Let
- $T>0$
- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be bounded and open
- $\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\|_{H^1(\Lambda,\:\mathbb R^d)}}$$ and $$H:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\|_{L^2(\Lambda,\:\mathbb R^d)}}=\overline V^{\left\|\;\cdot\;\right\|_{L^2(\Lambda,\:\mathbb R^d)}}\tag 1$$
- $\operatorname P_H$ denote the orthogonal projection from $L^2(\Lambda,\mathbb R^d)$ onto $H$
Please consider $$\left\{\begin{array}{rll}\displaystyle\frac{\partial u}{\partial t}-\nu\Delta u+\left(u\cdot\nabla\right)u+\frac1\rho\nabla p&=&f&&\text{in }(0,T)\times\Lambda\\\nabla\cdot u&=&0&&\text{in }(0,T)\times\Lambda\\u&=&0&&\text{on }(0,T)\times\partial\Lambda\\ u(0,\;\cdot\;)&=&u_0&&\text{in }\Lambda\end{array}\right.\tag 2$$ with $u:[0,T]\times\Lambda\to\mathbb R^d$, $p:[0,T]\times\overline\Lambda\to\mathbb R$, $f:[0,T]\times\overline\Lambda\to\mathbb R^d$ and $\nu,\rho>0$.
I want to find a mild solution of a stochastic variant of $(2)$. Therefore, I need to reformulate $(2)$ as an evolution equation in $H$, but how do I need to define the nonlinear part of the drift?
My problem is the following: Note that $\mathcal D(A):=\mathcal V\cap H^2(\Lambda,\mathbb R^d)$ is dense in $V$ and $$Au:=\text P_H\Delta u\;\;\;\text{for }u\in\mathcal D(A)$$ is a densely-defined linear symmetric operator on $H$ and $-A$ is the generator of a contraction $C^0$-semigroup on $H$.
Now, let $H_r:=\mathcal D(A^r)$ denote the domain of the fractional power of $A$ with exponent $r\in\mathbb R$. In order to apply the usual existence theory for mild solutions of SPDEs (see, for example, Assumption 2 on page 12 of A mild Itō formula for SPDEs), I need that the nonlinear part $F$ of the drift is a Lipschitz continuous mapping from $H_\gamma$ to $H_\alpha$ for some $\alpha,\gamma\in\mathbb R$ with $\gamma-\alpha<1$.
Now, I would like to define $F(u)=\operatorname P_H(u\cdot\nabla)u$, but this would require $(u\cdot\nabla)u$ to belong to $L^2(\Lambda,\mathbb R^d)$ (which is the case, for example, for $u\in H^2(\Lambda,\mathbb R^d)$ with $d\le 4$).
So, I'm confused how I need to define $F$ (and choose $\gamma$ and $\alpha$).
Let me note that I'm aware of the fact that, if $d\le 4$, $$b(u,v,w):=\int_\Lambda(u\cdot\nabla)v\cdot w\:{\rm d}x\;\;\;\text{for }u,v,w\in H_0^1(\Lambda,\mathbb R^d)$$ is a well-defined bounded trilinear form on $H_0^1(\Lambda,\mathbb R^d)$. Moreover, under suitable regularity assumptions on $\Lambda$, we can show that $H_{-1/2}=V'$ and $H_{1/2}=V$. Maybe that helps.
