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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" ( However, I can not verify his proof (Page 487 to Page 489).

Who can help me? Any comment will be deeply appreciated.

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Have a look at the following review article and the relevant references therein:

Yoshikazu Giga, Weak and strong solutions of the Navier-Stokes initial value problem, Publ. RIMS. Kyoto Univ., 19:887-910, 1983.

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Y. Giga, H. Sohr, Abstract L^{p} estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.

Also: The Navier-Stokes equations. An elementary analytic approach. Hermann Sohr.

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