By L. Garrido

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5) can be rewritten as σ= 1 u+ − u− u+ u− a(v) dv . 6) In other words, the propagation speed of the discontinuity connecting u− and u+ is the average of the wave speeds a(v), as v ranges in the interval between u− and u+ . Fig. 4 Application of the Whitham “equal area” rule to construct the correct position for the discontinuity. u B A x Finally, it turns out that the Rankine–Hugoniot condition corresponds to cutting off from the multi–valued profile two equal area lobes A and B as described in Fig.

8) we obtain d v t, x (t; xo ) = a (u) ∂t ∂x u + a (u) ∂x2 u = − a (u) ∂x u dt 2 2 = −v t, x (t; xo ) . The solution of the above equation of Ricatti type is v t, x (t; xo ) = v(0, xo ) . 1 + v(0, xo) t d By hypothesis, there are values of y for which v(0, y) = dy a(u(y) = s (y) ∈ R− . −1 For these values v t, x(t; y) blows up at time t = −s (y) . Therefore, v exists finite only for t < infy∈R {−s (y)−1 } = Tmax . 6) at least on a small time interval. On the other hand, we have seen that in the nonlinear case a (u) = 0 discontinuities may develop after a finite time.

5. Suppose that u ∈ C0c (R; R). e. 6). 0, Proof. By assumption uε ∈ C∞ (]0, T ] × R; R) ∩ C0 ([0, T ] × R; R). If U : R → R is a C2 convex function with entropy flux F, then uε satisfies ∂t U(uε ) + ∂x F(uε ) = U (uε ) ∂t uε + ∂x f (uε ) = ε U (uε ) Δ uε = ε Δ U(uε ) − ε U (uε ) (∂x uε )2 ≤ ε Δ U(uε ) . If we integrate over [0, T [×R the above inequality multiplied by a test function ϕ ∈ C∞ ([0, T [×R; R∗+ ), integrate by parts and pass to the limit as ε 0, then we obtain 36 3 One–Dimensional Scalar Conservation Laws 0 ≤ = → T R 0 T R 0 T R 0 ε Δ U(uε ) − ∂t U(uε ) − ∂x F(uε ) ϕ dx dt U(uε ) (ε Δ ϕ + ∂t ϕ ) + F(uε ) ∂x ϕ dx dt + U(u) ∂t ϕ + F(u) ∂x ϕ dx dt + R R U u(x) ϕ (0, x) dx U u(x) ϕ (0, x) dx .