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$\partial_t u+A(u) \partial_x u =0,\quad\partial_t v + A(u) \partial_x v =0,$

where $u=u(x,t),\quad v=v(x,t)\in\mathbb R^N$ are the unknowns and $A(u)$ is a
strictly hyperbolic matrix. Relying on the notion of weak solution
proposed by Dal Maso, LeFloch, and Murat
("Definition and weak stability of nonconservative products",
J. Math. Pures Appl. 74 (1995), 483--548),
we establish the existence of weak solutions for the corresponding
Cauchy problem, in the class of bounded functions with bounded variation.
The main steps in our proof are as follows:

(i) We solve the Riemann problem based on a prescribed family
of paths.

(ii) We derive a uniform bound on the total variation of
corresponding wave-front tracking approximations $u^h$, $v^h$.

(iii) We justify rigorously the passage to the limit in the
nonconservative product $A(u^h) \partial_x v^h$, based on the local
uniform convergence properties of $u^h$, by extending an argument
due to LeFloch and Liu ("Existence theory for nonlinear
hyperbolic systems in nonconservative form", Forum Math. 5
(1993), 261--280). Our results provide a generalization to the
existence theorem established earlier in the scalar case ($N=1$)
by the second author ("An existence and uniqueness result for two
nonstrictly hyperbolic systems", IMA Volumes in Math. and its
Appl. 27, "Nonlinear evolution equations that change type",
ed. B.L. Keyfitz and M. Shearer, Springer Verlag, 1990,
pp. 126--138.)

First, we generalize a technique due to Tzavaras and show that the boundary Riemann problem with diffusion admits a family of continuous solutions that remain uniformly bounded in the total variation norm. Careful estimates are necessary to cope with waves that collapse at the boundary and generate the boundary layer.

Second, we prove the convergence of these continuous solutions toward weak solutions of the Riemann problem when the diffusion parameter approaches zero. Following Dubois and LeFloch, we formulate the boundary condition in a weak form, based on a set of admissible boundary traces. Following Part I of this work, we identify and rigorously analyze the boundary set associated with the zero-diffusion method. In particular, our analysis fully justifies the use of the scaling $1/\varepsilon$ near the boundary (where $\varepsilon$ is the diffusion parameter), even in the characteristic case as advocated in Part I by the authors.

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