ABSTRACT
It is an obvious, well-known and nevertheless crucial observation that, by the characteristic mechanism (154),
non-entropy shocks of the shape S+ cannot appear evolutionary. (155)
Indeed, differentiating (154) in x yields
ux(x, t) = u′0(x−u(x,t)t)
1+u′0(x−u(x,t)t)t , so that u ′ 0 ≥ 0
=⇒ no blow-up of ux (“gradient catastrophe”) occurs. (156)
Recalling the necessary evolution property in (156), given a small δ > 0 and a bounded (say, for simplicity, in L1 and in L∞) solution u(x, t) of the Cauchy problem (17), (2), we construct its δ-deformation given explicitly by the characteristic method (154) as follows: (i) We perform a smooth δ-deformation of initial data u0 ∈ L1 ∩ L∞ by
introducing a suitable C1 function u0δ(x) such that ∫ |u0 − u0δ| < δ. (157)
By u1δ(x, t), we denote the unique local solution of the Cauchy problem, with data u0δ, so that, by (154), the continuous function u1δ(x, t) is defined algebraically on the maximal interval t ∈ [t0, t1(δ)), where we denote t0 = 0 and t1(δ) = Δ1δ. It is important that, here and later on, smooth deformations are performed in a small neighborhood of possible discontinuities only, leaving the the rest of smooth profiles untouchable, so that these evolve along the characteristics, as usual. Actually, this emphasizes the obvious fact that the shocks (on a set of zero
measure) occur as a result of nonlinear interaction of the areas with continuous solutions, which, hence, cannot be connected without discontinuities. (ii) Since at t = Δ1δ a shock of type S− (or possibly infinitely many shocks)
is supposed to occur, since otherwise we continue the algebraic procedure, we perform another suitable δ-deformation of the “data” u1δ(x,Δ1δ) to get a unique continuous solution u2δ(x, t) on the maximal interval t ∈ [t1(δ), t2(δ)), with t2(δ) = Δ1δ +Δ2δ, etc. . . . (k) With suitable choices of each δ-deformation of “data” at the moments
t = tj(δ), when ujδ(x, t) has a shock for j = 1, 2, ..., there exists a tk(δ) > 1 for some finite k = k(δ), where k(δ) → +∞ as δ → 0. It is easy to see that, for bounded solutions, k(δ) is always finite. A contradiction is obtained while assuming that tj(δ) → t¯ < 1 as j → ∞ for arbitrarily small δ > 0, meaning a kind of a “complete blow-up,” which is impossible for conservation laws, obeying the Maximum Principle.
