Abstract: The paper is concerned with the Cauchy problem for a nonlinear, strictly hyperbolic system with small viscosity: $$u_t+A(u)u_x=\ve\, u_{xx}, \qquad \qquad u(0,x)=\bar u(x).\eqno(*)$$ We assume that the integral curves of the eigenvectors $r_i$ of the matrix $A$ are straight lines. On the other hand, do not require the system ($*$) to be in conservation form, nor we make any assumption on genuine linearity or linear degeneracy of the characteristic fields.In this setting we prove that, for some small constant $\eta_0>0$ the following holds. For every initial data $\bar u\in \L^1$ with $\tv\{\bar u\}<\eta_0$, the solution $u^\ve$ of (*) is well defined for all $t>0$. The total variation of $u^\ve(t,\cdot)$ satisfies a uniform bound, independent of $t,\ve$. Moreover, as $\ve\to 0+$, the solutions $u^\ve(t,\cdot)$ converge to a unique limit $u(t,\cdot)$. The map $(t,\bar u) \mapsto S_t \bar u\doteq u(t,\cdot)$ is a Lipschitz continuous semigroup on a closed domain $\D\subset \L^1$ of functions with small total variation. This semigroup is generated by a particular Riemann Solver, which we explicitly determine.
The above results can also be applied to strictly hyperbolic systems on a Riemann manifold. Although these equations cannot be written in conservation form, we show that the Riemann structure uniquely determines a Lipschitz semigroup of ``entropic'' solutions, within a class of (possibly discontinuous) functions with small total variation. The semigroup trajectories can be obtained as the unique limits of solutions to a particular parabolic system, as the viscosity coefficient approaches zero.
The proofs rely on some new a priori estimates on the total variation of solutions for a parabolic system whose components drift with strictly different speeds.
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