Deriving a Bound for a System of Coupled PDEs Using the Energy Method

[physmatics]

Homework Statement

Hi!
Not sure if I'm posting in the right section, this problem is from a course in scientific computing. Anyway, we're considering a set of PDEs:

$$u_t + Au_x = 0 \quad 0<x<1, \ t>0 \\ u(x,0) = f(x) \quad 0 \leq x \leq1 \\ u_1(0,t) = 0 \quad t \geq 0 \\ u_2(1,t) = 0 \quad t \geq 0 \\ u = \begin{pmatrix} u_1 \\ u_2 \end{pmatrix}, \quad A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

Now, I want to compute a bound ||u(*, )|| in terms of f. We have the standard inner product and the norm, where ||u||^2 = (u,u).

The Attempt at a Solution

First and foremost, my idea is that I use what's called the energy method, where we multiply with u, integrate in space and apply the BCs, to find ||u||. What I can't wrap my head around though is the fact that A is a matrix and in reality I have two equations, but coupled since they both contain u1 and u2. Should I introduce two variables to multiply the equations with? How can I deal with the matrix while integrating? Or should I just give up the energy method completely and take on another approach?

Any help or small hints are much appreciated!

 

[gneill]

Hi Physmatics. You might find a better audience for your question in one of the mathematics homework forums. Maybe try
Calculus & Beyond Homework.

If you wish I can move your thread there. Let me know.

 

[physmatics]

Thank you for replying!
I figured out how to solve it on my own though, and if anyone is interested I can obviously describe the solution here :)