Low Mach Number Limit of Steady Euler Flows in Multi-Dimensional Nozzles

Abstract

In this paper, we consider the steady irrotational Euler flows in multidimensional nozzles. The first rigorous proof on the existence and uniqueness of the incompressible flow is provided. Then, we justify the corresponding low Mach number limit, which is the first result of the low Mach number limit on the steady Euler flows. We establish several uniform estimates, which does not depend on the Mach number, to validate the convergence of the compressible flow with extra force to the corresponding incompressible flow, which is free from the extra force effect, as the Mach number goes to zero. The limit is on the Hölder space and is unique. Moreover, the convergence rate is of order ε 2 , which is higher than the ones in the previous results on the low Mach number limit for the unsteady flow.

Low Mach Number Limit and Far Field Convergence Rates of Potential Flows in Multi-Dimensional Nozzles With an Obstacle Inside

“…The low Mach number limit for one dimensional problem was investigated in the BV space [7]. The first rigorous analysis of subsonic flows for the steady Euler equations past a body, in infinitely long nozzles, and largely open nozzle were obtained in [26], [27], and [40], respectively.…”

“…Now, one may introduce the critical speed for the flows. For each fixed 0 < θ ≤ 1, one can follow [27] to define q θ such that u < q θ if and only if M < θ. Specially, for the polytrotic case, define (41) µ 2 = (γ − 1)θ 2 2 + (γ − 1)θ 2 and q θ = µ 2 φ + h(1) , then the Bernoulli's function (40) can be written in the following form…”

“…Here we give the main ideas for the proof of the main results. Inspired by [27], the existence of incompressible and compressible subsonic flows in the domain Ω is first established via the variational method. Then some uniform estimates are obtained which also implies that the order of low Mach number limit is 2 .…”

“…Since the domain is unbounded, the truncation of the domain is used to to study problems (27) and (45).…”

“…The proof of Lemma 2 is the same to [27], so we omit the details here. Since φL and ϕ L are weak solutions of quasilinear elliptic equations of divergence form, similar to [15, Lemmas 6 and 7 ] and [27, Lemma 4.5], using the Nash-Moser iteration yields that there exists a positive constant K < L 4 such that (68)…”