Order Form - FENOMEC

8 CHAPTER I. FUNDAMENTAL CONCEPTS AND EXAMPLES
2. Shock formation and weak solutions
The existence and the uniqueness of (locally defined in time) smooth solutions of
strictly hyperbolic systems of conservation laws follow from standard compactness
arguments in Sobolev spaces. Generally speaking, smooth solutions u = u(x, t) of
nonlinear hyperbolic equations eventually loose their regularity at some finite critical
time, at which the derivative ∂ x u tends to infinity. This breakdown of smooth solutions
motivates the introduction of the concept of weak solutions, which allows us to
deal with discontinuous solutions of (1.1) such as shock waves.
First of all, in order to clarify the blow-up mechanism, we study the typical
case of Burgers equation, introduced in Example 1.3, and we discuss three different
approaches demonstrating the non-existence of smooth solutions. Let u = u(x, t) be
a continuously differentiable solution of (1.6) satisfying the initial condition (1.2) for
some smooth function u 0 . Suppose that this solution is defined for small times t, at
least.
Approach based on the implicit function theorem. Following the discussion
in Example 1.3, observe that the implicit function theorem fails to apply to (1.7)
when t is too large. More precisely, it is clear that the condition (1.8) always fails if
t is sufficiently large, except when
u 0 is a non-decreasing function. (2.1)
When (2.1) is satisfied, the transformation v ↦→ v − u 0 (x − vt) remains one-to-one for
all times and (1.7) provides the unique solution of (1.6) and (1.2), globally defined in
time.
Geometric approach. Given y 0 ∈ I R,thecharacteristic curve t ↦→ y(t) issuing
from y 0 is defined (locally in time, at least) by
y ′ (t) =u ( y(t),t ) , t ≥ 0,
y(0) = y 0 .
(2.2)
The point y 0 is referred to as the foot of the characteristic. Setting
v(t) :=u ( y(t),t )
and using (1.6) and (2.2) one obtains
v ′ (t) =0.
So, the solution is actually constant along the characteristic which, therefore, must
be a straight line. It is geometrically clear that two of these characteristic lines
will eventually intersect at some latter time, except if the u 0 satisfies (2.1) and the
characteristics spread away from each other and never cross.
Approach based on the derivative ∂ x u. Finally, we show the connection with
the well-known blow-up phenomena arising in solutions of ordinary differential equations.
Given a smooth solution u, consider its space derivative ∂ x u along a characteristic
t ↦→ y(t), that is, set
w(t) :=(∂ x u) ( y(t),t ) .