SBV-like regularity for general hyperbolic systems of conservation laws

2012/02/13 by Stefano Bianchini, Lei Yu, Bianchini, Stefano +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1202.2680

20 pages, no figure

arxiv created 2012/02/13 · arxiv updated 2012/02/14

Abstract

We prove the SBV regularity of the characteristic speed of the scalar hyperbolic conservation law and SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux function for general systems of conservation laws. More precisely, for the equation u_t + f(u)_x = 0, u : \R⁺ × \R → Ω⊂ \R^N, we only assume the flux f is C² function in the scalar case (N=1) and Jacobian matrix Df has distinct real eigenvalues in the system case (N≥ 2). Using the modification of the main decay estimate in Lau and localization method applied in \citeR, we show that for the scalar equation f'(u) belongs to SBV, and for system of conservation laws the scalar measure \[(D_u λ_i(u) ⋅ r_i(u) ) (l_i(u) ⋅ u_x )] has no Cantor part, where λ_i, r_i, l_i are the i-th eigenvalue, i-th right eigenvector and i-th left eigenvector of the matrix Df.

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