Vol. 28, No. 3, 1969

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A uniqueness theorem for weak solutions of symmetric quasilinear hyperbolic systems

Albert Emerson Hurd

Vol. 28 (1969), No. 3, 555–559
Abstract

The essentially bounded measurable (vector) function u(x,t) = (u1(x,t),,ur(x,t)) is called a weak solution of the initial-value problem for the system

∂u-  ∂𝒜-(x,t,u)
∂t +    ∂x    = 0

in the upper half-plane t 0 if it satisfies the usual integral identity (defining “weak”) together with the condition that, given a compact set D in t 0, there exists a function K(t) Lloc1([0,)) such that

u(x ,t)− u(x ,t)
-i-1------i-2---≦ K (t)
x1 − x2

holds a.e. for x1,x2 D and 0 < t < . It is shown that, if the matrix 𝒜∕∂u is symmetric and positive definite (a convexity condition), then weak solutions are uniquely determined by their initial conditions.

Mathematical Subject Classification
Primary: 35.57
Milestones
Received: 10 April 1967
Revised: 28 May 1968
Published: 1 March 1969
Authors
Albert Emerson Hurd