Abstract
This paper is a study of certain fully nonlinear 2x2 systems of partial differential equations in one space variable and time. The nonlinearity contains a term proportional to {vert_bar}{partial_derivative}U/{partial_derivative}x{vert_bar} where U - U(x,t) {element_of} {Re}{sup 2} is the unknown function and {vert_bar}.{vert_bar} is the Euclidean norm on {Re}{sup 2}; i.e., a term homogeneous of degree 1 in {partial_derivative}U/{partial_derivative}x and singular at the origin. Such equations are motivated by hypoplasticity. The paper introduces a notion of hyperbolicity for such equations and, in the hyperbolic case, proves existence of solutions for two initial value problems admitting similarity solutions: the Riemann problem and the scale-invariant problem. Uniqueness is addressed in a companion paper. 10 refs., 10 figs.
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