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My question is how to estimate the term $\Vert \nabla u \Vert_{L^{\infty}(\Omega)}$. Here we consider the 2D incompressible Navier Stokes equations:$$u_t -\Delta u+u\cdot \nabla u+\nabla p=f$$ and $$\nabla \cdot u=0.$$ $\Omega$ is a bounded domain in $\mathbb R^2$ with no-slip boundary condition. Since we are considering on a bounded domain, we can no longer consider the vorticity equation. I appreciate if some reference papers are pointed out. Thanks.

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