Global existence to the Cauchy problem for hyperbolic conservation laws with an isolated umbilic point

Felaco, Elisabetta; Rubino, Bruno; Sampalmieri, Rosella

doi:10.1090/S0033-569X-2013-01328-6

Authors: Elisabetta Felaco, Bruno Rubino and Rosella Sampalmieri
Journal: Quart. Appl. Math. 71 (2013), 629-659
MSC (2010): Primary 35L65, 35L80
DOI: https://doi.org/10.1090/S0033-569X-2013-01328-6
Published electronically: August 28, 2013
MathSciNet review: 3136988
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Abstract:

In this paper the existence of global weak solutions for a $2 \times 2$ system of non-strictly hyperbolic non-linear conservation laws is established for data in $L^{\infty }$.

The result is proven by means of viscous approximation and application of the compensated compactness method.

The presence of a degeneracy in the hyperbolicity of the system requires a careful analysis of the entropy functions, whose regularity is necessary to obtain an existence result. For this purpose we combine the classical techniques referring to a singular Euler-Poisson-Darboux equation with the compensated compactness method.

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