Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities
The author studies the global structure stability of the Lax's Riemann solution \(u=U(x/t)\), containing only shocks and contact discontinuities, of general \(n\times n\) quasilinear hyperbolic system of conservation laws, \[ {\partial u\over \partial t}+{\partial f(u)\over \partial x}=0, \] with piecewise constant initial data \(u=\widehat u_-\) for \(x\leq 0\) and \(u=\widehat u_+\) for \(x\geq 0\) at \(t=0\), where \(\widehat u_{\pm }\) are two constant vectors satisfying \(\widehat u_-\neq \widehat u_+\), \(u=(u_1,u_2,\dots ,u_n)^T\) is a unknown vector function of \((t,x)\), \(f:\mathbb{R}\)\(^n\to\mathbb{R}\)\(^n\) is a given \(C^1\) vector function of \(u\). In other words, it is shown the global existence and uniqueness of the piecewise \(C^1\)-solution \(u=u(t,x)\) of a class of generalized Riemann problem, which can be regarded as a perturbation of the corresponding Riemann problem, for the quasilinear hyperbolic system of conservation laws. The above stated solution has a global structure similar to that of the solution \(u=U(x/t)\). The main statement is that Lax's Riemann solution of a general \(n\times n\) quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.
scientific article; zbMATH DE number 1891269
scientific article