2022/12/31 by Daria Bolbot, Bolbot, Daria, Dimitrios Mitsotakis +3 · 1 citation
Mathematics · Physics and Astronomy · #35L65 #35Q40 #74J30 #76N30 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2301.00197
openalex publication_date 2022/12/31 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28
The aim is to assess the combined effect of diffusion and dispersion on shocks in the moderate dispersion regime. For a diffusive dispersive approximation of the equations of one-dimensional elasticity (or p-system), we study convergence of traveling waves to shocks. The problem is recast as a Hamiltonian system with small friction, and an analysis of the length of oscillations yields convergence in the moderate dispersion regime \eps, δ→ 0 with δ= o(\eps), under hypotheses that the limiting shock is admissible according to the Liu E-condition and is not a contact discontinuity at either end state. A similar convergence result is proved for traveling waves of the quantum hydrodynamic system with artificial viscosity as well as for a viscous Peregrine-Boussinesq system where traveling waves model undular bores, in all cases in the moderate dispersion regime.