Math295b Winter Term 2002

Assignment 3

1. Let 0 ≠ α ∈ ℝ and consider the initial value problem for Euler’s equation

∑_{k=1}^{n} x_{k}∂_{x_{k}}u = αu , u(x_{1}, …, x_{n−1}, 1) = g(x_{1}, …, x_{n−1}) .

Show that its solution satisfies the functional equation

u(λx) = λ^{α}u(x) , x ≠ 0 , λ > 0 .

What is the behavior of the solution at x = 0?

2. Consider the quasilinear initial value problem

\begin{cases}u_{t} + u u_{x} = 0 , & (t, x) ∈ (0, ∞) × ℝ \\ u(0, x) = g(x) , & x ∈ ℝ \end{cases}

Solve the equation and analyze the possible onset of singularities. What conditions on g would prevent the solution from developing singularities?

3. Let u ∈ C^{1}(𝔹(0, 1)) be a solution of

a(x, y)u_{x} + b(x, y)u_{y} = −u

and assume that a(x, y)x + b(x, y)y > 0 for x ∈ 𝕊^{1}. Show that u ≡ 0 then.

4. Consider the equation

\frac{∂R(u)}{∂y} + \frac{∂S(u)}{∂x} = 0 .

Any function u with

∫_{ℝ^{2}}[R(u)ϕ_{y} + S(u)ϕ_{x}] d(x, y) = 0 , ϕ ∈ C_{0}^{∞}(ℝ^{2})

is called weak solution. Assume that u is continuously differentiable away from some curve parametrized by (s(y), y) , y ∈ ℝ across which it has a jump discontinuity. Conclude that

s′(y) = \frac{S(u^{+}) − S(u^{−})}{R(u^{+}) − R(u^{−})}

where u^{±} indicate the one sided limits of u approaching the curve.

5. Consider the eikonal equation

c^{2}(u_{x}^{2} + u_{y}^{2}) = 1

Let γ_{t} be the level line [u(x, y) = t] of a solution u. Show that a point (x, y) moves in a direction perpendicular to γ_{t} at constant speed c (along a characteristic).

The Homework is due on March 8 2002