Soluciones globales para sistema isotérmico con fuente

In this short note, we are concerned with the global existence of solutions to the isothermal system with source, where the inhomogeneous terms f(x,t,ρ,u)=b(x,t)ρ+(a′(x)/a(x))*ρu^2+α(x,t)ρu|u| are appeared in the momentum equation. Our work extended the results in the previous papers “Resonance for the Isothermal System of Isentropic Gas Dynamics” (Proc. A.M.S.139(2011),2821-2826), “Global Existence and Stability to the Polytropic Gas Dynamics with an Outer Force” (Appl. Math. Let-ters, 95(2019), 35-40) and “Existence of Global Solutions for Isentropic GasFlow with Friction” (Nonlinearity, 33(2020), 3940-3969), where the global solution was obtained for the source f(x,t,ρ,u)=(a′(x)/a(x))*ρu^2, f(x,t,ρ,u)=b(x,t)ρ, f(x,t,ρ,u)=α(x,t)ρu|u| respectively.

An extension of Bakhvalov’s theorem for systems of conservation laws with damping

For [Formula: see text] systems of conservation laws satisfying Bakhvalov conditions, we present a class of damping terms that still yield the existence of global solutions with periodic initial data of possibly large bounded total variation per period. We also address the question of the decay of the periodic solution. As applications, we consider the systems of isentropic gas dynamics, with pressure obeying a [Formula: see text]-law, for the physical range [Formula: see text], and also for the “non-physical” range [Formula: see text], both in the classical Lagrangian and Eulerian formulation, and in the relativistic setting. We give complete details for the case [Formula: see text], and also analyze the general case when [Formula: see text] is small. Further, our main result also establishes the decay of the periodic solution.