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  {\Huge MTH 453/553 -- Homework 1}
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\begin{enumerate}
  
\item (20 points) [Solution to advection system]

Consider the wave equation 
\[ u_{tt} = c^2 u_{xx}\]
 written as a first order system of two equations in the form 
\[ \mb{y}_t+A\mb{y}_x = 0,\] where 
\[ A = \left[\begin{array}{cc}0 & -c\\ -c & 0\\ \end{array}\right ].\]
Diagonalize $A$ and decouple the system. Write this decoupled system in
terms of the
variable $\mb{w}$. Determine the characteristics and, hence,
$\mb{w}$. Knowing $\mb{w}$, determine $u(x,t)$ such that it satisfies
the initial data 
\[ u(x,0) = \eta(x),\qquad u_t(0,x)= \mu(x).\]

\item (20 points) [Well-posedness of advection-diffusion]

Consider the {\it advection-diffusion} equation
\[ u_t +a u_x = \nu u_{xx}\]
with constant coefficients $\nu >0$ and $a$. Show that the Cauchy problem is
well-posed. What happens as $\nu \rightarrow 0$ (in particular, what
happens to the bound on the norm of the solution)?


  
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