Existence Theory for Generalized Newtonian Fluids

Appendix C

Itô's formula in infinite dimensions

In this appendix we establish a version of Itô's formula which holds for weak solutions of SPDEs on a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$. Let u be a solution to the system

$\begin{array}{cc}\hfill {\int }_G\mathbf{u}(t)\cdot \mathit{\phi }\mathrm{d}x& ={\int }_G{\mathbf{u}}_0\cdot \mathit{\phi }\mathrm{d}x+{\int }_0^t{\int }_G\mathbf{H}:\mathrm{\nabla }\mathit{\phi }\mathrm{d}x\mathrm{d}\sigma \hfill \\ +{\int }_0^t{\int }_G\mathbf{h}\cdot \mathit{\phi }\mathrm{d}x\mathrm{d}t+{\int }_G{\int }_0^t\mathit{\phi }\cdot \Phi \mathrm{d}{\mathbf{W}}_{\sigma }\mathrm{d}x\hfill \end{array}$

(C.0.1)

for all $\phi \in C_0^{\infty }(G)$, where W is given by (9.1.1). We assume

(I1) ${\mathbf{u}}_0\in L^2(\mathrm{\Omega },{\mathcal{F}}_0,\mathbb{P};L^2(G))$;

(I2) $\mathbf{H}\in L^{p^{\prime }}(\mathrm{\Omega },\mathcal{F},\mathbb{P};L^{p^{\prime }}(Q))$ adapted ...