Order Form - FENOMEC

10 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES
Definition 2.2. (Concept of weak solution.) Given some initial data u 0 ∈ L ∞ ( I R,U)
we shall say that u ∈ L ∞ ( I R × I R + , U) isaweak solution to the Cauchy problem
(1.1) and (1.2) if
∫ +∞ ∫
(
u∂t θ + f(u) ∂ x θ ) ∫
dxdt + θ(0) u 0 dx = 0 (2.6)
0 IR
IR
for every function θ ∈ Cc
∞ ( I R × [0, +∞)) (the vector space of real-valued, compactly
supported, and infinitely differentiable functions).
Of course, if u is a continuously differentiable solution of (1.1) in the usual sense,
then by Green’s formula it is also a weak solution. The interest of the definition
(2.6) is that it allows u to be a discontinuous function. To construct weak solutions
explicitly we will often apply the following criterion.
Theorem 2.3. (Rankine-Hugoniot jump relations.) Consider a piecewise smooth
function u : I R × I R + →U of the form
{
u− (x, t), x < ϕ(t),
u(x, t) =
(2.7)
u + (x, t), x > ϕ(t),
where, setting Ω ± := { x ≷ ϕ(t) } , the functions u ± : Ω ± →U and ϕ : I R + → IR
are
continuously differentiable. Then, u is a weak solution of (1.1) ifandonlyifitisa
solution in the usual sense in both regions where it is smooth and, furthermore, the
following Rankine-Hugoniot relation holds along the curve ϕ:
−ϕ ′ (t) ( u + (t) − u − (t) ) + f(u + (t)) − f(u − (t)) = 0, (2.8)
where
u − (t) := lim
ε→0
u − (ϕ(t) − ε, t), u + (t) := lim
ε→0
u + (ϕ(t)+ε, t).
ε>0
ε>0
Proof. Given any function θ in Cc
∞ ( I R × (0, +∞)) let us rewrite (2.6) in the form
∑
∫∫
(
u± (x, t) ∂ t θ(x, t)+f(u ± (x, t)) ∂ x θ(x, t) ) dxdt =0.
± Ω ±
Applying Green’s formula in each region of smoothness Ω ± we obtain
∑
∫ +∞
(
± −ϕ ′ (t) u ± (t)+f ( u ± (t) )) θ(ϕ(t),t) dt =0,
±
which gives (2.8) since θ is arbitrary.
0
When u − and u + are constants and ϕ is linear, say ϕ(t) =λt,
{
u− , x < λt,
u(x, t) =
(2.9)
u + , x > λt,
Theorem 2.3 implies that (2.9) is a weak solution of (1.1) if and only if the vectors
u ± and the scalar λ satisfy the Rankine-Hugoniot relation
−λ (u + − u − )+f(u + ) − f(u − )=0. (2.10)
When u − ≠ u + the function in (2.9) is called the shock wave connecting u − to u +
and λ the corresponding shock speed.
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