Weak Convergence Methods for Nonlinear Partial Differential ...

Examples:
1. The p-system: {
∂ t u 1 − ∂ x u 2 = 0,
∂ t u 2 − ∂ x p(u 1 ) = 0,
( )
c − y
p a given function. Here, F(y) = 2
. This system arises in the study
−p(y 1 )
of nonlinear wave equation
when u 1 = ∂ t u, u 2 = ∂ x u.
∂ tt u − ∂ x (p(∂ x u)) = 0,
2. Euler’s equations for compressible gas flow:
Let
• ρ = mass density
• v = velocity
• E = e + 1 2 v2 energy per unit mass, where e is the ”internal energy”,
• p = p(ρ, e) pressure.
The last equation p = p(e, v) is a “constitutive equation”: p is assumed to
be a known function, which models the material specific properties.
Euler’s equations (in the variables: u = (u 1 , u 2 , u 3 ) = (ρ, ρv, ρE)) are
∂ t ρ + ∂ x (ρv) = 0
∂ t (ρv) + ∂ x (ρv 2 + p) = 0
∂ t (ρE) + ∂ x (ρEv + pv) = 0
(conservation of mass),
(conservation of momentum),
(conservation of energy).
For sufficiently small time intervals, it is not hard to prove that a single
conservation law has a classical solution. (One uses the method of characteristics
to construct it - cf. [Sch 10].) But already for quite simple PDEs (e.g. Burger’s
equation) such a solution does not exist for all times.
Recall: For a quasilinear equation of first order
the characteristic equations are
a(x, u) · ∇u = b(x, u)
dx j
dt = a j(x, y), j = 1, ..., n,
dy
dx
= b(x, y).
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