Let $\Omega\subset \mathbb{R}^3$ be an open set with smooth boundary $\partial \Omega$.
Consider the following linearized Navier-Stokes equations in $Q_T=\Omega\times (0,T)$ for an arbitrarily fixed $T\in (0,\infty)$,
$$
u_t-\Delta u+a(x,t)u+b\cdot \nabla u+\nabla p=f(x,t),\text{div } u=0
$$
with the initial and boundary conditions $u(x,0)=0, \left.u(x,t)\right|_{\partial \Omega\times (0,T)}=0$. Here $u(x,t)=(u^1(x,t),u^2(x,t),u^3(x,t))$ and $p(x,t)$ denote the unknown velocity and pressure respectively, $a(x,t)$ and $b(x,t)$ denote the given coefficients.
Question: Suppose that $$a\in L^r(0,T; L^s(\Omega)), b\in L^{r_1}(0,T; L^{s_1}(\Omega)),$$ where $2/r+3/s<2$, $2/r_1+3/s_1<1$,
and $f(x,t)\in C_0^\infty(\Omega\times (0,T))$, can we solve the above equations in arbitrary $L^p$? Can we get the estimates such as
$$\|u_t\|_{L^p(Q_T)}+\|D^2 u\|_{L^p(Q_T)}+\|u\|_{L^p(Q_T)}\leq \|f\|_{L^p(Q_T)}?$$
Solonnikov dealed with this problem in his paper "Estimates for solution of nonstationary Navier-Stokes equations" (http://www.springerlink.com/index/N8374858XNT22P11.pdf).
However, I can not verify his proof (Page 487 to Page 489).
Who can help me? Any comment will be deeply appreciated.
Best Answer
Have a look at the following review article and the relevant references therein: