Global Solutions to Isothermal System with Source.

Abstract

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 +\dfrac{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. Letters, 95(2019), 35-40) and “Existence of Global Solutions for Isentropic Gas Flow with Friction” (Nonlinearity, 33(2020), 3940-3969), where the global solution was obtained for the source f(x, t, ρ, u) = \dfrac{a'(x)}{a(x)}+ρu2, f(x, t, ρ, u) = b(x, t)ρ, f(x, t, ρ, u) = α(x, t)ρu|u| respectively.

Keywords

• Global L∞ solution
• isothermal system
• source terms
• compensated compactness