Methods of Vanishing Viscosity for Nonlinear ... - ACMAC
Methods of Vanishing Viscosity for Nonlinear ... - ACMAC
Methods of Vanishing Viscosity for Nonlinear ... - ACMAC
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Navier-Stokes Equations: Key Estimates III<br />
Then<br />
That is,<br />
∫<br />
ε 2 |ρx (t, x)| 2<br />
∫<br />
(γ − t ∫<br />
1)2<br />
dx + ε ρ γ−3 |ρ<br />
ρ(t, x) 3 x | 2 dxdτ<br />
2 0<br />
∫ [ ]<br />
ρx u t ∫ t ∫<br />
= −2ε<br />
dx + 2ε |u x | 2 dxdτ<br />
ρ<br />
τ=0<br />
0<br />
∫<br />
≤ ε2 |ρx (t, x)| 2 ∫<br />
2 ρ(t, x) 3 dx + |ρ0,x (x)| 2<br />
ε2 dx + C.<br />
ρ 0 (x) 3<br />
∫<br />
ε 2 |ρx (t, x)| 2 ∫ t ∫<br />
ρ(t, x) 3 dx + ε ρ γ−3 |ρ x | 2 dxdτ<br />
0<br />
∫<br />
≤ C<br />
(ε 2 |ρ0,x (x)| 2 )<br />
ρ 0 (x) 3 dx + 1 .<br />
Case II: u + ≠ u − : More technically involved.<br />
Gui-Qiang Chen (Ox<strong>for</strong>d) <strong>Vanishing</strong> <strong>Viscosity</strong>/Conservation Laws June 20–24, 2011 23 / 40
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