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Pointwise shock values do not affect the weak solution

Statement refuted

The claim refuted is that the distributional weak formulation determines the pointwise values of a piecewise $C^{1}$ solution along its shock curve. Let $u$ be a bounded distributional weak solution of $u_{t} + f (u)_{x} = 0$ on $Π_{T}$ that is piecewise $C^{1}$ with shock curve $Γ = {x = s (t)}$ (Distributional weak solutions of the Cauchy problem, Piecewise smooth shocks and one-sided traces). For any bounded measurable $θ ⁣ : (0, T) \to R$ , define $\tilde{u} (t, x) = u (t, x) (x \neq s (t)), \tilde{u} (t, s (t)) = θ (t) .$ Then $\tilde{u} = u$ almost everywhere and represents the same $L_{l o c}^{1}$ class, so it has the same weak formulation and the same initial datum, while its values on $Γ$ are completely arbitrary. More generally, any bounded measurable modification on a Lebesgue-null subset of $Π_{T}$ leaves the weak-solution class unchanged.

Facts & Assumptions

Given: a bounded piecewise $C^{1}$ distributional weak solution $u$ with shock curve $Γ = {x = s (t)}$ , a bounded measurable $θ$ , and the modification $\tilde{u}$ above.

[F1]

A bounded measurable function is a weak solution exactly when its $L_{l o c}^{1}$ class satisfies the integral identity; the identity pairs $u$ against test functions and therefore depends only on the class of $u$ modulo null sets (Distributional weak solutions of the Cauchy problem).

[F2]

The graph of the continuous $s$ is Borel in $(0, T) \times R$ , and each fixed-time spatial section is a singleton, of Lebesgue measure zero. Tonelli therefore gives zero space--time measure (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Measure-null sets and almost-everywhere statements relative to a measure). Modifications on this null set change no test integral (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree).

Proof

technique · direct

1.1F2given

The modification is measurable, bounded and a.e. equal. The set $Γ$ is null by [F2], and on its complement $\tilde{u} = u$ ; on $Γ$ the values $θ (t)$ are bounded and measurable, so $\tilde{u}$ is bounded and measurable and $\tilde{u} = u$ Lebesgue-a.e.

2.1F1F2step 1.1

The weak formulation is unchanged. Every test function in the weak identity is integrable against $∣ u ∣ + ∣ f (u) ∣$ on compact sets, and by [F2] the values on $Γ$ form a null set; hence each integral in the weak identity for $\tilde{u}$ equals the corresponding integral for $u$ , and the initial datum is likewise the same $L_{l o c}^{1}$ class. Since $u$ is a weak solution and the trace requirement depends only on the class, $\tilde{u}$ is a weak solution with the same datum.

3.1step 2.1F2∎

Arbitrary pointwise values on the shock. Choosing the constant functions $θ_{1} = 0$ and $θ_{2} = 1$ on $(0, T)$ gives two representatives of the same $L_{l o c}^{1}$ class that differ at every point of $Γ$ ; both satisfy the same weak formulation. Therefore the weak formulation cannot determine pointwise values on the shock curve, and the same argument applies to any bounded measurable modification on a Lebesgue-null subset of $Π_{T}$ .

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Sources

G. A. Chechkin and A. Yu. Goritsky (translated by B. Andreianov), “S. N. Kruzhkov’s lectures on first-order quasilinear PDEs,” in Analytical and Numerical Aspects of PDEs, de Gruyter 2009, complete lecture-notes text (standard reference, not scraped)
Alberto Bressan, “Hyperbolic Conservation Laws: An Illustrated Tutorial,” 2009, complete lecture notes (standard reference, not scraped)