Weak Convergence Methods for Nonlinear Partial Differential ...

In our case x n+1 = t, b ≡ 0, a n+1 = 1, (a 1 , ...a n ) = ∇F.
Since dx n+1
= 1, x
dt n+1 = 0 is x n+1 ≡ t. Also y ≡ y(0). The solution is given
by
y(t) = u(x(t), t)
and in particular u is constant along characteristics.
Example: Burgers equation u t + uu x = 0 (i.e.: a = (y, 1)). For initial values
given, e.g., by
⎧
⎪⎨ 1, x < 0
u(x, 0) = 1 − x, 0 ≤ x ≤ 1
⎪⎩
0, x ≥ 1
there are charactersitic curves t ↦→ x(t) that cross for times t > 0 (cf. Fig. 3.1)
and the solution is not defined unambiguously any longer. We therefore need a
weaker notion of solution. 3.1.
Figure 3.1: Crossing Characteristics.
Motivated by our earlier studies of weak solution, for a solution u and a test
function v ∈ Cc ∞ (R n × [0, ∞); R m ) we compute
0 =
∫ ∞
∫ ∞
= −
0 −∞
∫ ∞
∫ ∞
0
(∂ t u + div x F(u)) · v dxdt
−∞
and make the following
u · ∂ t v + F(u) : D x v dxdt −
∫ ∞
−∞
u(x, 0) ·v(x, 0) dx
} {{ }
=g(x)
Definition 3.1 u ∈ L ∞ (R n ×(0, ∞); R m ) is called and integral solution of (3.1)
if
∫ ∞
∫ ∞
0
−∞
for all v ∈ C ∞ c (R n × [0, ∞); R m ).
u · ∂ t v + F(u) : D x v dxdt +
56
∫ ∞
−∞
g(x) v(x, 0) dx = 0