Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

Measure-Valued Solution: Any Connected Component J of the
Support Is Bounded for γ ∈ (1, 3), II: Steps
As before: For any s 1 , s 3 ∈ J,
χ(s 1)χ(s 3)
χ(s 1) χ(s 3) ≥ 1;
χ(s) ≥ 0 is not identically zero and χ(s) → 0 as s → inf J, sup J,
=⇒ there exists s 2 such that χ ′ (s 2 ) > 0, χ(s 2 ) > 0.
Let s 3 > s 2 be points such that χ(s 3 ) > 0 and let s 1 → −∞. From the 1st
identity, χ(s1)χ(s2)
χ(s 1)
= χ(s 2 ) χ(s1)χ(s3)
χ(s 1) χ(s 3) + o(1), as s 1 → −∞.
[χ ′ (s)] + ≤ 2λ
s−s 1
χ(s).
From the 2nd equation, by throwing away the negative terms, we obtain
χ ′ (s 2 ) χ(s 1)χ(s 3 )
χ(s 1 ) χ(s 3 ) ≤ 2λ + 1 χ(s 1 )χ(s 2 )
+ o(1).
s 3 − s 1 χ(s 1 )
=⇒
Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 29 / 40