A
compression wave
with

(163) 
steepens with time and characteristic curves can cross: multiplevalued solution?
 The viscous Burgers' equation (158)
is a parabolic PDE and has a unique solution for all times .

Vanishing viscosity:
We look for a solution of the inviscid Burgers' equation that is a solution of
the viscous Burgers' equation in the limit
.
 Instead of a multiplevalued solution we get a discontinuity
where the characteristics end.

Discontinuities (shocks) are unavoidable
for
.
 Characteristics run into a shock from both sides.
 Discontinuities should be allowed in the initial conditions.
 We need to find weak solutions.