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\(L_1\) stability of conservation laws with coinciding Hugoniot and characteristic curves. (English) Zbl 0935.35090
The purpose of the paper is to study the well-posedness of the Cauchy problem for the nonlinear scalar hyperbolic conservation law in one spatial dimension. The authors restrict their considerations on the simplified situation where the Hugoniot curves and the characteristic curves coincide. They show that there exists a simple nonlinear functional, which is time-decreasing and equivalent to the \(L_1\) distance between the two solutions. This functional yields the \(L_1\) posedness of the Cauchy problem, which is the main result of the paper. The functional contains a Glimm functional and nonlinear coupling functional. For the construction of the linear functional, certain hyperbolic conservation laws are introduced, which yield particular solutions of the system. The theorem is proved using the random choice method and the wave tracing method. It is well known, that when the system is linearly degenerate, then the Hugoniot and characteristic curves coincide; but other important physical systems, not necessarily linearly degenerate, with this coinciding property, can be found.

35L45 Initial value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35L90 Abstract hyperbolic equations
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