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

TL;DR

This paper investigates the blowup mechanism of smooth solutions to 1D quasilinear hyperbolic systems with large initial data, revealing detailed singularity behaviors near blowup points for systems of size n≥3.

Contribution

It extends the understanding of blowup mechanisms from scalar and 2x2 systems to larger systems (n≥3), providing detailed analysis of singularity behaviors.

Findings

Blowup occurs through derivatives becoming unbounded while solutions remain small.

Decomposition along characteristic directions is effective for analysis.

Weighted energy estimates help describe the singularity behavior.

Abstract

For the first order 1D $n\times n$ quasilinear strictly hyperbolic system $\partial_tu+F(u)\partial_xu=0$ with $u(x, 0)=\varepsilon u_0(x)$, where $\varepsilon>0$ is small, $u_0(x)\not\equiv 0$ and $u_0(x)\in C_0^2(\mathbb R)$, when at least one eigenvalue of $F(u)$ is genuinely nonlinear, it is well-known that on the finite blowup time $T_{\varepsilon}$, the derivatives $\partial_{t,x}u$ blow up while the solution $u$ keeps to be small. For the 1D scalar equation or $2\times 2$ 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 $\partial_{t,x}u$ happens). In the present paper, for the $n\times n$ ($n\geq 3$) 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 $\partial_{t,x}u$ 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.