SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension

Abstract

We prove that if $t\mapsto u(t)\in \mathrm{BV}(\mathbb{R})$ is the entropy solution to a $N\times N$ 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 $\mathrm{SBV}$, 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 \[ u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N, \] 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 ${\mu }_{i,\mathrm{jump}}$ is positive.

References in corpus (1)

• SBV regularity for Hamilton-Jacobi equations in ${\mathbb{R}}^n$

Cited by in corpus (5)

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• New interaction estimates for the Baiti-Jenssen system
• Global structure of admissible BV solutions to piecewise genuinely nonlinear, strictly hyperbolic conservation laws in one space dimension