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A variational calculus for discontinuous solutions of systems of conservation laws. (English) Zbl 0846.35080
The paper is concerned with the Cauchy problem for the perturbed system of conservation laws in a single variable: \[ u_t+ [F(u)]_x= h(t, x, u),\quad u(0, x)= \overline u(x), \] where \(F: \mathbb{R}^n\to \mathbb{R}^n\) and \(h: [0, \infty)\times \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) are smooth functions. It is assumed that the system is strictly hyperbolic, and each characteristic field is either linearly degenerate or genuinely nonlinear in the sense of Lax. For initial conditions \(u(0, x)= \overline u^\varepsilon(x)\) depending on a small parameter \(\varepsilon\), the behavior of first-order variation of solutions \(u^\varepsilon(t, x)\) is studied when the solutions \(u^\varepsilon\) contain shocks, namely, discontinuities. Introducing “generalized tangent vectors” to solutions, consisting of measures whose absolutely continuous part is some \(L^1\) function and whose singular part is given by point charges, and then determining their evolution in time, the authors provide a description of corresponding solutions \(u^\varepsilon (t, \cdot)\) with first-order accuracy w.r.t. \(\varepsilon\) under suitable regularity assumptions and show that the perturbed solutions \(u^\varepsilon(t, \cdot)\) still admit a first-order approximation even after the interaction of two shocks of \(u^\varepsilon\).
Reviewer: C.Radu (Iaşi)

35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
35L67 Shocks and singularities for hyperbolic equations
Full Text: DOI
[1] DOI: 10.1512/iumj.1988.37.37021 · Zbl 0632.35041
[2] DOI: 10.1006/jdeq.1993.1111 · Zbl 0802.35095
[3] Bressan, A. 1995.A locally contractive metric for systems of conservation laws, IV Vol. XXII, 109–135. Pisa: Ann. Scuola Normale Sup. · Zbl 0867.35060
[4] A. Bressan and R M. Colombo, The semigroup generated by 2 x 2 conservation laws, Arch. Rational Mech. Anal., to appear · Zbl 0849.35068
[5] A.Bressan and A. Marson, A maximum principle for optimally controlled systems of conservation laws, Rend. Sem. Mat. Univ. Padova, to appear · Zbl 0935.49012
[6] DOI: 10.1512/iumj.1979.28.28011 · Zbl 0409.35057
[7] DOI: 10.1002/cpa.3160180408 · Zbl 0141.28902
[8] DOI: 10.1002/cpa.3160100406 · Zbl 0081.08803
[9] Rozdestvenskii B.L., A.M.S. Translations of Mathematical Monographs 55 (1983)
[10] DOI: 10.1016/0022-0396(91)90124-R · Zbl 0733.35072
[11] Li Ta-Tsien, Boundary Value Problems for Quasilinear Hyperbolic Systems (1985)
[12] Smoller J., Shock Waves and Reaction-Diffusion Equations (1983) · Zbl 0508.35002
[13] Ziemer W.P., Weakly Differential Functions (1989) · Zbl 0692.46022
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