have in particular for $\,\, \lambda = t\,$ : $\,\,u(x,\,t) \,=\, u(x-at,\,0) \,=\, u_0(x-at)
\,\,$ as illustrated on Figure 1.1. $ \hfill \square \kern0.1mm $

=5,0cm

[**Figure 1.1.**]{} The solution $\,\, u(x, t)\,\,$ of the advection equation

is constant along the characteristic lines.

=4,5cm

[**Figure 1.2.**]{} Initial-boundary value problem for the advection equation.

[1.2]{} $ \,\,\,\,  $ [ **Initial-boundary value problems for the advection equation.** ]{}

$\bullet \qquad \,\,\,\,\, $ The second step is concerned by the so-called initial-boundary value problem considered for $\, x>0 \,$ and $t>0 \,$ with some given initial condition $\,\, 
u_0(x) \,\,$ for $\, t=0 \,$ and a boundary condition $\,\, v_0(t) \,\,$ for $\,
x=0 \,$ :

(1.2.1) $\qquad \displaystyle 
{{\partial u}\over{\partial t}} \,\,+\,\, a \,\, {{\partial u }\over{\partial
x}}\,\,=\,\, 0 \,\,, \qquad t > 0 \,,\quad x > 0\,,  \qquad \,$ (equation)

(1.2.2) $\qquad \displaystyle 
u(x, \,0) \,\,= \,\, u_0(x) \,,\qquad \quad x > 0 \,,\,$ (initial condition)

(1.2.3) $\qquad \displaystyle 
u(0, \,t) \,\,\, = \,\, v_0(t) \,,\qquad \quad \,\, t > 0 \,,\,$ (boundary condition).

[**Proposition 1.2. $\quad$ Advection in the quadrant $\,\, x > 0 \,$ and $\,\, t > 0 \,.$** ]{}

We suppose that $\,\, a > 0 \,.$ 