2012/10/30 by Bingbing Ding, Ding, Bingbing, Ingo Witt +3
Mathematics · Physics and Astronomy · #35J70 #35L65 #35R35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #math.AP #msc:35J70 #msc:35L65 #msc:35R35
paper · pdf · doi:10.48550/arxiv.1210.7980
22 pages
arxiv created 2012/10/30 · openalex publication_date 2012/10/30 · arxiv updated 2012/10/31 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation \pt2u-\ds∑i=12\pi(ci2(u)\piu) =0, where ci(u)∈ C∞(\Bbb Rn), ci(0)≠ 0, and (c1'(0))2+(c2'(0))2≠ 0. This equation has an interesting physics background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition (u(0,x), \ptu(0,x))=(\ve u0(x), \ve u1(x)) with u0(x), u1(x)∈ C0∞(\Bbb R2), and \ve>0 is small, we will show that the classical solution u(t,x) stops to be smooth at some finite time T\ve. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives \nat,xu(t,x), while u(t,x) itself is continuous up to the blowup time T\ve.