### Abstract

The paper is concerned with the Cauchy problem for a nonlinear, strictly hyperbolic system with small viscosity: (*) u_{t} + A(u)u_{x} = ε u_{xx}, u(0,x) = ū(x). We assume that the integral curves of the eigenvectors r_{i} of the matrix A are straight lines. On the other hand, we do not require the system (*) to be in conservation form, nor do we make any assumption on genuine linearity or linear degeneracy of the characteristic fields. In this setting we prove that, for some small constant η_{0} > 0 the following holds. For every initial data ū ∈ L^{1} with Tot. Var. {ū} < η_{0}, the solution u^{ε} of (*) is well defined for all t > 0. The total variation of u^{ε}(t, ·) satisfies a uniform bound, independent of t, ε. Moreover, as ε → 0+, the solutions u^{ε}(t, ·) converge to a unique limit u(t, ·). The map (t, ū) → S_{t}ū; (approaches the limit) u(t, ·) is a Lipschitz continuous semigroup on a closed domain D ⊂ L^{1} of functions with small total variation. This semigroup is generated by a particular Riemann Solver, which we explicitly determine. The results above 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.

Original language | English (US) |
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Pages (from-to) | 1673-1713 |

Number of pages | 41 |

Journal | Indiana University Mathematics Journal |

Volume | 49 |

Issue number | 4 |

State | Published - Dec 1 2000 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Indiana University Mathematics Journal*,

*49*(4), 1673-1713.