2014/12/09 by Shuang Miao, Miao, Shuang, Pin Yu +1 · 4 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1412.3058
openalex publication_date 2014/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper is devoted to the study of shock formation of the 3-dimensional quasilinear wave equation - (1+3G′′(0) (∂tϕ)2)∂2t ϕ+Δϕ=0, where G′′(0) is a non-zero constant. We will exhibit a family of smooth initial data and show that the foliation of the incoming characteristic hypersurfaces collapses. Similar to 1-dimensional conservational laws, we refer this specific type breakdown of smooth solutions as shock formation. Since (⋆) satisfies the classical null condition, it admits global smooth solutions for small data. Therefore, we will work with large data (in energy norm). Moreover, no symmetry condition is imposed on the initial datum. We emphasize the geometric perspectives of shock formations in the proof. More specifically, the key idea is to study the interplay between the following two objects: (1) the energy estimates of the linearized equations of (⋆); (2) the differential geometry of the Lorentzian metric g=-\dfrac1(1+3G′′(0) (∂tϕ)2) d t2+dx12+dx22+dx32. Indeed, the study of the characteristic hypersurfaces (implies shock formation) is the study of the null hypersurfaces of g. The techniques in the proof are inspired by the work \citeCh-Shocks in which the formation of shocks for 3-dimensional relativistic compressible Euler equations with small initial data is established. We also use the short pulse method which is introduced in the study of formation of black holes in general relativity in \citeCh-BlackHoles and generalized in \citeK-R-09.