Chapter 1 contains what was originally published as [BLe]. In the framework of the
first point above, we prove that every Lax admissible weak solution of (1) coincides
with the corresponding SRS trajectory if and only if it has locally bounded variation
along a suitable family of space-like curves. By the uniqueness of SRS, there follows
a uniqueness result for (1) (2), within the class of solutions having the mentioned
property.
Chapters 2 and 3 are concerned with the L1 stability of wave patterns containing
some non-interacting large shock waves. We study the problem (1) with u in (2)
being a small L^ 1 ∩ BV perturbation of fixed Riemann data. We a priori assume that
the solution of the latter problem is given by a number of (arbitrarily large) Lax
compressive and Majda stable shocks of different characteristic families. We formulate
the BV and 11 Stability Conditions that express the expected mutual influence
of the large waves. By constructing suitable Glimm and Lyapunov functionals applicable
to our setting, we show that the former condition guarantees the existence
of a unique, global in time and space, 'admissible' solution to (1) (2); while the
latter condition is essential for the stablity of this same solution under (a class of)
perturbations of its initial data. This is carried out in Chapter 2, containing the
results of (Lel].
In Chapter 3 we present a revised version of the article (Le2], with some new
additions. Several authors had investigated the issue of wellposedness of (1) (2) in
various contexts, introducing different stability conditions. Some of them require
that the eigenvalues of suitable matrices related to wave transmissions - reflections
are smaller than 1 in absolute value, other refer to different algebraic properties of
the linearised system, such as for example existence of weights with whom the flow
of the system becomes a contraction. We explain and compare these conditions,
showing that the conditions of Chapter 2 generalize or unify them in appropriate
ways.
