Article: Data from F. Asakura et al provide new insights into mathematics.

According to recent research published in the Journal of Hyperbolic Differential Equations, "This article analyzes the existence of viscous shock profiles joining two states satisfying the Rankine-Hugoniot condition that comes from hyperbolic 2 x 2 systems of conservation laws having quadratic flux functions with an isolated umbilic point: the point where the characteristic speeds coincide and the Jacobian matrix of the flux functions is diagonalizable. The systems studied in this note are particularly in Schaeffer and Shearer's cases I and II which are relevant to the three-phase Buckley-Leverett model for oil reservoir flow."

"It is shown that any compressive ...

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