On the blowup mechanism of smooth solutions to 1D quasilinear strictly hyperbolic systems with large initial data

2022/04/18 by Li, Jun, Xu, Gang, Yin, Huicheng · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2204.08181

Abstract

For the first order 1D n× n quasilinear strictly hyperbolic system ∂_tu+F(u)∂_xu=0 with u(x, 0)=ε u₀(x), where ε>0 is small, u₀(x)\not≡ 0 and u₀(x)∈ C₀²(\mathbb R), when at least one eigenvalue of F(u) is genuinely nonlinear, it is well-known that on the finite blowup time T_ε, the derivatives ∂_t,xu blow up while the solution u keeps to be small. For the 1D scalar equation or 2× 2 strictly hyperbolic system (corresponding to n=1, 2), if the smooth solution u blows up in finite time, then the blowup mechanism can be well understood (i.e., only the blowup of ∂_t,xu happens). In the present paper, for the n× n (n≥ 3) strictly hyperbolic system with a class of large initial data, we are concerned with the blowup mechanism of smooth solution u on the finite blowup time and the detailed singularity behaviours of ∂_t,xu near the blowup point. Our results are based on the efficient decomposition of u along the different characteristic directions, the suitable introduction of the modulated coordinates and the global weighted energy estimates.

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

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