Systems of Conservation Laws with Incomplete Sets of Eigenvectors
Everywhere
Yuxi Zheng
Abstract
We present a class of systems of conservation laws which have repeated
eigenvalues and incomplete sets of eigenvectors in an open set of the state
space. These systems contain, as a prototypical example, a one-dimensional
$2\times 2$ system of equations with names such as transportation equations
in the context of flux-splitting schemes, pressureless equations in
isentropic gas dynamics, or adhesion particle dynamics equations
in astrophysics.
A distinctive feature of the prototypical example and our class of systems
is the singular concentration in one of the dependent variables in the form of
a weighted Dirac delta function at the same space-time location
as a shock wave forms, --the so-called delta-shock wave.
We describe these general systems from the
perspective of classical systems of conservation laws with strict hyperbolicity
and genuine nonlinearity. In the $2\times 2$ situation
($u_t+ f(u,v)_x =0, v_t + g(u,v)_x =0$), our general systems are such that
the following four conditions hold in an open set of the state space of
dependent variables:
(1) the eigenvalue is of multiplicity two,
(2) the system is not diagonalizable,
(3) the eigenvalue has vanishing directional derivative along a nonzero right
eigenvector,
(4) the eigenvalue has nonvanishing directional derivative along a nonzero
generalized right eigenvector.
We reduce these general systems to simple forms for continuous solutions,
and also for discontinuous solutions when a partial linearity (affinity)
condition is further assumed. Definition of distributional solutions
for these systems in the
space of bounded measures are made possible through the simple forms.
Solutions to Riemann problems for the systems are given.