$a_{i}\,=\,\left\\{\begin{array}[]{lll}\displaystyle
C_{i}+\frac{1}{N_{c}}C_{i+1},&&\text{for odd }i;\\\ {}\\\ \displaystyle
C_{i}+\frac{1}{N_{c}}C_{i-1},&&\text{for even }i,\end{array}\right.$ (83)
$C_{m}\,{\cal A}_{ij}^{k}(M_{1},M_{2})\,=\,{\cal
A}_{i}^{k}(C_{m},M_{1},M_{2})+{\cal A}_{j}^{k}(C_{m},M_{1},M_{2}),$ (84)

where the explicit expressions of the amplitude building blocks ${\cal
A}_{i}^{k}(C_{m},M_{1},M_{2})$ including contributions from the $B$ mesonic
WFs ${\phi}_{B2}$ are given in Appendix B.

##  Appendix B the amplitude building blocks

For the sake of simplification and convenience, shorthand is used for the
amplitude building blocks.

${\phi}_{B1,B2}\,=\,{\phi}_{B1,B2}(x_{1},b_{1})\,e^{-S_{B}},$ (85)
${\phi}_{P}^{a}\,=\,{\phi}_{P}^{a}(x_{2})\,e^{-S_{P}},$ (86)
${\phi}_{P}^{p,t}\,=\,r_{P}\,{\phi}_{P}^{p,t}(x_{2})\,e^{-S_{P}},$ (87)
${\phi}_{V}^{v}\,=\,f_{V}^{\parallel}\,{\phi}_{V}^{v}(x_{3})\,e^{-S_{V}},$
(88)
${\phi}_{V}^{t,s}\,=\,r_{V}\,f_{V}^{\perp}\,{\phi}_{V}^{t,s}(x_{3})\,e^{-S_{V}},$
(89) ${\cal C}\,=\,\frac{{\pi}\,C_{F}}{N_{c}^{2}}\,m_{B}^{4}\,f_{B}\,f_{P},$
(90)

where $r_{P}$ $=$ ${\mu}_{P}/m_{B}$ and $r_{V}$ $=$ $m_{V}/m_{B}$. For the
amplitude building block ${\cal A}_{i}^{j}(M_{1},M_{2})$, the subscript $i$
corresponds to the indices of Fig. 4, and the superscript $j$ refers to the
three possible Dirac structures ${\Gamma}_{1}{\otimes}{\Gamma}_{2}$ of the
operator $(\bar{q}_{1}q_{2})_{{\Gamma}_{1}}(\bar{q}_{3}q_{4})_{{\Gamma}_{2}}$,
namely $j$ $=$ $LL$ for $(V-A){\otimes}(V-A)$, $j$ $=$ $LR$ for
$(V-A){\otimes}(V+A)$ and $j$ $=$ $SP$ for $-2\,(S-P){\otimes}(S+P)$. The
explicit expressions of ${\cal A}_{i}^{j}(M_{1},M_{2})$ up to the order of
$r_{P}$ and $r_{V}$ are written as follows.

$\displaystyle{\cal A}^{LL}_{a}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{ab}({\alpha}_{g}^{V},{\beta}_{a}^{V},b_{1},b_{3})\,{\alpha}_{s}(t_{a}^{V})\,C_{i}(t_{a}^{V})$
(91)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}{\phi}_{V}^{v}\,(1+x_{3})+\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,(\bar{x}_{3}-x_{3})\big{]}$
$\displaystyle-{\phi}_{B2}\,\big{[}{\phi}_{V}^{v}-\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,x_{3}\big{]}\big{\\}}\,S_{t}(x_{3}),$
${\cal A}^{LR}_{a}(P,V)\,=\,-{\cal A}^{LL}_{a}(P,V),$ (92) $\displaystyle{\cal
A}^{SP}_{a}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{ab}({\alpha}_{g}^{V},{\beta}_{a}^{V},b_{1},b_{3})\,{\alpha}_{s}(t_{a}^{V})\,C_{i}(t_{a}^{V})$
(93) $\displaystyle
2\,r_{P}\,\big{\\{}{\phi}_{B1}\,\big{[}-{\phi}_{V}^{v}+{\phi}_{V}^{t}\,x_{3}-{\phi}_{V}^{s}\,(2+x_{3})\big{]}$
$\displaystyle+{\phi}_{B2}\,\big{[}{\phi}_{V}^{v}-{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{]}\big{\\}}\,S_{t}(x_{3}),$
$\displaystyle{\cal A}^{LL}_{a}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,f_{V}^{\parallel}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{ab}({\alpha}_{g}^{P},{\beta}_{a}^{P},b_{1},b_{2})\,{\alpha}_{s}(t_{a}^{P})$
(94)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}{\phi}_{P}^{a}\,(1+x_{2})+\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,(\bar{x}_{2}-x_{2})\big{]}$
$\displaystyle-{\phi}_{B2}\,\big{[}{\phi}_{P}^{a}-\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,x_{2}\big{]}\big{\\}}\,C_{i}(t_{a}^{P})\,S_{t}(x_{2}),$
${\cal A}^{LR}_{a}(V,P)\,=\,{\cal A}^{LL}_{a}(V,P),$ (95) ${\cal
A}^{SP}_{a}(V,P)\,=\,0,$ (96) ${\cal A}^{LL}_{b}(P,V)\,=\,2\,{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{ab}({\alpha}_{g}^{V},{\beta}_{b}^{V},b_{3},b_{1})\,{\alpha}_{s}(t_{b}^{V})\,C_{i}(t_{b}^{V})\,S_{t}(x_{1})\,{\phi}_{B1}\,{\phi}_{V}^{s},$
(97) ${\cal A}^{LR}_{b}(P,V)\,=\,-{\cal A}^{LL}_{b}(P,V),$ (98)
$\displaystyle{\cal A}^{SP}_{b}(P,V)$ $\displaystyle=$ $\displaystyle-{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{ab}({\alpha}_{g}^{V},{\beta}_{b}^{V},b_{3},b_{1})\,{\alpha}_{s}(t_{b}^{V})\,C_{i}(t_{b}^{V})\,S_{t}(x_{1})$
(99) $\displaystyle
2\,r_{P}\,\big{\\{}{\phi}_{B1}\,\big{[}{\phi}_{V}^{v}\,x_{1}+2\,{\phi}_{V}^{s}\bar{x}_{1}\big{]}+2\,{\phi}_{B2}\,{\phi}_{V}^{s}\,x_{1}\big{]}\big{\\}},$
${\cal A}^{LL}_{b}(V,P)\,=\,2\,{\cal
C}\,f_{V}^{\parallel}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{ab}({\alpha}_{g}^{P},{\beta}_{b}^{P},b_{2},b_{1})\,{\alpha}_{s}(t_{b}^{P})\,C_{i}(t_{b}^{P})\,S_{t}(x_{1})\,{\phi}_{B1}\,{\phi}_{P}^{p},$
(100) ${\cal A}^{LR}_{b}(V,P)\,=\,{\cal A}^{LL}_{b}(V,P),$ (101) ${\cal
A}^{SP}_{b}(V,P)\,=\,0,$ (102) $\displaystyle{\cal A}^{LL}_{c}(P,V)$
$\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{cd}({\alpha}_{g}^{V},{\beta}_{c}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{c}^{V})\,C_{i}(t_{c}^{V})\,S_{t}(x_{3})$
(103)
$\displaystyle{\phi}_{P}^{a}\,\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,{\phi}_{V}^{v}\,(\bar{x}_{2}-x_{1})+{\phi}_{B1}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,x_{3}\big{\\}}_{b_{1}=b_{3}},$
$\displaystyle{\cal A}^{LR}_{c}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{cd}({\alpha}_{g}^{V},{\beta}_{c}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{c}^{V})\,C_{i}(t_{c}^{V})\,S_{t}(x_{3})$
(104)
$\displaystyle{\phi}_{P}^{a}\,\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{[}{\phi}_{V}^{v}\,(x_{1}-\bar{x}_{2})+\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,x_{3}\big{]}-{\phi}_{B1}\,{\phi}_{V}^{v}\,x_{3}\big{\\}}_{b_{1}=b_{3}},$
$\displaystyle{\cal A}^{SP}_{c}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{cd}({\alpha}_{g}^{V},{\beta}_{c}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{c}^{V})\,C_{i}(t_{c}^{V})\,S_{t}(x_{3})$
(105)
$\displaystyle\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{v}-{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,(x_{1}-\bar{x}_{2})$
$\displaystyle-{\phi}_{B1}\,\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,x_{3}\big{\\}}_{b_{1}=b_{3}},$
$\displaystyle{\cal A}^{LL}_{c}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{cd}({\alpha}_{g}^{P},{\beta}_{c}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{c}^{P})\,C_{i}(t_{c}^{P})\,S_{t}(x_{2})$
(106)
$\displaystyle{\phi}_{V}^{v}\,\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,{\phi}_{P}^{a}\,(\bar{x}_{3}-x_{1})-{\phi}_{B1}\,\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,x_{2}\big{\\}}_{b_{1}=b_{2}},$
$\displaystyle{\cal A}^{LR}_{c}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{cd}({\alpha}_{g}^{P},{\beta}_{c}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{c}^{P})\,C_{i}(t_{c}^{P})\,S_{t}(x_{2})$
(107)
$\displaystyle{\phi}_{V}^{v}\,\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{[}{\phi}_{P}^{a}\,(\bar{x}_{3}-x_{1})-\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,x_{2}\big{]}+{\phi}_{B1}\,{\phi}_{P}^{a}\,x_{2}\big{\\}}_{b_{1}=b_{2}},$
$\displaystyle{\cal A}^{SP}_{c}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{cd}({\alpha}_{g}^{P},{\beta}_{c}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{c}^{P})\,C_{i}(t_{c}^{P})\,S_{t}(x_{2})$
(108)
$\displaystyle\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{(}{\phi}_{P}^{a}+{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,(\bar{x}_{3}-x_{1})$
$\displaystyle-{\phi}_{B1}\,\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,x_{2}\big{\\}}_{b_{1}=b_{2}},$
$\displaystyle{\cal A}^{LL}_{d}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{cd}({\alpha}_{g}^{V},{\beta}_{d}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{d}^{V})\,C_{i}(t_{d}^{V})\,S_{t}(x_{3})$
(109)
$\displaystyle{\phi}_{p}^{a}\,\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{[}{\phi}_{V}^{v}\,(x_{1}-x_{2})+\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,x_{3}\big{]}-{\phi}_{B1}\,{\phi}_{V}^{v}\,x_{3}\big{\\}}_{b_{1}=b_{3}},$
$\displaystyle{\cal A}^{LR}_{d}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{cd}({\alpha}_{g}^{V},{\beta}_{d}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{d}^{V})\,C_{i}(t_{d}^{V})\,S_{t}(x_{3})$
(110)
$\displaystyle{\phi}_{p}^{a}\,\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,{\phi}_{V}^{v}\,(x_{2}-x_{1})+{\phi}_{B1}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,x_{3}\big{\\}}_{b_{1}=b_{3}},$
$\displaystyle{\cal A}^{SP}_{d}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{cd}({\alpha}_{g}^{V},{\beta}_{d}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{d}^{V})\,C_{i}(t_{d}^{V})\,S_{t}(x_{3})$
(111)
$\displaystyle\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{v}-{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,(x_{2}-x_{1})$
$\displaystyle+{\phi}_{B1}\,\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,x_{3}\big{\\}}_{b_{1}=b_{3}},$
$\displaystyle{\cal A}^{LL}_{d}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{cd}({\alpha}_{g}^{P},{\beta}_{d}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{d}^{P})\,C_{i}(t_{d}^{P})\,S_{t}(x_{2})$
(112)
$\displaystyle{\phi}_{V}^{v}\,\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{[}{\phi}_{P}^{a}\,(x_{1}-x_{3})+\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,x_{2}\big{]}-{\phi}_{B1}\,{\phi}_{P}^{a}\,x_{2}\big{\\}}_{b_{1}=b_{2}},$
$\displaystyle{\cal A}^{LR}_{d}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{cd}({\alpha}_{g}^{P},{\beta}_{d}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{d}^{P})\,C_{i}(t_{d}^{P})\,S_{t}(x_{2})$
(113)
$\displaystyle{\phi}_{V}^{v}\,\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,{\phi}_{P}^{a}\,(x_{1}-x_{3})+{\phi}_{B1}\,\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,x_{2}\big{\\}}_{b_{1}=b_{2}},$
$\displaystyle{\cal A}^{SP}_{d}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{cd}({\alpha}_{g}^{P},{\beta}_{d}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{d}^{P})\,C_{i}(t_{d}^{P})\,S_{t}(x_{2})$
(114)
$\displaystyle\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{(}{\phi}_{P}^{a}+{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,(x_{3}-x_{1})$
$\displaystyle-{\phi}_{B1}\,\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,x_{2}\big{\\}}_{b_{1}=b_{2}},$
$\displaystyle{\cal A}^{LL}_{e}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{2}\,\mathbbm{d}b_{3}\,H_{ef}({\alpha}_{a}^{V},{\beta}_{e}^{V},b_{2},b_{3})\,{\alpha}_{s}(t_{e}^{V})\,C_{i}(t_{e}^{V})$
(115)
$\displaystyle\big{\\{}2\,{\phi}_{P}^{p}\,\big{[}{\phi}_{V}^{t}\,x_{3}+{\phi}_{V}^{s}\,(1+\bar{x}_{3})\big{]}-{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,\bar{x}_{3}\big{\\}}\,S_{t}(\bar{x}_{3}),$
${\cal A}^{LR}_{e}(P,V)\,=\,-{\cal A}^{LL}_{e}(P,V),$ (116)
$\displaystyle{\cal A}^{SP}_{e}(P,V)$ $\displaystyle=$ $\displaystyle 2\,{\cal
C}\,{\int}\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{2}\,\mathbbm{d}b_{3}\,H_{ef}({\alpha}_{a}^{V},{\beta}_{e}^{V},b_{2},b_{3})\,{\alpha}_{s}(t_{e}^{V})$
(117) $\displaystyle
C_{i}(t_{e}^{V})\,S_{t}(\bar{x}_{3})\,\big{\\{}{\phi}_{P}^{a}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,\bar{x}_{3}-2\,{\phi}_{P}^{p}\,{\phi}_{V}^{v}\big{\\}},$
$\displaystyle{\cal A}^{LL}_{e}(V,P)$ $\displaystyle=$ $\displaystyle-{\cal
C}\,{\int}\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{2}\,\mathbbm{d}b_{3}\,H_{ef}({\alpha}_{a}^{P},{\beta}_{e}^{P},b_{3},b_{2})\,{\alpha}_{s}(t_{e}^{P})\,C_{i}(t_{e}^{P})$
(118)
$\displaystyle\big{\\{}{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,\bar{x}_{2}+2\,{\phi}_{V}^{s}\,\big{[}{\phi}_{P}^{p}\,(1+\bar{x}_{2})+{\phi}_{P}^{t}\,x_{2}\big{]}\big{\\}}\,S_{t}(\bar{x}_{2}),$
${\cal A}^{LR}_{e}(V,P)\,=\,-{\cal A}^{LL}_{e}(V,P),$ (119)
$\displaystyle{\cal A}^{SP}_{e}(V,P)$ $\displaystyle=$ $\displaystyle 2\,{\cal
C}\,{\int}\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{2}\,\mathbbm{d}b_{3}\,H_{ef}({\alpha}_{a}^{P},{\beta}_{e}^{P},b_{3},b_{2})\,{\alpha}_{s}(t_{e}^{P})$
(120) $\displaystyle
C_{i}(t_{e}^{P})\,S_{t}(\bar{x}_{2})\,\big{\\{}2\,{\phi}_{P}^{a}\,{\phi}_{V}^{s}+{\phi}_{V}^{v}\,\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,\bar{x}_{2}\big{\\}},$
$\displaystyle{\cal A}^{LL}_{f}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{2}\,\mathbbm{d}b_{3}\,H_{ef}({\alpha}_{a}^{V},{\beta}_{f}^{V},b_{3},b_{2})\,{\alpha}_{s}(t_{f}^{V})\,C_{i}(t_{f}^{V})$
(121)
$\displaystyle\big{\\{}{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,x_{2}-2\,{\phi}_{V}^{s}\,\big{[}{\phi}_{P}^{p}\,(1+x_{2})-{\phi}_{P}^{t}\,\bar{x}_{2}\big{]}\big{\\}}\,S_{t}(x_{2}),$
${\cal A}^{LR}_{f}(P,V)\,=\,-{\cal A}^{LL}_{f}(P,V),$ (122)
$\displaystyle{\cal A}^{SP}_{f}(P,V)$ $\displaystyle=$ $\displaystyle 2\,{\cal
C}\,{\int}\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{2}\,\mathbbm{d}b_{3}\,H_{ef}({\alpha}_{a}^{V},{\beta}_{f}^{V},b_{3},b_{2})\,{\alpha}_{s}(t_{f}^{V})$
(123) $\displaystyle
C_{i}(t_{f}^{V})\,S_{t}(x_{2})\,\big{\\{}2\,{\phi}_{P}^{a}\,{\phi}_{V}^{s}\,-\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,{\phi}_{V}^{v}\,x_{2}\big{\\}},$
$\displaystyle{\cal A}^{LL}_{f}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{2}\,\mathbbm{d}b_{3}\,H_{ef}({\alpha}_{a}^{P},{\beta}_{f}^{P},b_{2},b_{3})\,{\alpha}_{s}(t_{f}^{P})\,C_{i}(t_{f}^{P})$
(124)
$\displaystyle\big{\\{}{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,x_{3}-2\,{\phi}_{P}^{p}\,\big{[}{\phi}_{V}^{t}\,\bar{x}_{3}-{\phi}_{V}^{s}\,(1+x_{3})\big{]}\big{\\}}\,S_{t}(x_{3}),$
${\cal A}^{LR}_{f}(V,P)\,=\,-{\cal A}^{LL}_{f}(V,P),$ (125)
$\displaystyle{\cal A}^{SP}_{f}(V,P)$ $\displaystyle=$ $\displaystyle 2\,{\cal
C}\,{\int}\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{2}\,\mathbbm{d}b_{3}\,H_{ef}({\alpha}_{a}^{P},{\beta}_{f}^{P},b_{2},b_{3})\,{\alpha}_{s}(t_{f}^{P})$
(126) $\displaystyle
C_{i}(t_{f}^{P})\,S_{t}(x_{3})\,\big{\\{}2\,{\phi}_{P}^{p}\,{\phi}_{V}^{v}\,-{\phi}_{P}^{a}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,x_{3}\big{\\}},$
$\displaystyle{\cal A}^{LL}_{g}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{gh}({\alpha}_{a}^{V},{\beta}_{g}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{g}^{V})\,C_{i}(t_{g}^{V})$
(127)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{t}-{\phi}_{P}^{t}\,{\phi}_{V}^{s}\big{)}\,(\bar{x}_{3}-x_{2})-{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(x_{1}+x_{2})$
$\displaystyle\qquad+\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{s}-{\phi}_{P}^{t}\,{\phi}_{V}^{t}\big{)}\,(x_{2}+\bar{x}_{3}-2\,\bar{x}_{1})+4\,{\phi}_{P}^{p}\,{\phi}_{V}^{s}\big{]}$
$\displaystyle+{\phi}_{B2}\,\big{[}\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,(x_{3}-x_{1})-2\,\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,{\phi}_{V}^{s}\big{]}\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{LR}_{g}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{gh}({\alpha}_{a}^{V},{\beta}_{g}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{g}^{V})\,C_{i}(t_{g}^{V})$
(128)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{t}-{\phi}_{P}^{t}\,{\phi}_{V}^{s}\big{)}\,(\bar{x}_{3}-x_{2})+{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(x_{1}+\bar{x}_{3})$
$\displaystyle\qquad-\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{s}-{\phi}_{P}^{t}\,{\phi}_{V}^{t}\big{)}\,(x_{2}+\bar{x}_{3}-2\,\bar{x}_{1})-4\,{\phi}_{P}^{p}\,{\phi}_{V}^{s}\big{]}$
$\displaystyle+{\phi}_{B2}\,\big{[}\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,(x_{3}-x_{1})$
$\displaystyle\qquad-2\,{\phi}_{P}^{p}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}-{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(x_{1}+\bar{x}_{3})\big{]}\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{SP}_{g}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{gh}({\alpha}_{a}^{V},{\beta}_{g}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{g}^{V})\,C_{i}(t_{g}^{V})$
(129)
$\displaystyle\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{[}{\phi}_{P}^{a}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,(x_{3}-x_{1})-\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,{\phi}_{V}^{v}\big{]}$
$\displaystyle+{\phi}_{B1}\,\big{[}{\phi}_{P}^{a}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}+\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,{\phi}_{V}^{v}\,(x_{2}-\bar{x}_{1})\big{]}\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{LL}_{g}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{gh}({\alpha}_{a}^{P},{\beta}_{g}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{g}^{P})\,C_{i}(t_{g}^{P})$
(130)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{t}-{\phi}_{P}^{t}\,{\phi}_{V}^{s}\big{)}\,(\bar{x}_{2}-x_{3})-{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(x_{1}+x_{3})$
$\displaystyle\qquad-\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{s}-{\phi}_{P}^{t}\,{\phi}_{V}^{t}\big{)}\,(\bar{x}_{2}+x_{3}-2\,\bar{x}_{1})-4\,{\phi}_{P}^{p}\,{\phi}_{V}^{s}\big{]}$
$\displaystyle+{\phi}_{B2}\,\big{[}\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,(x_{3}-\bar{x}_{1})$
$\displaystyle\qquad+2\,{\phi}_{P}^{p}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}+{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(x_{1}+x_{3})\big{]}\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{LR}_{g}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{gh}({\alpha}_{a}^{P},{\beta}_{g}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{g}^{P})\,C_{i}(t_{g}^{P})$
(131)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{t}-{\phi}_{P}^{t}\,{\phi}_{V}^{s}\big{)}\,(\bar{x}_{2}-x_{3})+{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(x_{1}+\bar{x}_{2})$
$\displaystyle\qquad+\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{s}-{\phi}_{P}^{t}\,{\phi}_{V}^{t}\big{)}\,(\bar{x}_{2}+x_{3}-2\,\bar{x}_{1})+4\,{\phi}_{P}^{p}\,{\phi}_{V}^{s}\big{]}$
$\displaystyle+{\phi}_{B2}\,\big{[}\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,(x_{3}-\bar{x}_{1})-2\,\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,{\phi}_{V}^{s}\big{]}\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{SP}_{g}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{gh}({\alpha}_{a}^{P},{\beta}_{g}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{g}^{P})\,C_{i}(t_{g}^{P})$
(132)
$\displaystyle\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{[}{\phi}_{P}^{a}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,(\bar{x}_{1}-x_{3})-\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,{\phi}_{V}^{v}\big{]}$
$\displaystyle+{\phi}_{B1}\,\big{[}{\phi}_{P}^{a}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}+\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,{\phi}_{V}^{v}\,(x_{1}-x_{2})\big{]}\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{LL}_{h}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{gh}({\alpha}_{a}^{V},{\beta}_{h}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{h}^{V})\,C_{i}(t_{h}^{V})$
(133)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{t}-{\phi}_{P}^{t}\,{\phi}_{V}^{s}\big{)}\,(\bar{x}_{3}-x_{2})+{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(\bar{x}_{3}-x_{1})$
$\displaystyle\qquad-\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{s}-{\phi}_{P}^{t}\,{\phi}_{V}^{t}\big{)}\,(x_{2}+\bar{x}_{3}-2\,x_{1})\big{]}$
$\displaystyle+{\phi}_{B2}\,\big{[}{\phi}_{P}^{a}\,{\phi}_{V}^{v}+\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,\big{]}\,(x_{1}-\bar{x}_{3})\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{LR}_{h}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{gh}({\alpha}_{a}^{V},{\beta}_{h}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{h}^{V})\,C_{i}(t_{h}^{V})$
(134)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{t}-{\phi}_{P}^{t}\,{\phi}_{V}^{s}\big{)}\,(\bar{x}_{3}-x_{2})+{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(x_{1}-x_{2})$
$\displaystyle\qquad+\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{s}-{\phi}_{P}^{t}\,{\phi}_{V}^{t}\big{)}\,(x_{2}+\bar{x}_{3}-2\,x_{1})\big{]}$
$\displaystyle+{\phi}_{B2}\,\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,(x_{1}-\bar{x}_{3})\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{SP}_{h}(P,V)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{2}\,H_{gh}({\alpha}_{a}^{V},{\beta}_{h}^{V},b_{1},b_{2})\,{\alpha}_{s}(t_{h}^{V})\,C_{i}(t_{h}^{V})$
(135)
$\displaystyle\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,{\phi}_{V}^{v}\,(x_{1}-x_{2})$
$\displaystyle+{\phi}_{B1}\,{\phi}_{P}^{a}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}(\bar{x}_{3}-x_{1})\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{LL}_{h}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{gh}({\alpha}_{a}^{P},{\beta}_{h}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{h}^{P})\,C_{i}(t_{h}^{P})$
(136)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{t}-{\phi}_{P}^{t}\,{\phi}_{V}^{s}\big{)}\,(\bar{x}_{2}-x_{3})+{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(\bar{x}_{2}-x_{1})$
$\displaystyle\qquad+\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{s}-{\phi}_{P}^{t}\,{\phi}_{V}^{t}\big{)}\,(\bar{x}_{2}+x_{3}-2\,x_{1})\big{]}$
$\displaystyle+{\phi}_{B2}\,\big{(}{\phi}_{P}^{p}+{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}-{\phi}_{V}^{s}\big{)}\,(x_{3}-x_{1})\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{LR}_{h}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{gh}({\alpha}_{a}^{P},{\beta}_{h}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{h}^{P})\,C_{i}(t_{h}^{P})$
(137)
$\displaystyle\big{\\{}{\phi}_{B1}\,\big{[}\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{t}-{\phi}_{P}^{t}\,{\phi}_{V}^{s}\big{)}\,(\bar{x}_{2}-x_{3})+{\phi}_{P}^{a}\,{\phi}_{V}^{v}\,(x_{1}-x_{3})$
$\displaystyle\qquad-\big{(}{\phi}_{P}^{p}\,{\phi}_{V}^{s}-{\phi}_{P}^{t}\,{\phi}_{V}^{t}\big{)}\,(\bar{x}_{2}+x_{3}-2\,x_{1})\big{]}$
$\displaystyle+{\phi}_{B2}\,\big{[}{\phi}_{P}^{a}\,{\phi}_{V}^{v}+\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}\,\big{]}\,(x_{3}-x_{1})\big{\\}}_{b_{2}=b_{3}},$
$\displaystyle{\cal A}^{SP}_{h}(V,P)$ $\displaystyle=$ $\displaystyle{\cal
C}\,{\int}\mathbbm{d}x_{1}\,\mathbbm{d}x_{2}\,\mathbbm{d}x_{3}\,\mathbbm{d}b_{1}\,\mathbbm{d}b_{3}\,H_{gh}({\alpha}_{a}^{P},{\beta}_{h}^{P},b_{1},b_{3})\,{\alpha}_{s}(t_{h}^{P})\,C_{i}(t_{h}^{P})$
(138)
$\displaystyle\big{\\{}\big{(}{\phi}_{B1}-{\phi}_{B2}\big{)}\,\big{(}{\phi}_{P}^{p}-{\phi}_{P}^{t}\big{)}\,{\phi}_{V}^{v}\,(x_{1}-\bar{x}_{2})$
$\displaystyle+{\phi}_{B1}\,{\phi}_{P}^{a}\,\big{(}{\phi}_{V}^{t}+{\phi}_{V}^{s}\big{)}(x_{3}-x_{1})\big{\\}}_{b_{2}=b_{3}},$
${\alpha}_{g}^{V}\,=\,m_{B}^{2}\,x_{1}\,x_{3},$ (139)
${\alpha}_{g}^{P}\,=\,m_{B}^{2}\,x_{1}\,x_{2},$ (140)
${\alpha}_{a}^{V}\,=\,m_{B}^{2}\,x_{2}\,\bar{x}_{3},$ (141)
${\alpha}_{a}^{P}\,=\,m_{B}^{2}\,\bar{x}_{2}\,x_{3},$ (142)
${\beta}_{a}^{V}\,=\,m_{B}^{2}\,x_{3},$ (143)
${\beta}_{a}^{P}\,=\,m_{B}^{2}\,x_{2},$ (144)
${\beta}_{b}^{V}\,=\,{\beta}_{b}^{P}\,=\,m_{B}^{2}\,x_{1},$ (145)
${\beta}_{c}^{V}\,=\,m_{B}^{2}\,x_{3}\,(x_{1}-\bar{x}_{2}),$ (146)
${\beta}_{c}^{P}\,=\,m_{B}^{2}\,x_{2}\,(x_{1}-\bar{x}_{3}),$ (147)
${\beta}_{d}^{V}\,=\,m_{B}^{2}\,x_{3}\,(x_{1}-x_{2}),$ (148)
${\beta}_{d}^{P}\,=\,m_{B}^{2}\,x_{2}\,(x_{1}-x_{3}),$ (149)
${\beta}_{e}^{V}\,=\,m_{B}^{2}\,\bar{x}_{3},$ (150)
${\beta}_{e}^{P}\,=\,m_{B}^{2}\,\bar{x}_{2},$ (151)
${\beta}_{f}^{V}\,=\,m_{B}^{2}\,x_{2},$ (152)
${\beta}_{f}^{P}\,=\,m_{B}^{2}\,x_{3},$ (153)
${\beta}_{g}^{V}\,=\,{\alpha}_{a}^{V}-m_{B}^{2}\,\bar{x}_{1}\,(x_{2}+\bar{x}_{3}),$
(154)
${\beta}_{g}^{P}\,=\,{\alpha}_{a}^{P}-m_{B}^{2}\,\bar{x}_{1}\,(\bar{x}_{2}+x_{3}),$
(155)
${\beta}_{h}^{V}\,=\,{\alpha}_{a}^{V}-m_{B}^{2}\,x_{1}\,(x_{2}+\bar{x}_{3}),$
(156)
${\beta}_{h}^{P}\,=\,{\alpha}_{a}^{P}-m_{B}^{2}\,x_{1}\,(\bar{x}_{2}+x_{3}),$
(157)
$t_{a,b}^{V}\,=\,{\max}({\alpha}_{g}^{V},{\beta}_{a,b}^{V},b_{1},b_{3}),$
(158)
$t_{a,b}^{P}\,=\,{\max}({\alpha}_{g}^{P},{\beta}_{a,b}^{P},b_{1},b_{2}),$
(159)
$t_{c,d}^{i}\,=\,{\max}({\alpha}_{g}^{i},{\beta}_{c,d}^{i},b_{2},b_{3}),$
(160)
$t_{e,f}^{i}\,=\,{\max}({\alpha}_{a}^{i},{\beta}_{e,f}^{i},b_{2},b_{3}),$
(161)
$t_{g,h}^{i}\,=\,{\max}({\alpha}_{a}^{i},{\beta}_{g,h}^{i},b_{1},b_{2}),$
(162)
$H_{ab}({\alpha},{\beta},b_{i},b_{j})\,=\,b_{i}\,b_{j}\,K_{0}\big{(}b_{i}\,\sqrt{\alpha}\big{)}\,\big{\\{}{\theta}\big{(}b_{i}-b_{j}\big{)}\,K_{0}\big{(}b_{i}\,\sqrt{\beta}\big{)}\,I_{0}\big{(}b_{j}\,\sqrt{\beta}\big{)}+\big{(}b_{i}\,{\leftrightarrow}\,b_{j}\big{)}\big{\\}},$
(163) $\displaystyle N_{c}\,H_{cd}({\alpha},{\beta},b_{1},b_{i})$
$\displaystyle=$ $\displaystyle
b_{1}\,b_{i}\,\big{\\{}{\theta}(b_{1}-b_{2})\,K_{0}\big{(}b_{1}\sqrt{{\alpha}}\big{)}\,I_{0}\big{(}b_{i}\sqrt{{\alpha}}\big{)}+\big{(}b_{1}\,{\leftrightarrow}\,b_{i}\big{)}\big{\\}}$
(164)
$\displaystyle\big{\\{}{\theta}({\beta})\,K_{0}\big{(}b_{i}\sqrt{\beta}\big{)}+i\,\frac{\pi}{2}\,{\theta}(-{\beta})\,\big{[}J_{0}\big{(}b_{i}\sqrt{-{\beta}}\big{)}+i\,Y_{0}\big{(}b_{i}\sqrt{-{\beta}}\big{)}\big{]}\big{\\}},$
$\displaystyle
H_{ef}({\alpha},{\beta},b_{i},b_{j})\,=\,-\frac{{\pi}^{2}}{4}\,b_{i}\,b_{j}\,\big{\\{}J_{0}\big{(}b_{i}\,\sqrt{\alpha}\big{)}+i\,Y_{0}\big{(}b_{i}\,\sqrt{\alpha}\big{)}\big{\\}}$
$\displaystyle\big{\\{}{\theta}\big{(}b_{i}-b_{j}\big{)}\,\big{[}J_{0}\big{(}b_{i}\,\sqrt{\beta}\big{)}+i\,Y_{0}\big{(}b_{i}\,\sqrt{\beta}\big{)}\big{]}\,J_{0}\big{(}b_{j}\,\sqrt{\beta}\big{)}+\big{(}b_{i}\,{\leftrightarrow}\,b_{j}\big{)}\big{\\}},$
(165) $\displaystyle N_{c}\,H_{gh}({\alpha},{\beta},b_{1},b_{i})$ (166)
$\displaystyle=$ $\displaystyle
b_{1}\,b_{i}\,\big{\\{}\frac{i\,{\pi}}{2}\,{\theta}\big{(}{\beta}\big{)}\,\big{[}J_{0}\big{(}b_{1}\,\sqrt{{\beta}}\big{)}+i\,Y_{0}\big{(}b_{1}\,\sqrt{{\beta}}\big{)}\big{]}+{\theta}\big{(}-{\beta}\big{)}\,K_{0}\big{(}b_{1}\,\sqrt{-{\beta}}\big{)}\big{\\}}$
$\displaystyle\frac{i\,{\pi}}{2}\,\big{\\{}{\theta}\big{(}b_{1}-b_{i}\big{)}\,\big{[}J_{0}\big{(}b_{1}\,\sqrt{\alpha}\big{)}+i\,Y_{0}\big{(}b_{1}\,\sqrt{\alpha}\big{)}\big{]}\,J_{0}\big{(}b_{i}\,\sqrt{\alpha}\big{)}+\big{(}b_{1}\,{\leftrightarrow}\,b_{i}\big{)}\big{\\}}.$

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$\displaystyle\widetilde{u}^{\tau}\big{|}\big{|}^{2}_{L^{2}(\mathbb{R}\times(0,T))}=\int_{0}^{T}\int_{-\delta}^{0}\left|\partial_{x}u^{\tau}(-x,t)\right|^{2}\,dxdt+\int_{0}^{T}\int_{0}^{X^{0}}\left|\partial_{x}u^{\tau}(x,t)\right|^{2}\,dxdt$
$\displaystyle=\int_{0}^{T}\int_{-\delta}^{0}\left|\partial_{x}\rho^{\tau}(-x,t)\right|^{2}\,dxdt+\int_{0}^{T}\int_{0}^{X^{0}}\left|\partial_{x}\rho^{\tau}(x,t)\,\chi(x)+\rho^{\tau}(x,t)\,\chi^{\prime}(x)\right|^{2}\,dxdt.$

Now, we directly obtain

$\displaystyle\big{|}\big{|}\partial_{x}\widetilde{u}^{\tau}\big{|}\big{|}^{2}_{L^{2}(\mathbb{R}\times(0,T))}\leq\left(3+2\left|\left|\chi^{\prime}\right|\right|^{2}_{L^{\infty}(0,X^{0})}\right)\,\left|\left|\rho^{\tau}\right|\right|^{2}_{L^{2}(0,T;H^{1}(0,X^{0}))},$

and applying Proposition 4.2 yields the existence of the constant $C_{2}$ such
that (79) holds.
Now for the point (ii), using similar arguments and standard properties
achieved by the convolution product we deduce that the function
$u^{\tau}_{\delta}$ satisfies the estimates (77)–(79). Furthermore, since
$\zeta$ is an even function, we directly deduce the equality
$u^{\tau}_{\delta}(0,t)=u^{\tau}(0,t)$ for a.e. $t\in(0,T)$. In order to
establish estimate (80), we first notice that

$\displaystyle\int_{0}^{T}\int_{0}^{X^{0}}\frac{(u^{\tau}_{\delta}-u^{\tau})^{2}}{\tau}$
$\displaystyle=\sum_{n=0}^{N_{T}-1}\int_{0}^{X^{0}}\left(u^{n+1}_{\delta}-u^{n+1}\right)^{2}(x)\,dx$
$\displaystyle=\sum_{n=0}^{N_{T}-1}\int_{0}^{X^{0}}\left[\int_{-\delta}^{\delta}\left(\widetilde{u}^{n+1}(x-y)-u^{n+1}(x)\right)\,\zeta_{\delta}(y)\,dy\right]^{2}\,dx$
$\displaystyle=\sum_{n=0}^{N_{T}-1}\int_{0}^{X^{0}}\left[\int_{-\delta}^{\delta}\left(\widetilde{u}^{n+1}(x-y)-\widetilde{u}^{n+1}(x)\right)\,\zeta_{\delta}(y)\,dy\right]^{2}\,dx.$

Then, thanks to the Cauchy-Schwarz inequality and the regularity of the
function $\zeta$, we get

$\displaystyle\int_{0}^{T}\int_{0}^{X^{0}}\frac{(u^{\tau}_{\delta}-u^{\tau})^{2}}{\tau}$
$\displaystyle\leq\sum_{n=0}^{N_{T}-1}\left(\int_{0}^{X^{0}}\int_{-\delta}^{\delta}\left|\widetilde{u}^{n+1}(x-y)-\widetilde{u}^{n+1}(x)\right|^{2}\,dy\,dx\right)\,\left(\int_{-\delta}^{\delta}\left|\zeta_{\delta}(y)\right|^{2}\,dy\right)$
$\displaystyle\leq\frac{||\zeta||^{2}_{L^{\infty}(-1,1)}}{\delta}\sum_{n=0}^{N_{T}-1}\int_{0}^{X^{0}}\int_{-\delta}^{\delta}\left|\widetilde{u}^{n+1}(x-y)-\widetilde{u}^{n+1}(x)\right|^{2}\,dy\,dx.$

Then, since the function $u^{\tau}$ and its extension $\widetilde{u}^{\tau}$
are Lipchitz continuous we have

$\displaystyle\int_{0}^{T}\int_{0}^{X^{0}}\frac{(u^{\tau}_{\delta}-u^{\tau})^{2}}{\tau}$
$\displaystyle\leq\frac{||\zeta||^{2}_{L^{\infty}(-1,1)}}{\delta}\sum_{n=0}^{N_{T}-1}\left|\left|\partial_{x}\widetilde{u}^{\tau}\right|\right|^{2}_{L^{\infty}(\mathbb{R}\times(0,T))}\int_{0}^{X^{0}}\int_{-\delta}^{\delta}y^{2}\,dy\,dx$
$\displaystyle\leq\frac{2\,X^{0}\,||\zeta||^{2}_{L^{\infty}(-1,1)}}{3}\sum_{n=0}^{N_{T}-1}\left|\left|\partial_{x}\widetilde{u}^{\tau}\right|\right|^{2}_{L^{\infty}(\mathbb{R}\times(0,T))}\delta^{2}.$

Hence, applying (78), we deduce that

$\displaystyle\int_{0}^{T}\int_{0}^{X^{0}}\frac{(u^{\tau}_{\delta}-u^{\tau})^{2}}{\tau}\leq\frac{2\,C_{1}\,X^{0}\,T\,||\zeta||^{2}_{L^{\infty}(-1,1)}}{3}\,\tau^{2\vartheta^{\prime}-2\vartheta-1},$

and we conclude the existence of the constant $C_{3}$ such that (80) holds.
Finally, the estimate (81) is consequence of (78) and the fact that
$\int\zeta^{\prime}_{\delta}(y)\,dy$ is bounded by $C/\delta$ for some
constant $C$ independent of $\delta$ which yields the existence of a constant
$C_{4}$, depending only on $\rho^{0}$ and $\zeta$, such that

$\displaystyle\left|\left|\partial^{2}_{x}u^{\tau}_{\delta}\right|\right|_{L^{\infty}(\mathbb{R}\times(0,T))}\leq
C_{4}\frac{\tau^{-\vartheta}}{\delta}=C_{4}\,\tau^{-\vartheta-\vartheta^{\prime}}.$

This completes the proof of Lemma 4.2. ∎

Let us now establish the semi-discrete variational inequality. In this
purpose, we consider a nonnegative function $\phi\in C^{\infty}_{0}([0,T))$
and $\eta\in C^{\infty}_{0}(\mathbb{R}_{+}\times[0,T))$ with
$\eta(0,t)\in[\rho_{-},\rho_{+}]$ for all $t\in[0,T)$. Then, taking
$u^{\tau}_{\delta}-\eta$ as a test function in (75), multiplying this equation
by $\phi$ and integrating in time we obtain

$\int_{\tau}^{T}\int_{0}^{X^{0}}\phi(t)\frac{u^{\tau}(x,t)-\sigma_{-\tau}u^{\tau}(x,t)}{\tau}\,\left(u^{\tau}_{\delta}-\eta\right)(x,t)\,dxdt-\int_{0}^{T}\phi(t)\,\dot{M^{\tau}}(t)\,\left(u^{\tau}_{\delta}-\eta\right)(0,t)\,dt\\\
+\int_{0}^{T}\int_{0}^{X^{0}}\phi(t)\partial_{x}u^{\tau}(x,t)\,\partial_{x}\left(u^{\tau}_{\delta}-\eta\right)(x,t)\,dxdt=\int_{0}^{T}\int_{0}^{X^{0}}\phi(t)\,g^{\tau}(x,t)\,\left(u^{\tau}_{\delta}-\eta\right)(x,t)\,dxdt\\\
+\sum_{n=0}^{N_{T}-1}\int_{n\tau}^{(n+1)\tau}\phi(t)\left(Q^{n+1}_{\tau}(\chi,u^{n+1}_{\delta})-Q^{n+1}_{\tau}(\chi,\eta(t))\right)\,dt,$

where we have used the linearity of the remaining term $Q^{n+1}_{\tau}$
(recall Proposition 3.2). Bearing in mind point (ii) of Lemma 4.2 we have
$u^{\tau}_{\delta}(0,t)=u^{\tau}(0,t)=\rho^{\tau}(0,t)$ for a.e. $t\in[0,T]$.
Hence, thanks to Proposition 3.3, we notice that

$\displaystyle-\int_{0}^{T}\phi(t)\,\dot{M^{\tau}}(t)\,\left(u^{\tau}_{\delta}-\eta\right)(0,t)\,dt=-\sum_{n=0}^{N_{T}-1}\int_{n\tau}^{(n+1)\tau}\phi(t)\,\frac{M\left(\rho^{n+1},\rho^{n}\right)}{\tau}\,\left(\rho^{n+1}(0)-\eta(0,t)\right)\,dt\geq
0.$

Therefore, it holds the following semi-discrete variational inequality:

(82)
$\int_{\tau}^{T}\int_{0}^{X^{0}}\phi(t)\,\frac{u^{\tau}(x,t)-\sigma_{-\tau}u^{\tau}(x,t)}{\tau}\,\left(u^{\tau}_{\delta}-\eta\right)(x,t)\,dxdt\\\
+\int_{0}^{T}\int_{0}^{X^{0}}\phi(t)\,\partial_{x}u^{\tau}(x,t)\,\partial_{x}\left(u^{\tau}_{\delta}-\eta\right)(x,t)\,dxdt\leq\int_{0}^{T}\int_{0}^{X^{0}}\phi(t)\,g^{\tau}(x,t)\,\left(u^{\tau}_{\delta}-\eta\right)(x,t)\,dxdt\\\
+\sum_{n=0}^{N_{T}-1}\int_{n\tau}^{(n+1)\tau}\phi(t)\,\left(Q^{n+1}_{\tau}(\chi,u^{n+1}_{\delta})-Q^{n+1}_{\tau}(\chi,\eta(t))\right)\,dt.$

We rewrite this inequality as

$\displaystyle A^{\tau}_{1}-A^{\tau}_{2}+A^{\tau}_{3}-A^{\tau}_{4}\leq
A^{\tau}_{5}+A^{\tau}_{6}-A^{\tau}_{7},$

with

$\displaystyle A^{\tau}_{1}$
$\displaystyle=\int_{\tau}^{T}\int_{0}^{X^{0}}\phi(t)\,\frac{u^{\tau}(x,t)-\sigma_{-\tau}u^{\tau}(x,t)}{\tau}\,u^{\tau}_{\delta}(x,t)\,dxdt,$
$\displaystyle A^{\tau}_{2}$
$\displaystyle=\int_{\tau}^{T}\int_{0}^{X^{0}}\phi(t)\,\frac{u^{\tau}(x,t)-\sigma_{-\tau}u^{\tau}(x,t)}{\tau}\,\eta(x,t)\,dxdt,$
$\displaystyle A^{\tau}_{3}$
$\displaystyle=\int_{0}^{T}\int_{0}^{X^{0}}\phi(t)\,\partial_{x}u^{\tau}(x,t)\,\partial_{x}u^{\tau}_{\delta}(x,t)\,dxdt,$
$\displaystyle A^{\tau}_{4}$
$\displaystyle=\int_{0}^{T}\int_{0}^{X^{0}}\phi(t)\,\partial_{x}u^{\tau}(x,t)\,\partial_{x}\eta(x,t)\,dxdt,$
$\displaystyle A^{\tau}_{5}$
$\displaystyle=\int_{0}^{T}\int_{0}^{X^{0}}\phi(t)\,g^{\tau}(x,t)\,\left(u^{\tau}_{\delta}-\eta\right)(x,t)\,dxdt,$
$\displaystyle A^{\tau}_{6}$
$\displaystyle=\sum_{n=0}^{N_{T}-1}\int_{n\tau}^{(n+1)\tau}\phi(t)\,Q^{n+1}_{\tau}(\chi,u^{n+1}_{\delta})\,dt,$
$\displaystyle A^{\tau}_{7}$
$\displaystyle=\sum_{n=0}^{N_{T}-1}\int_{n\tau}^{(n+1)\tau}\phi(t)\,Q^{n+1}_{\tau}(\chi,\eta(t))\,dt.$

In order to establish the variational inequality (10) it remains to pass to
the limit $\tau\downarrow 0$ in the above terms. This is the main objective of
the next result.

######  Proposition 4.8.

Let the assumptions of Theorem 1.1 hold and assume that $0<\tau<1$ is small
enough such that $\delta<X^{0}/2$. Moreover, we impose the following
conditions on $\vartheta$ and $\vartheta^{\prime}$

(83) $\displaystyle
3\vartheta+\vartheta^{\prime}<1,\quad\vartheta^{\prime}-\vartheta>1/2,\quad\vartheta^{\prime}>2\vartheta,\quad\vartheta<1/2.$

Then, for any nonnegative function $\phi\in C^{\infty}_{0}([0,T))$ and
$\eta\in C^{\infty}_{0}(\mathbb{R}_{+}\times[0,T))$ with
$\eta(0,t)\in[\rho_{-},\rho_{+}]$ for all $t\in[0,T)$, it holds

(84) $\displaystyle\liminf_{\tau\downarrow 0}A^{\tau}_{1}$
$\displaystyle\geq-\int_{0}^{T}\dot{\phi}\int_{0}^{X^{0}}\frac{u^{2}}{2}\,dxdt-\phi(0)\int_{0}^{X^{0}}\frac{u^{2}}{2}(x,0)\,dt,$
(85) $\displaystyle\lim_{\tau\downarrow 0}A^{\tau}_{2}$
$\displaystyle=-\int_{0}^{T}\dot{\phi}\int_{0}^{X^{0}}u\,\eta\,dxdt-\int_{0}^{T}\int_{0}^{X^{0}}\phi\,u\,\partial_{t}\eta\,dxdt-\phi(0)\int_{0}^{X^{0}}u(x,0)\,\eta(x,0)\,dx,$
(86) $\displaystyle\liminf_{\tau\downarrow 0}A^{\tau}_{3}$
$\displaystyle\geq\int_{0}^{T}\int_{0}^{X^{0}}\phi\,\left(\partial_{x}u\right)^{2}\,dxdt,$
(87) $\displaystyle\lim_{\tau\downarrow 0}A^{\tau}_{4}$
$\displaystyle=\int_{0}^{T}\int_{0}^{X^{0}}\phi\,\partial_{x}u\,\partial_{x}\eta\,dxdt,$
(88) $\displaystyle\lim_{\tau\downarrow 0}A^{\tau}_{5}$
$\displaystyle=\int_{0}^{T}\int_{0}^{X^{0}}\phi\,g\,(u-\eta)\,dxdt,$

and

(89) $\displaystyle\lim_{\tau\downarrow 0}A^{\tau}_{6}=\lim_{\tau\downarrow
0}A^{\tau}_{7}=0.$

###### Proof.

Let us first notice that $\vartheta=1/15$ and $\vartheta^{\prime}=3/4$ satisfy
the conditions (83). Let us also notice that the limits (85), (87), (88) and
$\lim_{\tau\downarrow 0}A^{\tau}_{7}=0$ are directly deduced thanks to
Proposition 4.3 and the techniques used in Section 4.3.1. Now, we rewrite the
term $A^{\tau}_{1}$ as

$\displaystyle A^{\tau}_{1}$
$\displaystyle=\int_{\tau}^{T}\int_{0}^{X^{0}}\phi\,\frac{u^{\tau}-\sigma_{-\tau}u^{\tau}}{\tau}u^{\tau}\,dxdt+\int_{\tau}^{T}\int_{0}^{X^{0}}\phi\,\frac{u^{\tau}-\sigma_{-\tau}u^{\tau}}{\tau}\,(u^{\tau}_{\delta}-u^{\tau})\,dxdt=A^{\tau}_{11}+A^{\tau}_{12}.$

Then, we split the term $A^{\tau}_{11}$ as

$\displaystyle A^{\tau}_{11}$
$\displaystyle=\frac{1}{2}\int_{\tau}^{T}\int_{0}^{X^{0}}\phi\,\frac{\left(u^{\tau}\right)^{2}-\left(\sigma_{-\tau}u^{\tau}\right)^{2}}{\tau}\,dxdt+\frac{1}{2}\int_{\tau}^{T}\int_{0}^{X^{0}}\phi\,\frac{\left(u^{\tau}-\sigma_{-\tau}u^{\tau}\right)^{2}}{\tau}\,dxdt=A^{\tau}_{111}+A^{\tau}_{112}.$

Rearranging the terms in $A^{\tau}_{111}$ we obtain

$\displaystyle
A^{\tau}_{111}=-\frac{1}{2}\int_{0}^{T-\tau}\int_{0}^{X^{0}}\frac{\sigma_{\tau}\phi-\phi}{\tau}\,\left(u^{\tau}\right)^{2}dxdt-\frac{1}{2\tau}\int_{0}^{\tau}\int_{0}^{X^{0}}\phi\left(u^{\tau}\right)^{2}dxdt+\frac{1}{2\tau}\int_{T-\tau}^{T}\int_{0}^{X^{0}}\phi\left(u^{\tau}\right)^{2}dxdt.$

Thus, as $\phi\in C^{\infty}_{0}([0,T))$ and since, as $\tau\downarrow 0$, we
have (up to a subsequence) $u^{\tau}\to u:=\chi\,\rho$ strongly in
$L^{2}(\mathbb{R}_{+}\times(0,T))$, we deduce that

(90) $\displaystyle\lim_{\tau\downarrow
0}A^{\tau}_{111}=-\int_{0}^{T}\dot{\phi}\int_{0}^{X^{0}}\frac{u^{2}}{2}\,dxdt-\phi(0)\int_{0}^{X^{0}}\frac{u^{2}}{2}(x,0)\,dt.$

Now, as $\phi$ is a nonnegative function and thanks to (80), we have

(91) $\displaystyle-A_{12}-A_{112}$
$\displaystyle\leq\frac{1}{2}\int_{0}^{T}\int_{0}^{X^{0}}\phi\,\frac{(u^{\tau}-u^{\tau}_{\delta})^{2}}{\tau}\,dxdt\leq\frac{C_{3}\,||\phi||_{L^{\infty}([0,T))}}{2}\,\tau^{2\vartheta^{\prime}-2\vartheta-1}\to
0,\quad\mbox{as }\tau\downarrow 0,$

where we have used the conditions (83) which imply that
$\vartheta^{\prime}-\vartheta>1/2$. Therefore, the limits (90)–(91) yield
(84). For $A^{\tau}_{3}$ we write

$\displaystyle
A^{\tau}_{3}=\int_{0}^{T}\int_{0}^{X^{0}}\phi\,\left(\partial_{x}u^{\tau}_{\delta}\right)^{2}\,dxdt+\int_{0}^{T}\int_{0}^{X^{0}}\phi\,\partial_{x}u^{\tau}_{\delta}\left(\partial_{x}u^{\tau}-\partial_{x}u^{\tau}_{\delta}\right)\,dxdt=A^{\tau}_{31}+A^{\tau}_{32}.$

We first notice, thanks to Lemma 4.2, that
$\partial_{x}u^{\tau}_{\delta}\rightharpoonup\partial_{x}u=\partial_{x}\left(\chi\rho\right)$
weakly in $L^{2}(\mathbb{R}\times(0,T))$ (where $\rho$ is the function
obtained in Proposition 4.3 which we extend continuously outside of
$[0,X^{0}]$). Then, using the lower semicontinuity of the functional $v\in
L^{2}((0,X^{0})\times(0,T))\mapsto\int_{0}^{T}\int_{0}^{X^{0}}v^{2}(x,t)\phi(t)\,dxdt$
for the weak topology in $L^{2}$ we have

(92) $\displaystyle\liminf_{\tau\downarrow
0}A^{\tau}_{31}\geq\int_{0}^{T}\int_{0}^{X^{0}}\phi\left(\partial_{x}u\right)^{2}\,dxdt.$

Then, we also rewrite the term $A^{\tau}_{32}$ as

$\displaystyle
A^{\tau}_{32}=\int_{0}^{T}\int_{\mathbb{R}}\phi\,\partial_{x}u^{\tau}_{\delta}\,\left(\partial_{x}\widetilde{u}^{\tau}-\partial_{x}u^{\tau}_{\delta}\right)\,dxdt-\int_{0}^{T}\int_{\mathbb{R}\setminus[0,X^{0}]}\phi\,\partial_{x}u^{\tau}_{\delta}\,\left(\partial_{x}\widetilde{u}^{\tau}-\partial_{x}u^{\tau}_{\delta}\right)\,dxdt=A^{\tau}_{321}+A^{\tau}_{322}.$

For $A^{\tau}_{321}$ we will use the following equality

$\displaystyle\int_{\mathbb{R}}h(x)\left(w\ast\zeta_{\delta}\right)(x)\,dx=\int_{\mathbb{R}}\left(h\ast\xi_{\delta}\right)(x)\,\left(w\ast\xi_{\delta}\right)(x)\,dx,\quad\forall
h,w\in L^{2}(\mathbb{R}),$

which can be proved by Fourier transform. Thus, recalling the definition of
the function
$\check{u}^{\tau}_{\delta}(x,t)=\left(\widetilde{u}^{\tau}\ast\xi_{\delta}\right)(x,t)$
we obtain

$\displaystyle
A^{\tau}_{321}=\int_{0}^{T}\phi\left(\int_{\mathbb{R}}\left(\partial_{x}\check{u}^{\tau}_{\delta}\right)^{2}\,dx-\int_{\mathbb{R}}\left(\partial_{x}\check{u}^{\tau}_{\delta}\ast\xi_{\delta}\right)^{2}\,dx\right)\,dt.$

Thanks to the nonnegativity of the function $\phi$, Jensen’s inequality, the
fact that $\delta\xi_{\delta}^{2}(y)\leq\xi_{\delta}(y)$ for all
$y\in\mathbb{R}$ we obtain

$\displaystyle A^{\tau}_{321}$
$\displaystyle\geq\int_{0}^{T}\phi(t)\int_{\mathbb{R}}\left(\left(\partial_{x}\check{u}^{\tau}_{\delta}\right)^{2}(x,t)-\delta\int_{-\delta/2}^{\delta/2}\left(\partial_{x}\check{u}^{\tau}_{\delta}\right)^{2}(x-y,t)\,\xi_{\delta}^{2}(y)\,dy\right)\,dxdt$
$\displaystyle\geq\int_{0}^{T}\phi(t)\int_{\mathbb{R}}\left(\left(\partial_{x}\check{u}^{\tau}_{\delta}\right)^{2}(x,t)-\int_{-\delta/2}^{\delta/2}\left(\partial_{x}\check{u}^{\tau}_{\delta}\right)^{2}(x-y,t)\,\xi_{\delta}(y)\,dy\right)\,dxdt$
$\displaystyle\geq\int_{0}^{T}\phi(t)\int_{\mathbb{R}}\left(\left(\partial_{x}\check{u}^{\tau}_{\delta}\right)^{2}-\left(\partial_{x}\check{u}^{\tau}_{\delta}\right)^{2}\ast\xi_{\delta}\right)(x,t)\,dxdt.$

In particular it holds

(93) $\displaystyle\lim_{\tau\downarrow 0}A^{\tau}_{321}\geq 0.$

For $A^{\tau}_{322}$ we first observe that
$\partial_{x}u^{\tau}_{\delta}(x,t)=0$ for a.e.
$(x,t)\in\mathbb{R}\setminus(-2\delta,X^{0}+\delta)$. Therefore, applying the
$L^{\infty}$ bounds established on $\partial_{x}\widetilde{u}^{\tau}$ and
$\partial_{x}u^{\tau}_{\delta}$ in Lemma 4.2, we get

$\displaystyle\left|A^{\tau}_{322}\right|\leq
6\,C_{1}^{2}\,||\phi||_{L^{1}(0,T)}\,\tau^{\vartheta^{\prime}-2\vartheta}.$

Moreover the conditions (83) imply in particular that
$\vartheta^{\prime}>2\vartheta$ and we conclude that

(94) $\displaystyle\lim_{\tau\downarrow 0}A^{\tau}_{322}=0.$

Collecting (92)–(94) yield (86). It remains to prove that

(95) $\displaystyle\lim_{\tau\downarrow 0}A^{\tau}_{6}=0,$

holds true. In this purpose, thanks to the proofs of Proposition 4.6 and
Proposition 4.7, we obtain the existence of a constant $C$ independent of
$\tau$ such that

$\displaystyle\left|A^{\tau}_{6}\right|\leq
C\,||\phi||_{L^{\infty}(0,T)}\,\left(\left|\left|\partial^{2}_{x}\left(\chi\,u^{\tau}_{\delta}\right)\right|\right|_{L^{\infty}(\mathbb{R}\times(0,T))}+\left|\left|\partial_{x}\left(\chi
u^{\tau}_{\delta}\right)\right|\right|_{L^{\infty}(\mathbb{R}\times(0,T))}\right)\,\tau^{1-2\vartheta}.$

Now, applying the estimates established in Lemma 4.2 and
$||\chi||_{L^{\infty}([0,X^{0}))}\leq 1$, we have

$\displaystyle\left|\left|\partial^{2}_{x}\left(\chi\,u^{\tau}_{\delta}\right)\right|\right|_{L^{\infty}(\mathbb{R}\times(0,T))}$
$\displaystyle\leq||\chi^{\prime\prime}||_{L^{\infty}([0,X^{0}))}\,\left(2\exp(b)-\exp(a)\right)+2C_{1}||\chi^{\prime}||_{L^{\infty}([0,X^{0}))}\tau^{-\vartheta}+C_{4}\tau^{-\vartheta-\vartheta^{\prime}},$

and

$\displaystyle\left|\left|\partial_{x}\left(\chi
u^{\tau}_{\delta}\right)\right|\right|_{L^{\infty}(\mathbb{R}\times(0,T))}\leq||\chi^{\prime}||_{L^{\infty}([0,X^{0}))}\,\left(2\exp(b)-\exp(a)\right)+C_{1}\tau^{-\vartheta}.$

Hence, we deduce that there exists a constant, still denoted $C$ and
independent of $\tau$, such that

$\displaystyle\left|A^{\tau}_{6}\right|\leq
C\,\tau^{1-3\vartheta-\vartheta^{\prime}}.$

Since we assume that $1-3\vartheta-\vartheta^{\prime}>0$, see (83), we
conclude that (95) holds. This completes the proof of Proposition 4.8. ∎

Now, passing to the limit $\tau\downarrow 0$ in (82) we conclude, thanks to
Proposition 4.8, that the weak solution $(\rho,M,X)$ of (1) satisfies the
variational inequality (10). Therefore, the triplet $(\rho,M,X)$ is a weak
solution to (1)–(2).

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