Methods of Vanishing Viscosity for Nonlinear ... - ACMAC

Measure-Valued Solution with Unbounded Support: γ > 3
Let A := ∪ {(u − ρ θ , ρ θ + u) : (ρ, u) ∈ supp ν}.
Let J = (s − , s + ) be any connected component of A.
Note that supp χ(s) = {(ρ, u) : u − ρ θ ≤ s ≤ u + ρ θ }.
Claim: Any connected component J of the support is bounded for γ > 3
Strategy: On the contrary, let inf{s : s ∈ J} = −∞.
Fix M 0 such that M 0 + 1 ∈ J and restrict s 2 ∈ (M 0 , M 0 + 1);
Choose sufficiently small s 1 ≤ −2|M 0 | to reach the contradiction.
New Observation:
Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 27 / 40