egin{equation*} egin{aligned} &quadint | abla(T_1-overline{T})^+|^2 m dx-int frac{3m_1}{2} abla psi_1 abla T_1(T_1-overline{T})^+ m dx +int abla m_1 abla overline{psi_1} abla T_1(T_1-overline{T})^+dx\ &=int frac{varepsilon m_1}{2}( abla psi_1)^2 T_1(T_1-overline{T})^+ m dx-int frac{3m_1}{2 au}(T_1-T_{1L})(T_1-overline{T})^+ dx, end{aligned} end{equation*} egin{equation*} egin{aligned} &quadint | abla(T_2-overline{T})^+ |^2 m dx-int frac{3m_2}{2} abla psi_2 abla T_2(T_2-overline{T})^+mathrm {dx} +int abla m_2 abla overline{psi_2} abla T_2(T_2-overline{T})^+ m dx\ &=int frac{mu m_2}{2}( abla psi_2)^2 T_2(T_2-overline{T})^+ m dx-int frac{3m_2}{2 au}(T_2-T_{2L})(T_2-overline{T})^+ m dx\ &=int frac{3m_1}{2} abla psi_1 abla T_1(T_1-overline{T})^+ m dxleq Cepsilonint | abla(T_1-overline{T})^+|^2+|(T_1-overline{T})^+|^2 m dx, end{aligned} end{equation*} egin{equation*} egin{aligned} &quad-int abla m_1 abla overline{psi_1} abla T_1(T_1-overline{T})^+ m dx +int frac{varepsilon m_1}{2 au}( abla psi_1)^2 T_1(T_1-overline{T})^+ m dx\ &=-int abla m_1 abla overline{psi_1} abla(T_1-overline{T}+overline{T})(T_1-overline{T})^+ m dx+int frac{varepsilon m_1}{2 au}( abla psi_1)^2 T_1(T_1-overline{T})^+ m dx \ &leq -int abla m_1 abla overline{psi_1} abla (T_1-overline{T})(T_1-overline{T})^+ m dx+int frac{varepsilon m_1}{2 au}( abla psi_1)^2 T_1(T_1-overline{T})^+ m dx\ &leq Cepsilon int abla m_1 abla overline{psi_1} abla ((T_1-overline{T})^+)^2 m dx+C(n_2varepsilon)epsilon^2int T_1(T_1-overline{T})^+ m dx \ &quad-int frac{3m_1}{2 au}(T_1-T_{1L})(T_1-overline{T})^+ m dx=-int frac{3m_1}{2 au}(T_1-overline{T}+overline{T}-T_{1L}(x))(T_1-overline{T})^+ m dx\ &=-int frac {3m_1}{2 au}(T_1-overline{T})(T_1-overline{T})^+ m dx-int frac{3m_1}{2 au}(overline{T}-T_{1L}(x))(T_1-overline{T})^+ m dx leq 0. end{aligned} end{equation*}
From Feilin Lan.