2019/01/15 by Myoungjean Bae, Bae, Myoungjean, Hyangdong Park +1
Mathematics · Physics and Astronomy · #35J47 #35J57 #35J66 #35Q31 #35R35 #74J40 #76N10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP #msc:35J47 #msc:35J57 #msc:35J66 #msc:35Q31 #msc:35R35 #msc:74J40 #msc:76N10
paper · pdf · doi:10.48550/arxiv.1901.04996
To be published in Journal of Differential Equations (2019); arXiv admin note: substantial text overlap with arXiv:1810.04411
arxiv created 2019/04/18 · arxiv updated 2019/04/19
We prove the existence of a subsonic axisymmetric weak solution (\bf u,ρ,p) with \bf u=ux\bf ex+ur\bf er+uθ\bf eθ to steady Euler system in a three-dimensional infinitely long cylinder N when prescribing the values of the entropy (=(p)/(ργ)) and angular momentum density (=ruθ) at the entrance by piecewise C2 functions with a discontinuity on a curve on the entrance of N. Due to the variable entropy and angular momentum density (=swirl) conditions with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity, nonzero swirl, and contains a contact discontinuity r=gD(x). We construct such a solution via Helmholtz decomposition. The key step is to decompose the Rankine-Hugoniot conditions on the contact discontinuity via Helmholtz decomposition so that the compactness of approximated solutions can be achieved. Then we apply the method of iteration to obtain a piecewise smooth subsonic flow with a contact discontinuity, nonzero vorticity, and nonzero angular momentum density. We also analyze the asymptotic behavior of the solution at far field.