Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959–1000] obtained the strong convergence of uniform L loc p bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is s...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2008-04, Vol.340 (1), p.558-568
Main Author: Cheng, Zhixin
Format: Article
Language:English
Subjects:
[formula omitted] solution
Dirac measure
Entropy solution
Exact sciences and technology
Lax entropy
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Sciences and techniques of general use
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Summary:Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959–1000] obtained the strong convergence of uniform L loc p bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein–Weierstrass theorem, and obtaining the strong convergence of uniform L ∞ or L loc p bounded viscosity solutions for scalar conservation law without convexity.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2007.08.050