2026/07/28 by Ralph Saxton, Katarzyna Saxton
Mathematics · Physics and Astronomy · #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Nonlinear Waves and Solitons #math-ph #math.AP #math.MP #msc:35L65 #msc:35L67 #msc:76E19 #msc:76L05
paper · pdf · doi:10.1007/s10659-026-10220-5
published as Journal of Elasticity (2026) · 36 pages
arxiv created 2026/07/30 · arxiv updated 2026/08/03
Several applications for systems of conservation laws of the form Ut + (Φ(U) U)x =0, U: Rt× Rx→ Rn , n≥ 2, with Φ(U) = ϕ(r, Θ): Rn→ R, r = |U|, and Θ= U/|U|∈ Sn-1, are obtained by imposing structural conditions to provide a classification framework for solutions dependent on the form of ϕ(U). By prescribing the evolution of a particular eigenvalue, we can categorize classes of such functions, ϕ, for which this evolution is met, into depending on either a scalar field z=rK(Θ), where K: Sn-1→ R, or on the vector field Θ, and find for which the amplitude of solutions may blow up in finite time. As a consequence, solutions to the corresponding Riemann problems can be divided into those with are classical and those which involve delta-shocks and/or vacuum states.