The Riemann problem for the planar motion of an elastic string
The object of the paper is to provide a constructive proof of the following result: Let T satisfy (i) T is of class \(C^ 3\), (ii) \(T(\xi)>0\) for all \(\xi\), (iii) \(T(1)=0\), (iv) there exists \(\xi_ J>1\) such that \(sgn T''(\xi)=sgn(\xi-\xi_ J)\), (v) there are exactly two solutions \(\xi_ m<\xi_ M\) of the equation \(T(\xi)/\xi=T(\xi)\). For any given \(W_ L\), \(W_ R\) in \({\mathbb{R}}^ 4\), the initial value problem \(r_{tt} = [T(| r_ x|)r_ x/| r_ x|]_ x\); \((r_ x,r_ t)(x,0)=W_ L\) if \(x<0\), \((r_ x,r_ t)(x,0)=W_ R\) if \(x>0\) has a unique weak solution among those functions \((r_ x,r_ t)\) depending piecewise smoothly on \(x/t\) and whose jump discontinuities satisfy certain entrophy admissible condition.
scientific article; zbMATH DE number 4023872