Establishment and sustainment of the structure of internal transport barriers (ITBs) is an important guarantee for the magnetic fusion plasmas. The related physics process for the establishing and sustaining of ITBs with $q_{min} \simeq$ 2 is simply summarized:the "off-axis sawteeth" (OAS) and double tearing modes instability, fast ions induced Alfvén eigenmodes, thermal pressure gradient induced low-frequency modes (LFMs) instability, etc. Firstly, the burst of "off-axis sawteeth" (OAS) is an important criterion for the evaluating of reversed $q$-profile with $q_{min} \simeq$ 2. The excitation conditions, classification and the structure of precursor mode of OAS are given in detailed, and the collapse event is triggered by the magnetic reconnection of $m$/$n$=2/1 double tearing modes (DTM). Secondly, the beta-induced Alfvén eigenmodes (BAEs) and reversed shear Alfvén eigenmodes (RSAEs) are easily excited by the fast ions during the oscillation of OAS. The toroidal mode numbers of the two kinds of Alfvén waves are 1 $\leq n \leq$ 5, which are located at 1.98 $\leq R \leq$ 2.07 m with normalized minor radius 0.2 $\leq \rho \leq$ 0.45. The excitation conditions are investigated for the condition of $q_{min} \simeq$ 2:and three different physical variables of thermal pressure gradient, fast ions distribution function, and the toroidal flow or flow shear are considered. Thirdly, the low-frequency modes (LFMs) instabilities are excited by the pressure gradient during the oscillation of OAS. The general fishbone like dispersion relationship (GFLDR) is adopted for solving the basic features of LFMs:① the frequency of LFMs scale with ions diamagnetic frequency; ② the LFMs has the Alfvén polarization direction; ③ the LFMs are reactive-type kinetic ballooning modes. The excitation of LFMs is not relied on the fast ions, which is taken place at higher pressure gradient regime $\alpha \propto (1 + \tau) (1 + \eta_i)$, $\tau=T_e/T_i$, $\eta_i=L_{n_i}/L_{T_i}$. In the end, the suppression of OAS and establishment of ITBs are achieved. Three important processes are contained for the condition of $q_{min} \simeq$ 2 in EAST:① the tangential injection (NBI1L) of NBI is more easier for the suppression of OAS in comparison with the perpendicular injection (NBI1R); ② the micro-instability can be suppressed during the oscillation of OAS, and the reversed shear $q$-profile is more favorable in the establishment of the structure of ITBs; ③ the establishment of ITBs is accompanied by the excitation of Alfvén waves instabilities (bigger toroidal mode number:1 $\leq n \leq$ 5), the sustainment of ITBs is accompanied by the thermal ions temperature gradient induced instability (median size:5 $\leq n \leq$ 10). Therefore, understanding the establishment and suppression of OAS, the excitation of Alfvén wave instability and the redistributed fast ions, the related instability of thermal pressure gradient, which are important for the establishment of ITBs.