A class of fully nonlinear 2x2 systems of partial differential equations

Journal Article · Sun Dec 31 04:00:00 EST 1995 · Communications in Partial Differential Equations

DOI:https://doi.org/10.1080/03605309508821126· OSTI ID:161661

Shearer, M ^[1]; Schaeffer, D G ^[2]

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1. North Carolina State Univ., Raleigh, NC (United States)
2. Duke Univ., Durham, NC (United States)

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.

• Sponsoring Organization: USDOE
• OSTI ID: 161661
• Journal Information: Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 7-8 Vol. 20; ISSN CPDIDZ; ISSN 0360-5302
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
99 GENERAL AND MISCELLANEOUS
ANALYTICAL SOLUTION
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
SHOCK WAVES
THERMOELASTICITY