Diffusive wave in the low Mach limit for compressible Navier–Stokes equations

Abstract

The low Mach limit for 1D non-isentropic compressible Navier-Stokes flow, whose density and temperature have different asymptotic states at infinity, is rigorously justified. The problems are considered on both well-prepared and ill-prepared data. For the well-prepared data, the solutions of compressible Navier-Stokes equations are shown to converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero when the difference between the states at ±∞ is suitably small. In particular, the velocity of diffusion wave is only driven by the variation of temperature. It is further shown that the solution of compressible Navier-Stokes system also has the same property when Mach number is small, which has never been observed before. The convergence rates on both Mach number and time are also obtained for the well-prepared data. For the ill-prepared data, the limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma. And the difference between the states at ±∞ can be arbitrary large in this case.(1.5) which satisfies the following equation ∂ t (P ε ) + div(u ε P ε ) + P ε divu ε = div(κ∇T ε ) + ε 2 [2µ|D(u ε )| 2 + λ|divu ε | 2 ].(1.6)

Diffusive wave in the low Mach limit for non-viscous and heat-conductive gas

“…This phenomenon is quite different from the constant case. Since the diffusion wave is independent of the viscosity µ = 0, we conjecture that the result of [13] is still valid without viscosity, that is µ = 0 in the system (1.1). Precisely speaking, we consider the nonviscous and heat-conductive gas in the following system v t − u x = 0, u t + P x = 0, (1.2)…”

“…Since the order of the termṽΨQ 1 with respect to (1 + t) is not enough if we estimate the righthand side of (2.13) directly, we need to use a weighted energy method. Similar to [13], define…”

“…As ε → 0, the limit of solutions of (1.3) is called the low Mach limit [16,17]. Similar to [1,13,26], we assume that the pressure is a small perturbation of a given constant stateP > 0, i.e. P =P + O(ε), (1.4) and without loss of generality, we further assumeP to be 1.…”

“…[19]), we control simultaneously the estimates for (φ y , ψ y ) H 1 by ζ y H 1 . Motivated by [13], a new type differential inequality is essentially used to obtain the desired a priori estimates.…”

“…Recently, Huang et.al. [13] began to study the case that the solutions have different end states and found that the solutions of compressible Navier-Stokes equations converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero, which is related to the thermal creep flow. That is, the flow in diffusion wave is only driven by the variation of temperature.…”