5.1 Bochner spaces

In the study of parabolic PDEs it is very useful to work with Banach space-valued mappings (see [143]). Let $(\mathcal{V},{\Vert \cdot \Vert }_{\mathcal{V}})$ be a Banach space. A mapping $u:[0,T]\to \mathcal{V}$ is called a step function iff for some $N\in \mathbb{N}$

$u(t)=\sum _{k=1}^N{\chi }_{A_k}(t)x_k,t\in [0,T],$

where ${\cup }_k{A}_k=[0,T]$, $A_k\cap A_j=\varnothing$ for $k\ne j$ and $x_k\in \mathcal{V}$ for $k=1,...,N$. A function $u:(0,T)\to \mathcal{V}$ is called Bochner measurable iff there is a sequence $(u_n)$ of step functions such that

$u_n(t)\to u(t)\text{in}\mathcal{V}$

for a.e. t. A function $u:(0,T)\to \mathcal{V}$ is called Bochner integrable iff there is a sequence $(u_n)$ of step functions such that (5.1.1) holds and

${\int }_0^T{\Vert u_n(t)-u(t)\Vert }_{\mathcal{V}}\mathrm{d}t\to 0,n\to \infty .$

The Bochner integral (as an element of $\mathcal{V}$) is defined as

${\int }_0^{T}u(t)\mathrm{d}t:=\underset{n}{\lim }{\int }_0^T{u}_n(t)\mathrm{d}t=\underset{n}{\lim }\sum _{k=1}^n{\mathcal{L}}^1(A_k^n)x_k^n.$

We define for $T>0$ and $p\in [1,\infty )$, the space $L^p(0,T;\mathcal{V})$ to be the set of all Bochner measurable functions $u:(0,T)\to \mathcal{V}$ such that

${\Vert u\Vert }_{L^p(0,T;\mathcal{V})}:={\left({\int }_0^T{\Vert u(t)\Vert }_{\mathcal{V}}^p\mathrm{d}t\right)}^{\frac{1}{p}}<\infty .$

The space $L^{\infty }(0,T;\mathcal{V})$ is the set of all Bochner measurable functions such that

${\Vert u\Vert }_{L^{\infty }(0,T;\mathcal{V})}:=\underset{{\mathcal{L}}^1(A)=0}{\inf }\underset{(0,T)\setminus A}{\sup }{\Vert u(t)\Vert }_{\mathcal{V}}<\infty .$

The space $L^p(0,T;\mathcal{V})$ for $p\in [1,\infty ]$ is a Banach space together with the norm given above.

For $u\in L^1(0,T;\mathcal{V})$ we consider the distribution

$C_0^{\infty }(0,T)\ni \phi \mapsto {\int }_0^{T}u(t){\phi }^{\prime }(t)\mathrm{d}t\in \mathcal{V}.$

Let $\mathcal{Y}$ be a Banach space with $\mathcal{V}\hookrightarrow \mathcal{Y}$ continuously. If there is $v\in L^1(0,T;\mathcal{Y})$ such that

${\int }_0^{T}u(t){\phi }^{\prime }(t)\mathrm{d}t=-{\int }_0^{T}v(t)\phi (t)\mathrm{d}t\text{for all}\phi \in C_0^{\infty }(0,T)$

we say that v is the weak derivative of u in $\mathcal{Y}$ and write $v={\partial }_{t}u$. The space $W^{1,p}(0,T;\mathcal{V})$ consists of those functions from $L^p(0,T;\mathcal{V})$ having weak derivatives in $L^p(0,T;\mathcal{V})$. It is a Banach function space together with the norm

${\Vert u\Vert }_{W^{1,p}(0,T;\mathcal{V})}^p:={\Vert u\Vert }_{L^p(0,T;\mathcal{V})}^p+{\Vert {\partial }_{t}u\Vert }_{L^p(0,T;\mathcal{V})}^p.$

Obviously this can be iterated to define the space $W^{k,p}(0,T;\mathcal{V})$.

In order to study the time regularity of functions from Bochner spaces we need to define different notations of continuity.

Obviously, we have the inclusions

$C^{\alpha }([0,T];\mathcal{V})\subset C([0,T];\mathcal{V})\subset C_w([0,T];\mathcal{V})$

for any $\alpha \in (0,1]$. The following variant of Sobolev's Theorem holds.

The following theorem shows how to obtain compactness in Bochner spaces. The original version is due to Aubin and Lions (see [14] and [109]) but does not include the case $p=1$. The following more general version can be found in [128].

In the context of stochastic PDEs we will be confronted with functions having only fractional derivatives in time. We define for $p\in (1,\infty )$ and $\alpha \in (0,1)$ the norm

${\Vert u\Vert }_{W^{\alpha ,p}(0,T;\mathcal{V})}^p:={\Vert u\Vert }_{L^p(0,T;\mathcal{V})}^p+{\int }_0^T{\int }_0^T\frac{{\Vert u({\sigma }_1)-u({\sigma }_2))\Vert }_{\mathcal{V}}^p}{|{\sigma }_1-{\sigma }_2{|}^{1+\alpha p}}\mathrm{d}{\sigma }_1\mathrm{d}{\sigma }_2.$

The space $W^{\alpha ,p}(0,T;\mathcal{V})$ is now defined as the subspace of $L^p(0,T;\mathcal{V})$ consisting of the functions having finite $W^{\alpha ,p}(0,T;\mathcal{V})$-norm. It can be shown that this is a complete space and we have $W^{1,p}(0,T;\mathcal{V})\subset W^{\alpha ,p}(0,T;\mathcal{V})\subset L^p(0,T;\mathcal{V})$. The following version of Theorem 5.1.22 holds (see [73]).

The following interpolation result is a special case of [8, Thm. 3.1]

5.2 Basics on parabolic Lipschitz truncation

In this section we show how a Lipschitz truncation for non-stationary problems can be constructed. We follow the ideas of [65] (see [101] for a similar approach). Let $Q_0=I_0\times {\mathcal{B}}_0\subset \mathbb{R}\times {\mathbb{R}}^d$ be a space time cylinder. Let $\mathbf{u}\in L^{\sigma }(I_0,W^{1,\sigma }({\mathcal{B}}_0))$ and $\mathbf{H}\in L^{\sigma }(Q_0)$ satisfy

${\int }_{Q_0}\mathbf{u}\cdot {\partial }_t\mathit{\xi }\mathrm{d}x\mathrm{d}t={\int }_{Q_0}\mathbf{H}:\mathrm{\nabla }\mathit{\xi }\mathrm{d}x\mathrm{d}t\text{for all}\mathit{\xi }\in C_0^{\infty }(Q_0).$

Let $\alpha >0$. We say that $Q^{\prime }=I^{\prime }\times {\mathcal{B}}^{\prime }\subset \mathbb{R}\times {\mathbb{R}}^d$ is an α-parabolic cylinder if $r_{I^{\prime }}=\alpha r_{{\mathcal{B}}^{\prime }}^2$. For $\kappa >0$ we define the scaled cylinder $\kappa Q^{\prime }:=(\kappa I^{\prime })\times (\kappa {\mathcal{B}}^{\prime })$. By ${\mathcal{Q}}^{\alpha }$ we denote the set of all α-parabolic cylinders. We define the α-parabolic maximal operators ${\mathcal{M}}^{\alpha }$ and ${\mathcal{M}}_s^{\alpha }$ for $s\in [1,\infty )$ by

$\begin{array}{cc}\hfill ({\mathcal{M}}^{\alpha }f)(t,x)& :=\underset{Q^{\prime }\in {\mathcal{Q}}^{\alpha }:(t,x)\in Q^{\prime }}{\sup }{⨍}_{Q^{\prime }}|f(\tau ,y)|\mathrm{d}\tau \mathrm{d}y,\hfill \\ \hfill {\mathcal{M}}_s^{\alpha }f(t,x)& :={({\mathcal{M}}^{\alpha }(|f{|}^s(t,x)))}^{\frac{1}{s}}.\hfill \end{array}$

It is standard [134] that for all $f\in L^p({\mathbb{R}}^{d+1})$

${\Vert {\mathcal{M}}_s^{\alpha }f\Vert }_{L^q({\mathbb{R}}^{d+1})}⩽c{\Vert f\Vert }_{L^q({\mathbb{R}}^{d+1})}\forall q\in (s,\infty ],$

${\mathcal{L}}^{d+1}(\{x\in {\mathbb{R}}^d:|{\mathcal{M}}_s^{\alpha }f(x)|>\lambda \})⩽c\frac{{\Vert f\Vert }_q}{{\lambda }^q}\forall q\in [s,\infty ),\forall \lambda >0,$

where the constants are independent of α. Another important tool is a parabolic Poincaré estimate for u in terms of $\mathrm{\nabla }\mathbf{u}$ and H, see Theorem B.1 of [65]: Let $Q_r^{\alpha }=I_r\times {\mathcal{B}}_r\in {\mathcal{Q}}^{\alpha }$ and any $\mathbf{u}\in L^1(I_r;W^{1,1}({\mathcal{B}}_r))$ with ${\partial }_t\mathbf{u}=\mathrm{div}\mathbf{H}$ in ${\mathcal{D}}^{\prime }(Q_r^{\alpha })$ for some $\mathbf{H}\in L^1(Q_r^{\alpha })$. Then the following holds

${⨍}_{Q_r^{\alpha }}|\mathbf{u}-{\mathbf{u}}_{Q_r^{\alpha }}|\mathrm{d}x\mathrm{d}t⩽cr{⨍}_{Q_r^{\alpha }}(|\mathrm{\nabla }\mathbf{u}|+\alpha |\mathbf{H}|)\mathrm{d}x\mathrm{d}t,$

where c only depends on the dimension.

In order to define the Lipschitz truncation we have to cut large values of $\mathrm{\nabla }\mathbf{u}$ and H. We define the “bad set” as

${\mathcal{O}}_{\lambda }^{\alpha }:=\{{\mathcal{M}}_{\sigma }^{\alpha }({\chi }_{Q_0}|\mathrm{\nabla }\mathbf{u}|)>\lambda \}\cup \{\alpha {\mathcal{M}}_{\sigma }^{\alpha }({\chi }_{Q_0}|\mathbf{H}|)>\lambda \}.$

This is the set where we have to change u. In contrast to the stationary case discussed in Section 1.3, a straightforward argument is not available. So we follow the more flexible strategy based on a Whitney covering as done at the end of Section 3.1. According to Lemma 3.1 of [65] there exists an α-parabolic Whitney covering $\{Q_i\}$ of ${\mathcal{O}}_{\lambda }^{\alpha }$.

For each $Q_i$ we define $A_i:=\{j:Q_j\cap Q_i\ne \varnothing \}$. Note that $\mathrm{\#}{A}_i\le {120}^{d+2}$ and $r_j\sim r_i$ for all $j\in A_i$.

Now we define

${\mathbf{u}}_{\lambda }^{\alpha }:=\mathbf{u}-\sum _{i\in \mathcal{I}}{\phi }_i(\mathbf{u}-{\mathbf{u}}_i),$

where ${\mathbf{u}}_i:={\mathbf{u}}_{Q_i}:={⨍}_{Q_i}\mathbf{u}\mathrm{d}x\mathrm{d}t$. (In order to obtain a truncation with suitable properties up to the boundary one has to involve cut-off function as can be seen in [65] and Chapter 6. We neglect this for brevity.) We show first that the sum in (5.2.8) converges absolutely in $L^1(Q_0)$:

${\int }_{Q_0}|\mathbf{u}-{\mathbf{u}}_{\lambda }^{\alpha }|\mathrm{d}x⩽c\sum _i{\int }_{Q_i}|\mathbf{u}-{\mathbf{u}}_i|\mathrm{d}x\mathrm{d}t⩽c\sum _i{\int }_{Q_i}|\mathbf{u}|\mathrm{d}x\mathrm{d}t⩽c{\int }_{Q_0}|\mathbf{u}|\mathrm{d}x\mathrm{d}t,$

where we used (PP1) and the finite intersection property of $Q_i$ (PW4). We proceed by showing the estimate for the gradient

$\begin{array}{cc}\hfill & {\int }_{{\mathcal{O}}_{\lambda }^{\alpha }}|\mathrm{\nabla }(\mathbf{u}-{\mathbf{u}}_{\lambda }^{\alpha })|\mathrm{d}x\mathrm{d}t⩽c\sum _i{\int }_{Q_i}|\mathrm{\nabla }({\phi }_i(\mathbf{u}-{\mathbf{u}}_i))|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ⩽c\sum _i{\int }_{Q_i}|\mathrm{\nabla }\mathbf{u}|+\left|\frac{\mathbf{u}-{\mathbf{u}}_i}{r_i}\right|\mathrm{d}x\mathrm{d}t⩽c\sum _i{\int }_{Q_i}|\mathrm{\nabla }\mathbf{u}|+\alpha \left|\mathbf{H}\right|\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ⩽c{\int }_{Q_0}|\mathrm{\nabla }\mathbf{u}|+\alpha \left|\mathbf{H}\right|\mathrm{d}x\mathrm{d}t,\hfill \end{array}$

where we used (PP3), (5.2.6) and (PW4). This shows that the definition in (5.2.8) makes sense. In particular we have

${\mathbf{u}}_{\lambda }^{\alpha }=\{\begin{array}{cc}\mathbf{u}\hfill & \text{in}{Q}_0\setminus {\mathcal{O}}_{\lambda }^{\alpha },\hfill \\ {\sum }_i{\phi }_i{\mathbf{u}}_i\hfill & \text{in}{Q}_0\cap {\mathcal{O}}_{\lambda }^{\alpha }.\hfill \end{array}$

The truncation ${\mathbf{u}}_{\lambda }^{\alpha }$ has better regularity properties than u; indeed, $\mathrm{\nabla }{\mathbf{u}}_{\lambda }^{\alpha }$ is bounded by λ.

The next lemma will control the time error we obtain when we use the truncation as a test function. We will only consider this from a formal point of view ignoring the technical difficulties connected with the distributional character of the time-derivative of u. We refer to [65, Thm. 3.9.(iii)] for a rigorous treatment.

5.3 Existence results for power law fluids

The flow of a homogeneous incompressible fluid in a bounded body $G\subset {\mathbb{R}}^d$ ($d=2,3$) during the time interval $(0,T)$ is described by the following set of equations

$\{\begin{array}{cc}\hfill \rho {\partial }_t\mathbf{v}+\rho (\mathrm{\nabla }\mathbf{v})\mathbf{v}=\mathrm{div}\mathbf{S}-\mathrm{\nabla }\pi +\rho \mathbf{f}\hfill & \text{in}Q,\hfill \\ \hfill \mathrm{div}\mathbf{v}=0\hfill & \text{in}Q,\hfill \\ \hfill \mathbf{v}=0\hfill & \text{on}\partial G,\hfill \\ \hfill \mathbf{v}(0,\cdot )={\mathbf{v}}_0\hfill & \text{in}G.\hfill \end{array}$

See for instance [23]. Here the unknown quantities are the velocity field $\mathbf{v}:Q\to {\mathbb{R}}^d$ and the pressure $\pi :Q\to \mathbb{R}$. The function $\mathbf{f}:Q\to {\mathbb{R}}^d$ represents a system of volume forces and ${\mathbf{v}}_0:G\to {\mathbb{R}}^d$ the initial datum, while $\mathbf{S}:Q\to {\mathbb{R}}_{sym}^{d\times d}$ is the stress deviator and $\rho >0$ is the density of the fluid. Equation (5.3.11)₁ and (5.3.11)₂ describe the conservation of balance and the conservation of mass respectively. Both are valid for all homogeneous incompressible liquids and gases. In order to describe a specific fluid one needs a constitutive law relating the viscous stress tensor S to the symmetric gradient $\mathit{\epsilon }(\mathbf{v}):=\frac{1}{2}(\mathrm{\nabla }\mathbf{v}+\mathrm{\nabla }{\mathbf{v}}^T)$ of the velocity. In the simplest case this relation is linear, i.e.,

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

where $\nu >0$ is the viscosity of the fluid. In this case we have $\mathrm{div}\mathbf{S}=\nu \mathrm{\Delta }\mathbf{v}$ and (5.3.11) is the famous Navier–Stokes equation. Its mathematical treatment started with the work of Leray and Ladyshenskaya (see [106]). The existence of a weak solution (where derivatives are to be understood in a distributional sense) can be established by nowadays standard arguments. However the regularity issue (i.e. the existence of a strong solution) is still open.

As already motivated at the beginning of Chapter 4, a much more flexible model is

$\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))=\nu (|\mathit{\epsilon }(\mathbf{v})|)\mathit{\epsilon }(\mathbf{v}),$

where ν is the generalized viscosity function. Of particular interest is the power law model

$\mathbf{S}(\mathit{\epsilon }(\mathbf{v}))={\nu }_0{(1+|\mathit{\epsilon }(\mathbf{v})|)}^{p-2}\mathit{\epsilon }(\mathbf{v})$

where ${\nu }_0>0$ and $p\in (1,\infty )$, cf. [13,23]. We recall that the case $p\in [\frac{3}{2},2]$ covers many interesting applications.

In the following we give a historical overview concerning the theory of weak solutions to (5.3.11) and sketch the proofs, cf. [29]. It can be understood as the non-stationary counterpart to Section 1.4.

Monotone operator theory (1969).

Due to the appearance of the convective term $(\mathrm{\nabla }\mathbf{v})\mathbf{v}=\mathrm{div}(\mathbf{v}\otimes \mathbf{v})$ the equations for power law fluids (the constitutive law is given by (5.3.11)) highly depend on the value of p. The first results were achieved by Ladyshenskaya and Lions for $p⩾\frac{3d+2}{d+2}$ (see [106] and [109]). They showed the existence of a weak solution in the space

$L^p(0,T;W_{0,\mathrm{div}}^{1,p}(G))\cap L^{\infty }(0,T;L^2(G)).$

The weak formulation reads as

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

for all $\mathit{\phi }\in C_{0,\mathrm{div}}^{\infty }([0,T)\times G)$ with S given by (5.3.14). In the case $p⩾\frac{3d+2}{d+2}$ it follows from parabolic interpolation that $(\mathrm{\nabla }\mathbf{v})\mathbf{v}\cdot \mathbf{v}\in L^1(Q)$. So the weak solution is also a test-function and the existence proof is based on monotone operator theory and compactness arguments.

Let us assume that

$p>\frac{3d+2}{d+2}$

and that we have a sequence of approximated solutions, i.e.,

$({\mathbf{v}}_n)\subset L^p(0,T;W_{0,\mathrm{div}}^{1,p}(G))\cap L^{\infty }(0,T;L^2(G))$

solving (5.3.15). A sequence of approximated solutions can be obtained, for instance via a Galerkin–Ansatz (see [111], Chapter 5). We want to pass to the limit. Assume further that

${\partial }_t{\mathbf{v}}_n\in L^{p^{\prime }}(0,T;W_{\mathrm{div}}^{-1,p^{\prime }}(G))\cong L^p{(0,T;W_{0,\mathrm{div}}^{1,p}(G))}^{\prime }.$

Then ${\mathbf{v}}_n$ is also an admissible test-function (using (5.3.16)). We gain uniform a priori estimates and (after choosing an appropriate subsequence and applying Korn's inequality)

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

A parabolic interpolation implies

${\mathbf{v}}_n\rightharpoonup \mathbf{v}\text{in}{L}^{p\frac{d+2}{d}}(Q).$

As in the stationary case, (5.3.14) yields together with (5.3.17)

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

A main difference to the stationary problem is the compactness of the velocity. Due to (5.3.17), (5.3.18) and (5.3.15) we can control the time derivative and have

${\partial }_t{\mathbf{v}}_n\rightharpoonup {\partial }_t\mathbf{v}\text{in}{L}^{p^{\prime }}(0,T;W_{\mathrm{div}}^{-1,p^{\prime }}(\mathrm{\Omega })).$

Combining (5.3.17) and (5.3.20) the Aubin–Lions Compactness Theorem (cf. Theorem 5.1.22) yields

${\mathbf{v}}_n\to \mathbf{v}\text{in}{L}^{\min \{p^{\prime },p\}}(0,T;L_{\mathrm{div}}^2(\mathrm{\Omega }))$

and together with (5.3.18)

${\mathbf{v}}_n\to \mathbf{v}\text{in}{L}^q(Q)\forall q<p\frac{d+2}{d}.$

Plugging the convergences (5.3.17)–(5.3.21) together we can pass to the limit in the approximate equation in all terms except for $\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_n))$. As done in section 1.4 we have to apply arguments from monotone operator theory and show

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

This follows along the same line as in the stationary case; the only term which needs a comment is the integral involving the time derivative. Here we have in addition to the terms from the stationary case the integral

$\begin{array}{cc}\hfill & -{\int }_0^T\langle {\partial }_t{\mathbf{v}}_n,{\mathbf{v}}_n-\mathbf{v}\rangle \mathrm{d}t\hfill \\ \hfill & =-{\int }_0^T\frac{d}{dt}({\int }_{\mathrm{\Omega }}|{\mathbf{v}}_n-\mathbf{v}{|}^2\mathrm{d}x)\mathrm{d}t-{\int }_0^T\langle {\partial }_t\mathbf{v},{\mathbf{v}}_n-\mathbf{v}\rangle \mathrm{d}t\hfill \\ \hfill & ⩽-{\int }_0^T\langle {\partial }_t\mathbf{v},{\mathbf{v}}_n-\mathbf{v}\rangle \mathrm{d}t⟶0,n\to \infty ,\hfill \end{array}$

using ${\mathbf{v}}_n(0)=\mathbf{v}(0)={\mathbf{v}}_0$ a.e., (5.3.17) and (5.3.20). As the integrand in (5.3.22) is non-negative the claim follows.

$L^{\infty }$ -truncation (2007).

The classical results have been improved by Wolf to the case $p>\frac{2d+2}{d+2}$ via $L^{\infty }$-truncation. In this situation we have $(\mathrm{\nabla }\mathbf{v})\mathbf{v}\in L^1(Q)$ and therefore we can test with functions from $L^{\infty }(Q)$. The basic idea (which was already used in the stationary case in [78] together with the bound $p⩾\frac{2d}{d+1}$) is to approximate v by a bounded function ${\mathbf{v}}_L$ which is equal to v on a large set and whose $L^{\infty }$-norm can be controlled by L. Now we will present the approach developed in [140]. Note that the $L^{\infty }$-truncation has been used in the parabolic context before in [80] and [39]. Different from [140], both these papers deal with periodic and Navier's slip boundary conditions, respectively. So, the problems connected with the harmonic pressure do not occur.

Let us assume that

$p>\frac{2d+2}{d+2}$

and the existence of approximate solutions ${\mathbf{v}}_n$ to (5.3.15) with uniform a priori estimates in

$L^p(0,T;W_{0,\mathrm{div}}^{1,p}(G))\cap L^{\infty }(0,T;L^2(G)).$

Note that test-functions have to be bounded as we only have $(\mathrm{\nabla }\mathbf{v})\mathbf{v}\in L^1(Q)$ due to (5.3.23) and a parabolic interpolation. We have again the convergences (5.3.17)–(5.3.21) so we only have to establish the limit in $\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_n))$. As the solution is not a test-function anymore we have to use some truncation. The $L^{\infty }$-truncation destroys the solenoidal character of a function and a correction via the Bogovskiĭ  operator does not give the right sign when testing the time-derivative. So one has to introduce the pressure. In [140] this is done locally for the difference of approximate equation and limit equation. Due to the localization one has to use cut-off functions which we neglect in the following as they only produce additional terms of lower order. We have

$-{\int }_Q{\mathbf{u}}_n\cdot {\partial }_t\mathit{\phi }\mathrm{d}x\mathrm{d}t=-{\int }_Q{\mathbf{H}}_n^1:\mathrm{\nabla }\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q\mathrm{div}{\mathbf{H}}_n^2\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t$

for all $\mathit{\phi }\in C_{0,\mathrm{div}}^{\infty }(Q)$ with

$\begin{array}{cc}\hfill {\mathbf{H}}_n^1& :=\mathbf{S}(\mathit{\epsilon }({\mathbf{v}}_n))-\tilde{\mathbf{S}}\rightharpoonup 0\text{in}{L}^{p^{\prime }}(Q),\hfill \\ \hfill {\mathbf{H}}_n^2& :={\mathbf{v}}_n\otimes {\mathbf{v}}^n-\mathbf{v}\otimes \mathbf{v}\rightharpoonup 0\text{in}{L}^{\sigma }(Q),\hfill \\ \hfill \mathrm{\nabla }{\mathbf{H}}_n^2& ={\mathbf{v}}_n\otimes {\mathbf{v}}_n-\mathbf{v}\otimes \mathbf{v}\rightharpoonup 0\text{in}{L}^{\sigma }(Q),\hfill \end{array}$

where $\sigma :=\frac{p(d+2)}{p(d+2)-(2d+2)}\in (1,\infty )$, cf. (5.3.23). Now we can introduce a pressure ${\pi }_n$ and decompose it into ${\pi }_n={\pi }_n^h+{\pi }_n^1+{\pi }_n^2$ such that

$\begin{array}{cc}\hfill & -{\int }_Q({\mathbf{u}}_n-\mathrm{\nabla }{\pi }_n^h)\cdot {\partial }_t\mathit{\phi }\mathrm{d}x\hfill \\ \hfill & =-{\int }_Q({\mathbf{H}}_n^1-{\pi }_n^{1}I):\mathrm{\nabla }\mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_Q\mathrm{div}({\mathbf{H}}_n^2-{\pi }_n^{2}I)\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t\hfill \end{array}$