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:
∫ M0+1 χ(s 1)χ(s 2)
M 0 χ(s 1)
Lions-Perthame-Tadmor’s argument:
=⇒ ∫ M 0 +1
χ(s 1 )χ(s 2 )
M 0 χ(s 1 )
ds 2 ≤ C(λ)|s 1 | λ , λ < 0.
χ(s 1)χ(s 2)
χ(s 1)
≥ χ(s 2 ) a.e. s 1 , s 2 ∈ J, s 1 < s 2 .
ds 2 ≥ ∫ M 0 +1
M 0
χ(s 2 )ds 2 = C(M 0 , λ) > 0
Gui-Qiang Chen (Oxford) Vanishing Viscosity/Conservation Laws June 20–24, 2011 27 / 40