Abstract
This chapter discusses some results on the uniqueness of solutions to systems of conservation laws of the form U t + f (U) x = 0, –∞ < X < ∞, where u = u (x,t) ɛ Rn and f is a smooth nonlinear mapping from Rn to Rn. The chapter presents the assumption that this equation is strictly hyperbolic, that is, the Jacobian ∇f of f has n real and distinct eigenvalues: λ1 (u) < … < λn (u). Systems of this form arise in continuum mechanics. The equations of inviscid fluid dynamics, for example, form a system of three equations; the components of u represent densities of mass, momentum, and total energy, while the equations express the physical laws of the conservation of the corresponding three quantities. Other examples are provided by the equations of shallow water waves, magneto-hydrodynamics and in certain special cases, elasticity.
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