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The Entropy Condition and Uniqueness

Once a shock forms, the weak solution is no longer unique — the same data can spawn many. We meet the extra rule, the entropy condition, that throws out the impostors and keeps the one piece of physics nature actually picks.

A jump condition is not enough

In the last guide we earned the Rankine-Hugoniot condition: a discontinuity in a scalar conservation law u_t + f(u)_x = 0 must travel at the speed s = (f(u_R) - f(u_L)) / (u_R - u_L), the slope of the secant line joining the two states. That single equation looks like it ought to settle everything — give me the left and right values, and out pops the shock speed. But there is a quiet scandal hiding underneath it.

The scandal is this: the jump condition is satisfied by jumps that run the wrong way too. Imagine data that is already a smooth ramp — low on the left, high on the right — the kind of spreading-out flow that physically should just fan open. The Rankine-Hugoniot algebra is perfectly happy to instead glue that data into a sharp discontinuity standing in the middle, and to call it a solution. Both the smooth fan and the standing jump satisfy the same conservation law in the weak sense. Uniqueness, the thing every well-posed problem promised us, has quietly evaporated.

Two competing weak solutions, side by side

Let us make the scandal painfully concrete with Burgers' equation, u_t + (u^2 / 2)_x = 0, whose flux is f(u) = u^2 / 2 and whose characteristic speed is f'(u) = u — points carry their own value as their velocity. Take the data u_L = 0 on the left and u_R = 1 on the right: slow stuff behind, fast stuff ahead. Because the fast values are in front and outrunning the slow ones, the characteristics spread apart, opening a gap that no characteristic reaches. Something must fill it.

Two honest answers now compete for this single data set. Candidate A is a standing shock: keep u = 0 to the left of a jump and u = 1 to its right, and let the jump travel at the Rankine-Hugoniot speed s = (f(1) - f(0)) / (1 - 0) = 1/2. That satisfies the jump condition exactly, so it is a perfectly legal weak solution. Candidate B is a continuous fan that fills the gap: u = 0 for x <= 0, then u = x/t for 0 < x < t, then u = 1 for x >= t. It too solves the PDE, with no jump anywhere. Same data, two different solutions — the equation alone simply cannot choose between them.

Candidate B is a rarefaction wave: instead of forcing a jump, it lets the gap fill with a continuous fan of values, u = x/t, that interpolates smoothly from 0 up to 1. It is built precisely so a fresh characteristic threads through every point of the missing wedge. Candidate A is a discontinuity that the jump condition certifies but that physics rejects. We need a tie-breaker — and crucially, the tie-breaker must rule for B and against A.

The entropy condition: characteristics must run into the shock

Here is the picture that decides everything. Draw the shock as a line in the (x,t)-plane and draw the characteristics on each side as little arrows. A discontinuity is physically admissible only if characteristics flow into it from both sides — information piles into the shock and is destroyed there, like cars crowding into a traffic jam. A jump that has characteristics emanating from it, spraying information outward, is a forbidden 'expansion shock': you cannot manufacture information out of a bare discontinuity.

Turn that picture into algebra. The characteristic speed just behind the shock is f'(u_L), just ahead it is f'(u_R), and the shock itself crawls at speed s. 'Characteristics run in from both sides' means the speed behind exceeds s and s exceeds the speed ahead. For a convex flux like Burgers' this is the Lax entropy condition: f'(u_L) > s > f'(u_R). For Burgers' that simplifies to u_L > u_R — a shock is admissible only when the value behind is larger than the value ahead.

Lax entropy condition (convex flux f):    f'(u_L)  >  s  >  f'(u_R)

   characteristics behind:  speed f'(u_L)  -->|
                                              |  shock at speed s
   characteristics ahead:                  |<--  speed f'(u_R)
         (both arrows aim AT the shock line:  admissible)

Burgers (f = u^2/2, f' = u):   condition becomes   u_L > u_R

   u_L=0, u_R=1  -->  0 > 1 is FALSE  -->  Candidate A is rejected
                      (so the rarefaction fan B is the right answer)
The Lax entropy condition in one line, and why it throws out the expansion shock for u_L=0, u_R=1.

Now run our two candidates through the gate. For u_L = 0, u_R = 1 the Lax test demands 0 > 1 — flatly false — so Candidate A, the standing shock, is inadmissible and gets thrown out. Candidate B, the rarefaction, has no jump at all and survives. The tie is broken exactly the way we hoped: the smooth physical fan wins, the spurious shock loses. Flip the data to u_L = 1, u_R = 0 and the verdict flips too — now 1 > 0 holds, the shock is admissible, and it is the unique physical answer.

Where the rule comes from: a whisper of viscosity

The Lax condition is not pulled from a hat. It is the fingerprint left by physics we deleted. Real fluids and real traffic have a little diffusion — a tiny smoothing term. So consider the regularized equation u_t + f(u)_x = epsilon u_xx with a small epsilon > 0. This parabolic equation has the smoothing power of the heat equation: it is perfectly well-posed, has smooth unique solutions, and never forms a true discontinuity. The cliff that would become a shock is rounded into a thin, steep layer instead.

Now let epsilon shrink to zero. The thin smooth layer steepens toward a genuine jump, and the limit you reach is the vanishing-viscosity solution. The deep theorem is that this limit is unique, and it is exactly the entropy-admissible weak solution — Candidate B for spreading data, the shock for compressing data. The entropy condition is therefore not an arbitrary referee's call. It is the scar that the limit of true physics leaves behind, the only weak solution that is the shadow of a smooth one.

Uniqueness restored, and what to carry forward

With the entropy condition bolted on, the bookkeeping finally balances. Among the swarm of weak solutions, exactly one is entropy-admissible, and that one exists and depends continuously on the data. For scalar conservation laws in any number of dimensions this is the content of the celebrated Kruzhkov uniqueness theorem: entropy weak solutions are unique. We have closed the loop opened back when smooth characteristics first crossed and a shock was born.

  1. Given left/right states across a candidate discontinuity, compute the Rankine-Hugoniot speed s = (f(u_R) - f(u_L)) / (u_R - u_L).
  2. Compute the characteristic speeds behind and ahead, f'(u_L) and f'(u_R).
  3. Check the Lax inequality f'(u_L) > s > f'(u_R): do characteristics flow INTO the jump from both sides?
  4. If yes, the shock is admissible and is your solution. If no, reject it and fill the gap with a rarefaction fan instead.

Hold on to two honest caveats. First, the clean Lax form f'(u_L) > s > f'(u_R) is the convex-flux version; for a non-convex flux f you need the stronger Oleinik chord condition, and a single jump may have to be replaced by a composite of shocks and fans. Second, the comforting Kruzhkov theory is a scalar story — for systems of conservation laws, existence and uniqueness are far more delicate and parts remain genuinely open. With the selection rule in hand, the very next guide assembles everything — entropy shocks and rarefaction fans together — into the complete answer for piecewise-constant data: the Riemann problem.