Global well-posedness of the 3-dimensional Navier-Stokes initial value problem in $L^p\cap L^2$ with $3<p<\infty$

By using the continuous induction method, we prove that the initial value problem of the three dimensional Navier-Stokes equations is globally well-posed in $L^p(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$ for any $3<p<\infty$. The proof is rather simple.