Here are some reading notes for the paper [LW05] (see //References// below) by Tatsien Li and Libin Wang. === Idea for prove global well-posedness In [LW05], they used the following continuity method to prove the global well-posedness. **Continuity Method** (see pp. 21 in [Tao06] for more details) Let $I$ be a time interval, and for each $t\in I$ suppose we have two statements,a a //hypothesis// $H(t)$ and a //conclusion// $C(t)$. Suppose we can verify the following four assersions: *(Hypothesis implies conclusion) If $H(t)$ is true for some time $t\in I$, then $C(t)$ is also true for that time $t$. *(Conclusion is stronger than hypothesis) If $C(t)$ is true for some $t\in I$, then $H(t')$ is true for all $t'\in I$ in a neighborhood of $t$. *(Conclusion is closed) If $t_1,t_2,\cdots$ is a sequence of times in $I$ which converges to another time $t\in I$, and $C(t_n)$ is true for all $t_n$, then $C(t)$ is true. *(Base case) $H(t)$ is true for at least one time $t\in I$. Then $C(t)$ is true for all $t\in I$. In the following, we give some details about calculating some preliminary results in Section 2 of [LW05]. === How to prove (2.9) in [LW05] $$\begin{split} \frac{d v_i}{d_i t} &= \frac{\partial (l_i u)}{\partial t} + \lambda_i \frac{\partial (l_i u)}{\partial x}~\\ &=l_i\p_t u + \lambda_i l_i\p_x u + \p_t(l_i)u+\lambda_i\p_x(l_i)u,~\\ \end{split}$$ using (1.1) to dispense the first two terms, and noticing (2.6), which implies $$\begin{split} \frac{d v_i}{d_i t} &=\sum_j \big(\p_t(l_i)v_jr_j + \lambda_i\p_x(l_i)v_jr_j\big)~\\ &=(\text{using (1.5)})~\\ &=-\sum_j\big( l_i\p_t(r_j) v_j + \lambda_il_i\p_xr_jv_j \big) ~\\ &=-\sum_j\big(l_i\grad r_j \p_t u v_j + \lambda_i l_i \grad r_j \p_xuv_j\big) ~\\ &=\sum_jl_i\grad r_j v_j \big( A(u)-\lambda_i\big)\p_xu,~\\ \end{split}$$ which equals the right-hand side of (2.9) by using (2.7). === How to prove (2.13) $$\begin{split} d[v_i(dx-\lambda_idt)] &=\p_tv_i dt\wedge dx + \p_xv_idx\wedge dx - \p_t(v_i\lambda_i)dt\wedge dt - \p_x(v_i\lambda_i)dx\wedge dt ~\\ &=(\p_t v_i + \p_x(v_i\lambda_i))dt\wedge dx~\\ &=(\text{using (2.9), we establish the equality}). \end{split}$$ === How to prove (2.18) Differentiating (1.3) with respect to $x$ gives $$\begin{split} A(u)_x r_j = (\lambda_jr_j)_x - A(u)r_{j,x}. \end{split}$$ $$\begin{split} \frac{d w_i}{d_i t} &=l_i\p_x u_x + \lambda_il_i\p_x u_x + \p_tl_i u_x + \lambda_i \p_xl_i u_x; \end{split}$$ we use the same arguments as for proof of (2.9) to deal with the last two terms; and we find $$\p_tl_i u_x + \lambda_i \p_xl_i u_x = (\lambda_k-\lambda_i)l_i\grad r_j r_k w_jw_k;$$ for the first two terms, we differentiate (1.1) with respect to $x$, and we obtain that: $$\begin{split} l_i\p_x u_x + \lambda_il_i\p_x u_x &= -l_i A(u)_x\p_x u =-l_iA(u)_x r_jw_j ~\\ &=-l_i\big( (\lambda_jr_j)_x - A(u)r_{j,x}\big) w_j ~\\ &=-l_i\big( \lambda_{j,x}r_j + (\lambda_j-\lambda_i) r_{j,x}\big)w_j ~\\ &=\big(-\grad \lambda_j \delta_{ij} + (\lambda_il_i - \lambda_jl_i)\grad r_j\big) u_x w_j; \end{split}$$ using (2.7) and the above two equalities, we can establishes (2.18). === References: [LW05] T. Li and L. Wang, //Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems//, Discrete and Continuous Dynamical Systems, vol. **12**, Number **1**, January **2005**, pp. 59-78 [Tao06] Terence Tao, Nonlinear dispersive equations: local and global analysis, CBMS regional conference series in mathematics, July 2006.