Order Form - FENOMEC

26 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES
where µ U (u) is a non-positive, locally bounded measure depending on the solution
u under consideration. Clearly, the measure µ U (u) cannot be prescribed arbitrarily
and, in particular, must vanish on the set of continuity points of u.
Definition 5.3. (Traveling waves.) Consider a propagating jump discontinuity connecting
two states u − and u + at some speed λ. A function u ε (x, t) =w(y) with
y := (x − λt)/ε is called a traveling wave of (3.1) connecting u − to u + at the speed
λ if it is a smooth solution of (3.1) satisfying
w(−∞) =u − , w(+∞) =u + , (5.9)
and
lim
|y|→+∞ w′ (y) = lim
|y|→+∞ w′′ (y) =...=0. (5.10)
For instance consider, in (3.1), conservative regularizations of the form
R ε = ( S(u ε ,εu ε x,ε 2 u ε xx,...) ) x
with the natural condition S(u, 0,...) = 0 for all u. Then the traveling waves w of
(3.1) are given by the ordinary differential equation
−λw ′ + f(w) ′ = S(w, w ′ ,w ′′ ,...) ′
or, equivalently, after integration over intervals (−∞,y]by
S(w, w ′ ,w ′′ ,...)=f(w) − f(u − ) − λ (w − u − ). (5.11)
It is straightforward (but fundamental) to check that:
Theorem 5.4. If w is a traveling wave solution, then the pointwise limit
( x − λt
) {
u− , x < λt,
u(x, t) := lim w =
(5.12)
ε→0 ε u + , x > λt,
is a weak solution of (1.1) satisfying the entropy inequality (3.8). In particular, the
Rankine-Hugoniot relation (2.10) follows from (5.11) by letting y → +∞.
Moreover, the solution u satisfies the kinetic relation (5.8) where the dissipation
measure is given by
µ U (u) =Mδ x−λt ,
∫
M := − S(w(y),w ′ (y),w ′′ (y),...) · D 2 U(w) w ′ (y) dy,
IR
where δ x−λt denotes the Dirac measure concentrated on the line x − λt=0.
We will see in Chapter III that traveling wave solutions determine the kinetic
relation:
• For the scalar equation with cubic flux (Example 4.2) the kinetic relation can
be determined explicitly; see Theorem III-2.3.
• For the more general model in Example 4.3 a careful analysis of the existence of
traveling wave solutions leads to many interesting properties of the associated
kinetic function (monotonicity, asymptotic behavior); see Theorem III-3.3.
This analysis allows one to identify the terms Φ, φ, andµ U (u) in (5.4), (5.5),
and (5.7), respectively.
• Systems of equations such as those in Examples 4.6 and 4.7 can be covered by
the same approach; see Remark III-5.4 and the bibliographical notes.
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