Similarity Solutions of Shock Formation for First-order Strictly Hyperbolic Systems

2025/11/01 by Eshima, Jun, Deike, Luc, Stone, Howard A.
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2511.00672

Abstract

Shocks due to hyperbolic partial differential equations (PDEs) appear throughout mathematics and science. The canonical example is shock formation in the inviscid Burgers' equation (∂ u)/(∂ t)+u(∂ u)/(∂ x)=0. Previous studies have shown that when shocks form for the inviscid Burgers' equation, for positions and times close to the shock singularity, the dynamics are locally self-similar and universal, i.e., dynamics are equivalent regardless of the initial conditions. In this paper, we show that, in fact, shock formation is self-similar and universal for general first-order strictly hyperbolic PDEs in one spatial dimension, and the self-similarity is like that of the inviscid Burgers' equation. An analytical formula is derived and verified for the self-similar universal solution.

Citations

• Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks
• Formation of singularities in one‐dimensional nonlinear wave propagation

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