M-D Riemann problem for a class of quasilinear hyperbolic system and its perturbation (Q1895528)

The Cauchy problem of the following quasilinear hyperbolic system is discussed: \[ U_t+ F(U)_x+ G(U)_y= 0,\tag{1} \] where \(U= {^t(u, v)}\), \(F={^t(u^2, uv)}\), \(G={^t(uv, v^2)}\). The system is nonstrictly hyperbolic and possibly arises in some physical problems such as oil recovery and elastic theory. The initial data of our problem are given as \[ U|_{t= 0}= U^0(x, y),\tag{2} \] where \(U^0(x, y)\) has discontinuity on three curves \(\Gamma_i\) \((i= 1, 2, 3)\) issuing from the origin, and equals \(U_i(x, y)\), smooth function in the domain \(\Omega_i\). Generally, the solution of this problem has a flowery singularity structure issuing from the origin; hence the solution cannot be regarded as a perturbation of a solution to one-dimensional problem. Therefore, the initial problem is called essentially multidimensional problem. In this note, the author is going to study the problem (1), (2) in rather general assumptions. We only require that the ratio \(v^0/u^0\) be constant in each domain \(\Omega_i\). Then the solution is not self-similar, but it can be obtained as a perturbation of a corresponding self-similar solution.