# SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one Space Dimension

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## Abstract

We prove that if \({t \mapsto u(t) \in BV(\mathbb{R})}\) is the entropy solution to a *N* × *N* strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields

$$u_t + f(u)_x = 0,$$

*u*(

*t*) is in SBV, i.e. its distributional derivative

*u*

_{ x }is a measure with no Cantorian part.

The proof is based on the decomposition of *u* _{ x }(*t*) into waves belonging to the characteristic families

$$u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal{M}(\mathbb{R}), \, \tilde r_i(t) \in \mathbb{R}^N,$$

*v*

_{ i }in regions bounded by characteristics. To this aim, a new interaction measure

*μ*

_{ i,jump}is introduced, controlling the creation of atoms in the measure

*v*

_{ i }(

*t*).

The main argument of the proof is that for all *t* where the Cantorian part of *v* _{ i } is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure *μ* _{ i,jump} is positive.

## Keywords

Shock Front Hyperbolic System Riemann Problem Entropy Solution Continuous Part
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