Unbounded Solutions of One-Dimensional Conservation Laws with Asymmetrical Power-Type Flux Function

Abstract

The paper considers a problem for the quasilinear first-order partial differential equation in the case of one spatial variable. Flow function is assumed to be the asymmetric power function with a single inflection point at zero, the initial conditions are of the unbounded exponential and power types. Piecewise smooth generalized entropy solutions are constructed using the characteristic method. These solutions are defined within the entire half-plane t > 0, they have a countable number of discontinuity lines, and are changing their sign while passing through each discontinuity line. The characteristics are straight lines and the discontinuity lines are obtained by enveloping the characteristics using the Legendre transform. Explicit formulas are obtained for the discontinuity lines. In the case of a power-type initial condition, hyperbolas are the discontinuity lines. While in the case of exponential initial function, they are the logarithmic curves. The Rankine --- Hugoniot condition is used to continue solution beyond the discontinuity line. Due to the flux function properties, the entropy increase condition is satisfied automatically. Thus, the obtained solutions are the generalized entropy by construction. The paper proves one-sided periodicity of the constructed solution with respect to the spatial variable in the case of an exponential initial condition

The work was supported by the Ministry of Science and Higher Education of the Russian Federation (project no. FSFN-2024-0004)

Please cite this article in English as:

Gargyants L.V. Unbounded solutions of one-dimensional conservation laws with asymmetrical power-type flux function. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2024, no. 5 (116), pp. 4--14 (in Russ.). EDN: QKCUJL

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