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Communications in Mathematical Physics

, Volume 313, Issue 1, pp 1–33 | Cite as

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

  • Stefano Bianchini
  • Laura CaravennaEmail author
Article

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

$$u_t + f(u)_x = 0,$$
then up to a countable set of times \({\{t_n\}_{n \in \mathbb{N} }}\) the function 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

$$u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal{M}(\mathbb{R}), \, \tilde r_i(t) \in \mathbb{R}^N,$$
and the balance of the continuous/jump part of the measures 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.Centro De Giorgi, Scuola Normale SuperiorePisaItaly

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