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

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.

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Correspondence to Laura Caravenna.

Additional information

This work has been supported by the ERC Starting Grant CONSLAW.

Communicated by P. Constantin

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Bianchini, S., Caravenna, L. SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one Space Dimension. Commun. Math. Phys. 313, 1–33 (2012). https://doi.org/10.1007/s00220-012-1480-5

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Keywords

  • Shock Front
  • Hyperbolic System
  • Riemann Problem
  • Entropy Solution
  • Continuous Part