Front tracking approximations and existence of solutions to conservation law PDEs

Question

This question is about page 38-42 of these notes on censervation laws, more precisely PDEs of the form $u_t + [f(u)]_x =0.$ In this section of the note, the author provides a proof of the existence of an entropic solution to the differential equation with initial condition $u_0$, where the total variation of $u_0$ is sufficiently small. The method of proof is to consider approximation of $u_0$ by piecewise linear functions. Here is another reference for this method of proof: Ronald JDiPerna. Global existence of solutions to nonlinear hyperbolic systems of conservation laws.

Page 38-42 is about how to deal with the issue that we may get infinitely many wavefronts (discontinuities) within finite time. A simplified front-tracking process is given. given three states $u_l,u_m,u_r$ (left, middle, right) related by $u_m = \Psi_j(\sigma) u_l, u_r =\Psi_{j'}(\sigma') u_m,$ one can define the auxiliary right state (bottom of page 39) $$ \tilde u_r = \Psi_j(\sigma) \circ \Psi_{j'}(\sigma') (u_l), \text{for }j> j' $$

The semigroup $\Psi$ describes how different states $u$ are related to each other. A definition of $\Psi$ can be found on page 20 of the notes. The larger the value of $j,$ the faster the characteristic speed of the wavefront.

So we kind of "re-order" the way we apply $\Psi$ according to which of $j,j'$ is bigger. Then we introduce a nonphysical wave front to connect $\tilde u_r$ and $u_r$ together.

Now I am having difficulties understanding the following things: Why do we need to reorder $\Psi$ in the definition of $\tilde u?$ Apparently this is to ensure that the solution is piecewise constant (so no rarefaction waves will appear). But why does this ensure that we do not have a rarefaction wave?

At the end of page 39, the notes say that this is due to euqation $(4.25)$ on page 37:

Because of (4.25), the piecewise constant function $\tilde v$ contains exactly two wave-fronts...

but I am unable to see why $(4.25)$ is useful here. Why and how are we utilizing $(4.25)?$

I really wish to include more detail in this question, but if I quote an equation, I need to include the definition of all the terms/symbols, and there are simply too many of them to be included here. Please feel free to ignore the details which I have not included. The most important thing is: why this construction works? An explanation to just this question would be extremely helpful.

Answer 1

Let's tackle the two parts separately. Suppose both families are genuinely non-linear.

Why do we need to reorder Ψ in the definition of 𝑢̃?

To see this, go back to the Riemann problem, which was solved by moving across the curves of each of the characteristic families, in increasing order. The order is very important because it ensures that the characteristic speeds increase (by the assumption that the system is strictly hyperbolic) and that a solution to the Riemann problem is well-defined. Hence, unless you reorder $\Psi$, you wouldn't obtain a solution to the Riemann Problem with data $u_l,\tilde{u}_r$.

But why does this ensure that we do not have a rarefaction wave?

Let the wave strength of the incoming fronts be $\sigma,\sigma^{\prime}$ respectively. If they're both shock fronts, then $\sigma,\sigma^{\prime}<0$, and so by construction $u_l,\tilde{u_r}$ are connected by two shock fronts of the families (i.e one intermediate state). However, even if one of the fronts is a rarefaction front, say $\sigma$, then $\sigma<\epsilon$, and hence there is still only one intermediate state connecting $u_l$ and $\tilde{u}_r$ (since the rarefaction is replaced by a piece-wise constant rarefaction fan in the construction).

Thus, if we define $\tilde{u}_m=\Psi_{j^{\prime}}(\sigma^{\prime})(u_l)$, the Riemann problem with states $u_l,\tilde{u}_r$ is (approximately) solved by a simple function taking the three values $u_l,\tilde{u}_m,\tilde{u}_r$, regardless of what the colliding fronts were. If $u_{m},u_{r}$ was a shock/rarefaction front, then $u_l,\tilde{u}_m$ will be a shock/rarefaction front respectively, and mutatis mutandis for $u_{l},u_{m}$ and $\tilde{u_m},\tilde{u}_r$. Finally, if either family is linearly degenerate, then there is nothing to prove.

I am not sure where 4.25 comes in, though.