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## 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:

[tex] 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}

[/tex]

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!