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Distributions as initial values in a triangular hyperbolic system of conservation laws. (English) Zbl 1470.46064

Summary: The present paper concerns the system \(u_t+[\varphi(u)]_x=0\), \(v_t+[\psi(u)v]_x=0\) having distributions as initial conditions. Under certain conditions, and supposing \(\phi,\psi:\mathbb{R}\rightarrow\mathbb{R}\) functions, we explicitly solve this Cauchy problem within a convenient space of distributions \(u,v\). For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, \(\delta\)-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader’s convenience, a brief survey of the distributional product is also included.

MSC:

46F10 Operations with distributions and generalized functions
35D99 Generalized solutions to partial differential equations
35L67 Shocks and singularities for hyperbolic equations
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