\begin{equation*} \begin{aligned} &\quad\int |\nabla(T_1-\overline{T})^+|^2 \rm dx-\int \frac{3m_1}{2}\nabla \psi_1\nabla T_1(T_1-\overline{T})^+ \rm dx +\int \nabla m_1\nabla \overline{\psi_1}\nabla T_1(T_1-\overline{T})^+dx\\ &=\int \frac{\varepsilon m_1}{2}(\nabla \psi_1)^2 T_1(T_1-\overline{T})^+ \rm dx-\int \frac{3m_1}{2\tau}(T_1-T_{1L})(T_1-\overline{T})^+ dx, \end{aligned} \end{equation*} \begin{equation*} \begin{aligned} &\quad\int |\nabla(T_2-\overline{T})^+ |^2 \rm dx-\int \frac{3m_2}{2}\nabla \psi_2\nabla T_2(T_2-\overline{T})^+\mathrm {dx} +\int \nabla m_2\nabla \overline{\psi_2}\nabla T_2(T_2-\overline{T})^+\rm dx\\ &=\int \frac{\mu m_2}{2}(\nabla \psi_2)^2 T_2(T_2-\overline{T})^+ \rm dx-\int \frac{3m_2}{2\tau}(T_2-T_{2L})(T_2-\overline{T})^+ \rm dx\\ &=\int \frac{3m_1}{2}\nabla \psi_1\nabla T_1(T_1-\overline{T})^+ \rm dx\leq C\epsilon\int |\nabla(T_1-\overline{T})^+|^2+|(T_1-\overline{T})^+|^2 \rm dx, \end{aligned} \end{equation*} \begin{equation*} \begin{aligned} &\quad-\int \nabla m_1\nabla \overline{\psi_1}\nabla T_1(T_1-\overline{T})^+ \rm dx +\int \frac{\varepsilon m_1}{2\tau}(\nabla \psi_1)^2 T_1(T_1-\overline{T})^+\rm dx\\ &=-\int \nabla m_1\nabla \overline{\psi_1}\nabla(T_1-\overline{T}+\overline{T})(T_1-\overline{T})^+\rm dx+\int \frac{\varepsilon m_1}{2\tau}(\nabla \psi_1)^2 T_1(T_1-\overline{T})^+ \rm dx \\ &\leq -\int \nabla m_1\nabla \overline{\psi_1}\nabla (T_1-\overline{T})(T_1-\overline{T})^+\rm dx+\int \frac{\varepsilon m_1}{2\tau}(\nabla \psi_1)^2 T_1(T_1-\overline{T})^+ \rm dx\\ &\leq C\epsilon \int \nabla m_1\nabla \overline{\psi_1}\nabla ((T_1-\overline{T})^+)^2\rm dx+C(n_2\varepsilon)\epsilon^2\int T_1(T_1-\overline{T})^+ \rm dx \\ &\quad-\int \frac{3m_1}{2\tau}(T_1-T_{1L})(T_1-\overline{T})^+ \rm dx=-\int \frac{3m_1}{2\tau}(T_1-\overline{T}+\overline{T}-T_{1L}(x))(T_1-\overline{T})^+ \rm dx\\ &=-\int \frac {3m_1}{2\tau}(T_1-\overline{T})(T_1-\overline{T})^+\rm dx-\int \frac{3m_1}{2\tau}(\overline{T}-T_{1L}(x))(T_1-\overline{T})^+ \rm dx \leq 0. \end{aligned} \end{equation*}
From Feilin Lan.
