Formation and construction of large variational shock waves for 1-D n× n quasilinear hyperbolic conservation systems

In the paper [Li Jun, Xu Gang, Yin Huicheng, On the blowup mechanism of smooth solutions to 1D quasilinear strictly hyperbolic systems with large variational initial data, Nonlinearity 38 (2025), No.2, 025016], for the 1-D n× n (n 3) strictly hyperbolic system ∂ₜv+F(v)∂ₓv=0 with some classes of large variational initial data $v(x, 0)$, the geometric blowup mechanism and the detailed singularity behaviours of ∂x,tv near the blowup point are studied when the n× n matrix $F(v)$ admits at least one genuinely nonlinear eigenvalue. In this paper, we focus on the formation and construction of a large variational shock wave from the blowup point for 1-D n× n quasilinear hyperbolic conservation law system ∂ₜu+∂ₓf(u)=0 when some smooth simple wave solution is generic non-degenerate before the formation of singularity and the corresponding eigenvalue is genuinely nonlinear.