_0^{\infty }(0,T)$ and $\mathit{\psi }\in C_{0,\mathrm{div}}^{\infty }(G)$. Setting $\mathbf{F}:=\mathrm{\nabla }{\mathrm{\Delta }}^{-1}\mathbf{f}\in L^{q^{\prime }}(Q)$ we have

$\begin{array}{cc}\hfill {\int }_Q\mathbf{v}\cdot \mathit{\psi }{g}^{\prime }\mathrm{d}x\mathrm{d}t& ={\int }_Q\mathbf{S}(\mathit{\epsilon }(\mathbf{v})):\mathrm{\nabla }\mathit{\psi }g\mathrm{d}x\mathrm{d}t-{\int }_Q\mathbf{v}\otimes \mathbf{v}:\mathrm{\nabla }\mathit{\psi }g\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_Q\mathbf{F}:\mathrm{\nabla }\mathit{\psi }g\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & =:-{\int }_Q\mathbf{Q}:\mathrm{\nabla }\mathit{\psi }g\mathrm{d}x\mathrm{d}t\hfill \end{array}$

which is equivalent to

${\int }_0^T({\int }_G\mathbf{v}\cdot \mathit{\psi }\mathrm{d}x)g^{\prime }\mathrm{d}t=-{\int }_0^T({\int }_G\mathbf{Q}:\mathrm{\nabla }\mathit{\psi }\mathrm{d}x)g\mathrm{d}t.$

If we define

$\begin{array}{cc}\hfill \alpha (t)& :={\int }_G\mathbf{v}(t,\cdot )\cdot \mathit{\psi }\mathrm{d}x,\hfill \\ \hfill \beta (t)& :={\int }_G\mathbf{Q}:\mathrm{\nabla }\mathit{\psi }\mathrm{d}x,\hfill \end{array}$

we obtain from (7.1.23)

${\int }_0^T\alpha g^{\prime }\mathrm{d}t=-{\int }_0^T\beta g\mathrm{d}t$

for all $g\in C_0^{\infty }(0,T)$. Since α and β belong to $L^1(0,T)$ this implies ${\alpha }^{\prime }=\beta$. Hence the following holds

$\alpha (t)=\alpha (0)+{\int }_0^t\beta (s)\mathrm{d}s,t\in (0,T).$

Now we define

$\tilde{\mathbf{Q}}(t):={\int }_0^t\mathbf{Q}(s)ds$

and follow from (7.1.25)

${\int }_G((\mathbf{v}(t,\cdot )-\mathbf{v}(0,\cdot ))\cdot \mathit{\psi }+\tilde{\mathbf{Q}}(t):\mathrm{\nabla }\mathit{\psi })\mathrm{d}x=0$

for all $\mathit{\psi }\in W_{0,\mathrm{div}}^{1,q}(G)$. Here we took into account $\tilde{\mathbf{Q}}(t)\in L^{q^{\prime }}(G)$ which is a consequence of $q>\frac{11}{5}$ and (7.1.4). De Rahm's Theorem implies the existence of $\tilde{\pi }(t)\in L_0^{q^{\prime }}(G)$ for which

${\int }_G((\mathbf{v}(t,\cdot )-\mathbf{v}(0,\cdot ))\cdot \mathit{\psi }+\tilde{\mathbf{Q}}:\mathrm{\nabla }\mathit{\psi })\mathrm{d}x={\int }_G\tilde{\pi }(t)\mathrm{div}\mathit{\psi }\mathrm{d}x$

for all $\mathit{\psi }\in W_0^{1,q}(G)$. For $u\in L^q(G)$ we set $\mathit{\psi }={\mathrm{Bog}}_G(u-{(u)}_G)\in W_{0,\mathrm{div}}^{1,q}(G)$ so that

$\begin{array}{cc}\hfill {\int }_G\tilde{\pi }(t)u\mathrm{d}x& ={\int }_G\tilde{\pi }(t)(\mathrm{div}\mathit{\psi }+{(u)}_G)\mathrm{d}x={\int }_G\tilde{\pi }(t)\mathrm{div}\mathit{\psi }\mathrm{d}x\hfill \\ \hfill & ={\int }_G((\mathbf{v}(t,\cdot )-\mathbf{v}(0,\cdot ))\cdot \mathit{\psi }+\tilde{\mathbf{Q}}(t):\mathrm{\nabla }\mathit{\psi })\mathrm{d}x.\hfill \end{array}$

Due to $\mathbf{v}\in C_w([0,T];L^2(G))$ (see (7.1.19)) and $\tilde{Q}\in C([0,T];L^{q^{\prime }}(G))$ we obtain

$\underset{t\to t_0}{\lim }{\int }_G\tilde{\pi }(t)u\mathrm{d}x={\int }_G\tilde{\pi }(t_0)u\mathrm{d}x$

for all $u\in L^q(G)$, hence $\tilde{\pi }\in C_w([0,T];L^{q^{\prime }}(G))$. Equation (7.1.25) finally implies

$\begin{array}{cc}\hfill & {\int }_Q\mathbf{S}(\mathit{\epsilon }(\mathbf{v})):\mathrm{\nabla }\mathit{\phi }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ={\int }_Q\mathbf{v}\otimes \mathbf{v}:\mathrm{\nabla }\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q\mathbf{f}\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_Q\mathbf{v}\cdot {\partial }_t\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_G{\mathbf{v}}_0\cdot \mathit{\phi }(0,\cdot )\mathrm{d}x+{\int }_Q\tilde{\pi }{\partial }_t\mathrm{div}\mathit{\phi }\mathrm{d}x\mathrm{d}t\hfill \end{array}$

for all φ of the class

$Y:=\mathrm{span}\{g\mathit{\psi },g\in C_0^{\infty }[0,T),\mathit{\psi }\in C_0^{\infty }(G)\}$

which is dense (see Lemma 5.1.1).  □

The “original” pressure term π can be obtained by setting $\pi :={\partial }_t\tilde{\pi }$. But without further information about the regularity with respect to time of the quantities involved in the equation it only exists in the sense of distributions.

We start with an approximate system whose solution is known to exist. Let ${\mathbf{v}}_m\in L^q(0,T;W_{0,\mathrm{div}}^{1,q}(G))\cap L^{\infty }(0,T;L^2(G))$ be a solution to

$\begin{array}{cc}\hfill & {\int }_Q\mathbf{S}(\mathit{\epsilon }(\mathbf{v})):\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t+\frac{1}{m}{\int }_Q|\mathit{\epsilon }(\mathbf{v}){|}^{q-2}\mathit{\epsilon }(\mathbf{v}):\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ={\int }_Q\mathbf{f}\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q\mathbf{v}\otimes \mathbf{v}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t+{\int }_Q\mathbf{v}{\partial }_t\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_G{\mathbf{v}}_0\mathit{\phi }(0)\mathrm{d}x\hfill \end{array}$

for all $\mathit{\phi }\in C_{0,\mathrm{div}}^{\infty }([0,T)\times G)$, where $q>\max \{\frac{11}{5},p\}$.

The existence of ${\mathbf{v}}_m$ follows from Theorem 7.1.28. Since we are allowed to test with ${\mathbf{v}}_m$, we find

$\begin{array}{cc}\hfill & \frac{1}{2}{\Vert {\mathbf{v}}_m(t)\Vert }_{L^2}^2+{\int }_0^t{\int }_G\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m)):\mathit{\epsilon }({\mathbf{v}}_m)\mathrm{d}x\mathrm{d}\sigma +\frac{1}{m}{\int }_0^t{\int }_G|\mathit{\epsilon }({\mathbf{v}}_m){|}^q\mathrm{d}x\mathrm{d}\sigma \hfill \\ \hfill & =\frac{1}{2}{\Vert {\mathbf{v}}_0\Vert }_{L^2}^2+{\int }_0^t{\int }_G\mathbf{f}:{\mathbf{v}}_{m}dx\mathrm{d}\sigma ,\hfill \end{array}$

for all $t\in (0,T)$. By coercivity and Korn's inequality we obtain

${\int }_Q\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m)):\mathit{\epsilon }({\mathbf{v}}_m)\mathrm{d}x\mathrm{d}t⩾c({\int }_Q|\mathrm{\nabla }{\mathbf{v}}_m{|}^p\mathrm{d}x\mathrm{d}t-1)$

thus

${\Vert m^{-1/q}\mathit{\epsilon }({\mathbf{v}}_m)\Vert }_{q,Q}+{\Vert {\mathbf{v}}_m\Vert }_{L^{\infty }(0,T;L^2)}^2+{\Vert \mathrm{\nabla }{\mathbf{v}}_m\Vert }_{p,Q}⩽c.$

Hence we find a function $\mathbf{v}\in L^p(0,T;W_{0,\mathrm{div}}^{1,p}(G))\cap L^{\infty }(0,T;L^2(G))$ for which (passing to a subsequence)

$\begin{array}{ccc}\hfill \mathrm{\nabla }{\mathbf{v}}_m& \rightharpoonup \mathrm{\nabla }\mathbf{v}\hfill & \text{in}{L}^p(Q),\hfill \\ \hfill {\mathbf{v}}_m& \stackrel{⁎}{\rightharpoonup }\mathbf{v}\hfill & \text{in}{L}^{\infty }(0,T;L^2(G)),\hfill \\ \hfill \frac{1}{m}|\mathit{\epsilon }({\mathbf{v}}_m){|}^{q-2}\mathit{\epsilon }({\mathbf{v}}_m)& \to 0\hfill & \text{in}{L}^{q^{\prime }}(Q).\hfill \end{array}$

Since $\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m))$ is bounded in $L^{p^{\prime }}(Q)$ by (7.2.30), there exist $\tilde{\mathbf{S}}\in L^{p^{\prime }}(Q)$ with

$\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m))\rightharpoonup \tilde{\mathbf{S}}\text{in}{L}^{p^{\prime }}(Q).$

Let us have a look at the time derivative. From equation (7.2.28) we get the uniform boundedness of ${\partial }_t{\mathbf{v}}_m$ in $L^{p^{\prime }}(0,T;W_{\mathrm{div}}^{-3,2}(G))$ and weak convergence of ${\partial }_t{\mathbf{v}}_m$ to ${\partial }_t\mathbf{v}$ in the same space (for a subsequence). This shows by using the compactness of the embedding $W_{0,\mathrm{div}}^{1,p}(G)\hookrightarrow L_{\mathrm{div}}^{2{\sigma }_2}(G)$ for some ${\sigma }_2>1$ (which follows from our assumption $p>\frac{6}{5}$, resp. $p>\frac{2n}{n+2}$) and the Aubin–Lions theorem (see Theorem 5.1.23) that ${\mathbf{v}}_m\to \mathbf{v}$ in $L^{\sigma }(0,T;L_{\mathrm{div}}^{2{\sigma }_2}(G))$. This and the boundedness in $L^{\infty }(0,T;L^2(G))$ imply that for some $\sigma >1$

${\mathbf{v}}_m\to \mathbf{v}\text{in}{L}^s(0,T;L^{2\sigma }(G))\text{for all}s<\infty .$

As a consequence we have

${\mathbf{v}}_m\otimes {\mathbf{v}}_m\to \mathbf{v}\otimes \mathbf{v}\text{in}{L}^s(0,T;L^{\sigma }(G))\text{for all}s<\infty .$

Overall, we get our limit equation

$\begin{array}{cc}\hfill {\int }_Q\tilde{\mathbf{S}}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t& ={\int }_Q\mathbf{f}\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q\mathbf{v}\otimes \mathbf{v}:\mathit{\epsilon }(\mathit{\phi })\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_Q\mathbf{v}{\partial }_t\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_G{\mathbf{v}}_0\mathit{\phi }(0)\mathrm{d}x\hfill \end{array}$

for all $\mathit{\phi }\in C_{0,\mathrm{div}}^{\infty }([0,T)\times G)$.

The entire forthcoming effort is to prove $\tilde{\mathbf{S}}=\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))$ almost everywhere. We start with the difference of the equation of ${\mathbf{v}}_m$ and the limit equation which is

$\begin{array}{cc}\hfill & -{\int }_Q({\mathbf{v}}_m-\mathbf{v})\cdot {\partial }_t\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q(\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m))-\tilde{\mathbf{S}}):\mathrm{\nabla }\mathit{\phi }\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ={\int }_Q({\mathbf{v}}_m\otimes {\mathbf{v}}_m-\mathbf{v}\otimes \mathbf{v}+m^{-1}|\mathit{\epsilon }({\mathbf{v}}_m){|}^{q-2}\mathit{\epsilon }({\mathbf{v}}_m)):\mathrm{\nabla }\mathit{\phi }\mathrm{d}x\mathrm{d}t\hfill \end{array}$

for all $\mathit{\phi }\in C_{0,\mathrm{div}}^{\infty }([0,T)\times G)$. We define ${\mathbf{u}}_m:={\mathbf{v}}_m-\mathbf{v}$. Then by (7.2.33)

$\begin{array}{cc}\hfill & {\mathbf{u}}_m\rightharpoonup \mathbf{0}\text{in}{L}^p(0,T;W_{0,\mathrm{div}}^{1,p}(G)),\hfill \\ \hfill & {\mathbf{u}}_m\to \mathbf{0}\text{in}{L}^{2\sigma }(Q),\hfill \\ \hfill & {\mathbf{u}}_m{\rightharpoonup }^{⁎}\mathbf{0}\text{in}{L}^{\infty }(0,T;L^2(G)).\hfill \end{array}$

Thus, we can write (7.2.36) as

${\int }_Q{\mathbf{u}}_m\cdot {\partial }_t\mathit{\phi }\mathrm{d}x\mathrm{d}t={\int }_Q{\mathbf{H}}_m:\mathrm{\nabla }\mathit{\phi }\mathrm{d}x\mathrm{d}t$

for all $\mathit{\phi }\in C_{0,\mathrm{div}}^{\infty }(Q)$, where ${\mathbf{H}}_m:={\mathbf{H}}_m^1+{\mathbf{H}}_m^2$ with

$\begin{array}{cc}\hfill {\mathbf{H}}_m^1& :=\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m))-\tilde{\mathbf{S}},\hfill \\ \hfill {\mathbf{H}}_m^2& :={\mathbf{v}}_m\otimes {\mathbf{v}}_m-\mathbf{v}\otimes \mathbf{v}+m^{-1}|\mathit{\epsilon }({\mathbf{v}}_m){|}^{q-2}\mathit{\epsilon }({\mathbf{v}}_m).\hfill \end{array}$

Moreover, (7.2.31) and (7.2.33) imply

${\Vert {\mathbf{H}}_m^1\Vert }_{p^{\prime }}⩽c$

as well as

${\mathbf{H}}_m^2\to 0\text{in}{L}^{\sigma }(Q).$

Now take any cylinder $Q_0⋐(0,T)\times G$. Now, (7.2.37), (7.2.38), (7.2.39) and (7.2.40) ensure that we can apply Corollary 6.1.4. In particular, for suitable $\zeta \in C_0^{\infty }(\frac{1}{6}{Q}_0)$ with ${\chi }_{\frac{1}{8}{Q}_0}⩽\zeta ⩽{\chi }_{\frac{1}{6}{Q}_0}$ Corollary 6.1.4 implies

$\underset{m\to \infty }{\lim \sup }|\int ({\mathbf{H}}_m^1:\mathrm{\nabla }({\mathbf{v}}_m-\mathbf{v}))\zeta {\chi }_{{\mathcal{O}}_{m,k}^{\complement }}\mathrm{d}x\mathrm{d}t|⩽c{2}^{-k/p}.$

In other words

$\underset{m\to \infty }{\lim \sup }|\int ((\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m))-\tilde{\mathbf{S}}):\mathrm{\nabla }({\mathbf{v}}_m-\mathbf{v}))\zeta {\chi }_{{\mathcal{O}}_{m,k}^{\complement }}\mathrm{d}x\mathrm{d}t|⩽c{2}^{-k/p}.$

Now, the boundedness of $\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))$ and $\tilde{\mathbf{S}}$ in $L^{p^{\prime }}(\frac{1}{6}{Q}_0)$ and Theorem 6.1.25 (h) and (g) give

$\underset{m\to \infty }{\lim \sup }|\int ((\tilde{\mathbf{S}}-\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))):\mathrm{\nabla }({\mathbf{v}}_m-\mathbf{v}))\zeta {\chi }_{{\mathcal{O}}_{m,k}^{\complement }}\mathrm{d}x\mathrm{d}t|⩽c{2}^{-k/p}.$

This and the previous estimate imply

$\underset{m\to \infty }{\lim \sup }|\int ((\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m))-\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))):\mathrm{\nabla }({\mathbf{v}}_m-\mathbf{v}))\zeta {\chi }_{{\mathcal{O}}_{m,k}^{\complement }}\mathrm{d}x\mathrm{d}t|⩽c{2}^{-k/p}.$

Let $\theta \in (0,1)$. Then by Hölder's inequality and Theorem 6.1.25 (g)

$\begin{array}{cc}\hfill & \underset{m\to \infty }{\lim \sup }\int {((\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m))-\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))):\mathrm{\nabla }({\mathbf{v}}_m-\mathbf{v}))}^{\theta }\zeta {\chi }_{{\mathcal{O}}_{m,k}}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ⩽c|{\mathcal{O}}_{m,k}{|}^{1-\theta }⩽c{2}^{-(1-\theta )\frac{k}{p}}.\hfill \end{array}$

This, the previous estimate and Hölder's inequality lead to

$\underset{m\to \infty }{\lim \sup }\int {((\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m))-\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))):\mathrm{\nabla }({\mathbf{v}}_m-\mathbf{v}))}^{\theta }\zeta \mathrm{d}x\mathrm{d}t⩽c{2}^{-(1-\theta )\frac{k}{p}}.$

For $k\to \infty$ the right-hand-side converges to zero. Now, the monotonicity of S implies that $\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_m))\to \mathbf{S}(\mathit{\epsilon }(\mathbf{v}))$ a.e. in $\frac{1}{8}{Q}_0$. This concludes the proof of Theorem 7.0.27.  □