Exercises: Wasilij Barsukow
www.mathematik.uni-wuerzburg.de/∼barsukow/
Due: Tue., Nov 17, 2015; 16:00
Introduction to conservation laws
Gabriella Puppo
University of Würzburg
Assignment 5
Exercise 1. (Spherically symmetric self-steepening)
A possible1 extension of Burgers’ equation
6 P.
q : R+
0 ×R→R
∂t q + q∂x q = 0
(1)
to d spatial dimensions is
1
∂t q + ∇ · (q ⊗ q) = 0
2
d
d
q : R+
0 ×R →R
(2)
i) Consider smooth positive initial data q(0, x) = q0 (x) and determine the lifetime of a differentiable
solution of (1). To do this, find the collision time of two characteristics starting at x0 and x0 + δx in
the limit δx → 0.
In multi-d in addition to self-steepening an initial perturbation gets diluted thus approaching more and
more a linearized regime where it is governed by the wave equation. We want to study the balance between
these two p
opposing effects. Assume a spherically symmetric solution – it fulfills the radial version of (2)
with r := x21 + . . . + x2d
∂t q +
1
∂r (rd−1 q 2 ) = 0
2rd−1
(3)
Here, to simplify notation, q(t, x ∈ Rd ) ∈ Rd has been identified with its radial component q(t, r) ∈ R
depending on t and r only.
ii) Solve (3) with the method of characteristics.
iii) Consider spherically symmetric smooth positive initial data q(0, r) analogously to i) and compute the
lifetime of a differentiable solution as a function of d. What is the condition for the shock never to
develop?
Exercise 2. (Cole-Hopf transform)
Consider the viscous Burgers’ equation
4 P.
q : R+
0 ×R→R
∂t q + q∂x q = η∂x2 q
(4)
i) Defining
q(t, x) =: −2η
(∂x φ)(t, x)
φ(t, x)
+
φ : R+
0 ×R→R
show that φ fulfills the heat equation ∂t φ = η∂x2 φ.
ii) Using Exercise 3 of Assignment 1 write down an explicit solution of (4) for the initial data q(0, x) =:
q0 (x).
iii) Study the limit η → 0 with the method of steepest descent as in the aforementioned Exercise, assuming
that g 0 (y) = 0 has only one solution. What does this condition mean?
the direct extension is ∂t q + (q · ∇)q = 0. The 3-d analogue of q∂x q = 12 ∂x q 2 however is the identity
1
(q · ∇)q = ∇(|q|2 ) − q × (∇ × q) such that this extended Burgers’ equation cannot be directly rephrased as a conservation
2
law.
1 Actually,
1
Total: 10 P.