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\centerline{\large \bf On the Riemann problem for strictly hyperbolic systems}

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\centerline{Stefano Bianchini}
\centerline{\it IAC - CNR, Rome, Italy }
\centerline{e-mail bianchin@iac.rm.cnr.it }
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%   Text of the abstract

We consider the construction and the properties of the Riemann solver
for the strictly hyperbolic system

u_t + f(u)_x = 0      ~~~~~~~~~~~        (*)

We first prove a general regularity theorem on the
admissible i-curves, depending on the number of
inflection points of the flux f. Namely, if there is only one inflection
point, the i-curve is continuously differentiable.
If the i-th eigenvalue of the Jacobian matrix Df
is genuinely nonlinear, it is known that the i-curve is twice continuously
differentiable. However, in the general case
we give an example where the i-curve is only
Lipschitz continuous,if the flux f has two inflection points.

Moreover, using the same analysis performed in the
vanishing viscosity case, we show a general way for constructing the
i-curves, and we prove a stability result on the solution to the Riemann
problem. In particular we prove the uniqueness of the admissible
curves for  (*).                                              

Finally we apply the construction to various approximations to
the hyperbolic system (*): 
vanishing viscosity, relaxation schemes and
the semidiscrete upwind scheme. In particular, when the system is in
conservation form, we obtain the existence of smooth travelling
profiles for all small admissible jumps.  

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