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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Measure-Valued Solution: Any Connected Component J <strong>of</strong> the<br />
Support Is Bounded <strong>for</strong> γ ∈ (1, 3), I: Strategy<br />
On the contrary, suppose that J is unbounded from below.<br />
Let M 0 = sup{s : s ∈ J} ∈ (−∞, ∞].<br />
Let s 1 , s 2 , s 3 ∈ (−∞, M 0 ) with s 1 < s 2 < s 3 . The commutation relation =⇒<br />
(s 2 − s 1 ) χ(s 1)χ(s 2 )<br />
χ(s 1 )<br />
+ (s 3 − s 2 ) χ(s 3)χ(s 2 )<br />
χ(s 3 )<br />
= (s 3 − s 1 )χ(s 2 ) χ(s 1)χ(s 3 )<br />
χ(s 1 ) χ(s 3 ) .<br />
Differentiating this equation in s 2 and dividing by (s 3 − s 1 ), we obtain<br />
χ ′ (s 2 ) χ(s 1)χ(s 3 )<br />
χ(s 1 ) χ(s 3 )<br />
= s 2 − s 1 χ(s 1 )χ ′ (s 2 )<br />
+ s 3 − s 2 χ(s 3 )χ ′ (s 2 )<br />
s 3 − s 1 χ(s 1 ) s 3 − s 1 χ(s 3 )<br />
+ 1 χ(s 1 )χ(s 2 )<br />
− 1 χ(s 3 )χ(s 2 )<br />
.<br />
s 3 − s 1 χ(s 1 ) s 3 − s 1 χ(s 3 )<br />
Strategy:<br />
Gui-Qiang Chen (Ox<strong>for</strong>d) <strong>Vanishing</strong> <strong>Viscosity</strong>/Conservation Laws June 20–24, 2011 28 / 40
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