Local small-data Riemann solver near a strictly hyperbolic state

Define $\mathcal{W}(u_L,\sigma )={\mathrm{\Psi }}_m({\sigma }_m;{\mathrm{\Psi }}_{m-1}({\sigma }_{m-1};\cdots {\mathrm{\Psi }}_1({\sigma }_1;u_L)\cdots ))$ and $F(u_L,u_R,\sigma )=\mathcal{W}(u_L,\sigma )-u_R$. By [F2] and [F3], $F$ is $C^1$, $F(\bar{u},\bar{u},0)=0$, and the columns of $D_{\sigma }F(\bar{u},\bar{u},0)$ are $r_1(\bar{u}),\dots ,r_m(\bar{u})$.

These eigenvectors are linearly independent by strict hyperbolicity, so $D_{\sigma }F$ is invertible. Applying [F4] with parameter $(u_L,u_R)$ gives neighborhoods $P$ of $(\bar{u},\bar{u})$ and $Q$ of $0$, and a unique $C^1$ vector $\sigma (u_L,u_R)\in Q$ for every pair in $P$. If $u_L=u_R$, then $\sigma =0$ solves the composition equation by [F2], so this is the unique strength vector. Choose $\eta >0$ with $(-\eta ,\eta {)}^m\subseteq Q$. Since the IFT solution $\sigma (u_L,u_R)$ is continuous and vanishes at $(\bar{u},\bar{u})$, shrink $\delta >0$ so both radius-$\delta$ state balls lie in $P$ and every corresponding IFT solution lies in $(-\eta ,\eta {)}^m$.

Shrink these neighborhoods so all intermediate states stay in $N$. Since $N$ is a neighborhood of $\bar{u}$ and there are finitely many continuous wave curves, this is possible. The characteristic-speed gaps at $\bar{u}$ are positive; continuity of eigenvalues, shock speeds and rarefaction speeds lets us further shrink the neighborhoods until the speed range of each family lies in a disjoint interval ordered by $i$.

For each strength, use the rarefaction fan from [F8] when ${\sigma }_{i}D{\lambda }_i{r}_i>0$, the Lax shock from [F6] when it is negative, and the contact from [F7] for a linearly-degenerate family. Each shock/contact satisfies Rankine–Hugoniot, hence is a weak jump by [F5].

The family speed intervals from step 3.1 are disjoint and ordered, so the elementary waves paste into a single-valued self-similar profile from $u_L$ to $u_R$. Each fan satisfies the equation classically by [F8]; integration by parts across each shock/contact gives boundary coefficient $[f]-s[u]=0$ by [F5], [F6], and [F7], and the continuous fan edges contribute no jump term. Thus the pasted profile is a distributional weak solution. Its wave speeds are bounded, so its nonconstant region has length at most a constant times $t$ and its $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ trace is the Riemann step as $t\downarrow 0$. The endpoint states match by the composition equation in step 1.1. Uniqueness of $\sigma$ from step 2.1 gives uniqueness of this ordered local elementary-wave construction, not of arbitrary weak solutions.