  In this article, we study the Kapustin-Witten equations on a closed,
simply-connected, four-dimensional manifold which were introduced by Kapustin
and Witten. We use the Taubes' compactness theorem in arXiv:1307.6447v4 to
prove that if $(A,\phi)$ is a smooth solution of Kapustin-Witten equations and
the connection $A$ is closed to a $generic$ ASD connection $A_{\infty}$, then
$(A,\phi)$ must be a trivial solution. We also prove that the moduli space of
the solutions of Kapustin-Witten equations is non-connected if the connections
on the compactification of moduli space of ASD connections are all $generic$.
At last, we extend the results for the Kapustin-Witten equations to other
equations on gauge theory such as the Hitchin-Simpson equations and Vafa-Witten
on a compact K\"{a}hler surface.
