Abstract
For the isentropic compressible fluids in one-space dimension, we prove that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy. Moreover, we obtain the same result for the nonisentropic compressible gas flow, that is, for the fluid dynamics of the Navier-Stokes equations coupled with a transport equation of entropy. These results generalize those in Guo and Jiang's work (2006) where the one-dimensional compressible fluids with constant viscosity are considered.
Highlights
Self-similar solutions have attracted much attention in mathematical physics because understanding them is fundamental and important for investigating the well-posedness, regularity, and asymptotic behavior of differential equations in physics
For the isentropic compressible fluids in one-space dimension, we prove that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy
Since the pioneering work of Leray [1], self-similar solutions of the Navier-Stokes equations for incompressible fluids have been widely studied in different settings (e.g., [2, page 207]; [3, page 120]; [4,5,6,7,8,9,10]; [11, Chapter 23]; [12,13,14,15,16,17,18,19,20])
Summary
Self-similar solutions have attracted much attention in mathematical physics because understanding them is fundamental and important for investigating the well-posedness, regularity, and asymptotic behavior of differential equations in physics. Guo and Jiang [21] considered (1) with constant viscosity, μ(ρ) ≡ μ > 0, and linear densitydependent pressure, P(ρ) = aρ, where a > 0 is a constant, and proved that there exist neither forward nor backward selfsimilar solutions with finite total energy Their investigation generalized the results for 3D incompressible fluids in Necas et al.’s work [6] to the 1D compressible case with P(ρ) = ργ, where γ = 1. The following global-energy estimate on (−∞, ∞) × [0, T] holds uniformly with respect to all strong solutions; that is, for every T > 0, there exists a positive constant C(T) depending only on T, ρ0(x), and u0(x) such that sup [0,T].
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