2016/09/07 by Moon-Jin Kang, Alexis Vasseur, Kang, Moon-Jin +3 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1609.01825
openalex publication_date 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a L2-contraction of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift. The shift function depends on the time and space variables. It solves a parabolic equation with inhomogeneous coefficients reflecting the perturbation. We consider a suitably small L2-perturbation around a viscous planar shock wave of arbitrarily large strength. However, we do not impose any condition on the anti-derivative variables of the perturbation around shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in the L2 norm, then the L2-contraction holds true for the viscous shock wave up to a shift function which may depend on the temporal and spatial variables. Moreover, as the time t tends to infinity, the L2-contraction holds true up to a time-dependent shift function. In particular, if we choose some special initial perturbation, then we can prove a L2 convergence of the solutions towards the associated shock profile up to a time-dependent shift.