How does the low Mach number limit behave with no-slip boundary and large temperature variations?

The limit keeps a divergence constraint, not incompressibility

The paper studies the full compressible Navier-Stokes equations in a smooth bounded domain in three dimensions, with density, velocity, internal energy, temperature and pressure as unknowns, and with an ideal polytropic gas equation of state. The Mach number ε is the ratio of a characteristic flow velocity to a characteristic sound speed, and the goal is to describe what happens as ε tends to zero.

A formal asymptotic expansion suggests that solutions converge to a limit (r, ω, π) satisfying a low Mach number system. Its third equation is γCv div ω = div(κ∇(1/r)), where γ is the ratio of specific heats and κ is the thermal conductivity. When heat conduction is absent, or when the limiting temperature is spatially constant, the right-hand side vanishes and the constraint reduces to div ω = 0, the familiar incompressible limit. With heat conduction and large temperature variations, the constraint stays nontrivial: the paper's convergence theorem yields div ω = (κ/2)Δθ together with (1+θ)^{-1}(θ_t + ω·∇θ) = (κ/2)Δθ. So the low Mach number approximation filters acoustic compressibility while retaining temperature and conduction effects, which is exactly why the general model is harder than the incompressible limit.

Large temperature variations destroy the skew-symmetric cancellation

For isentropic flows the pressure depends only on density, and changes of variables such as ρ^ε = 1 + ερ^ε_1 or ρ^ε = e^{ερ^ε_1} produce a skew-symmetric leading-order operator of the form ε^{-1}(div, ∇). That structure makes the singular terms cancel in the L² energy identity and yields ε-uniform estimates.

When κ = 0 the same skew-symmetric form ε^{-1}(div, ∇) is available, and the singular terms still cancel. But for κ > 0 with large temperature variations, the penalization operator is no longer skew-symmetric in the original variables, because the singular pressure term couples nonlinearly with both density and temperature. The paper introduces the change of variables p^ε = 1 + εq^ε and Θ^ε = 1 + θ^ε, and then the modified velocity v^ε := u^ε − (κ/2)∇θ^ε, which restores skew symmetry for the singular acoustic part. This transformation does not fully decouple v from temperature: the equation for v still contains the third-order term −β∇Δθ^ε, and under the no-slip condition the tangential trace of v equals −(κ/2)∇_τθ^ε, so it depends on the tangential temperature gradient.

No-slip walls remove the vorticity boundary relation

In a bounded domain, acoustic waves interact with the boundary, so Fourier-based methods used in whole-space or periodic settings are not directly available, and normal differentiation produces boundary terms that basic energy estimates do not control. High-order normal derivatives therefore cannot be estimated by direct differentiation.

The paper is explicit that the no-slip condition cannot provide the vorticity boundary relation used in slip-boundary settings. In the vorticity-slip and Navier-slip cases, high-order estimates are closed through a div-curl decomposition of v, and the estimate for the curl part relies crucially on the boundary condition satisfied by curl v. Under no-slip, that mechanism is unavailable, so the high-order normal estimates from those settings cannot be transferred directly. The paper states that the low Mach number problem for the full Navier-Stokes system with no-slip boundary conditions and large temperature variations had remained open.

A divergence-free Stokes lifting closes the estimate

The central technical device is a divergence-free Stokes lifting of the boundary trace of v_t, the time derivative of the modified velocity. Taking v_t as a test function in the differentiated momentum equation produces a boundary term involving (n·∇v_t)·v_t that cannot be controlled. To remove it, the authors solve a Stokes problem with boundary data g = −(κ/2)∇_τθ_t and ℓ_0 = 0, obtaining a lifting ℓ whose existence, uniqueness and estimates follow from the cited Stokes theory.

They then use Z := v_t − ℓ as a new test function. By construction Z vanishes on the boundary and div Z = div v_t in the domain. Because ℓ is divergence-free with zero normal trace, the correction does not alter the cancellation of the singular pressure terms. Combining the resulting energy estimate with the bounds for ℓ gives the desired ε-uniform estimate for v_t. The paper also notes that constructing special test functions this way is applicable to other problems, citing Stokes-based and Bogovskii-based constructions elsewhere.

Two skew-symmetric structures split the estimate into two blocks

The proof combines two skew-symmetric structures. One couples temperature with weighted unknowns and yields uniform estimates for the time derivative of temperature; the other couples pressure with the modified velocity and controls the time derivative of velocity and higher-order spatial derivatives. Concretely, the system satisfied by (q, v, θ) is used to estimate the high-order norms of ∇²div v and ∇²Δθ together with the low-order norms of v_t in L^∞_t(L²) and L²_t(H¹). A second system is used to estimate θ_t in L^∞_t(H¹) and L²_t(H²).

The authors introduce the exponential transformation p = e^{εσ}, Θ = e^{ϑ}, equivalent to the linear transformation through εq = e^{εσ} − 1 and θ = e^{ϑ} − 1, and set η := (1/2)(εσ − ϑ) and u^ε := εu. In the variables (η, u^ε, ϑ) the principal singular part becomes skew-symmetric and has the same form as in the small-temperature-variation case. The reformulated system also has the advantage that the coefficient of ϑ_t is constant. The estimates are organized through the quantities M₁(t) and M₂(t), which control (η_t, u^ε_t, ϑ_t) and the second-order time derivatives weighted by powers of ε.

The uniform bound holds on a Mach-independent time interval

The uniform estimate theorem assumes initial data bounded in H² for (ρ^ε_0, u^ε_0, q^ε_0), in H³ for θ^ε_0, and in L² or H¹ for the time derivatives up to second order, with additional ε-weighted conditions on (div u^ε_t, ∇q^ε_t)(0) and on (ε²u^ε_tt, ε²q^ε_tt, εθ^ε_tt)(0). It also assumes ρ^ε_0 ≥ M_0^{-1} > 0 and Θ^ε_0 ≥ A > 0 for positive constants M_0 and A independent of ε, and requires β > 0.

Under these hypotheses there exist positive constants ε_0, T and C, all independent of ε in the stated sense, such that the unique strong solution satisfies M^ε(t) ≤ C for every ε in (0, ε_0] and t in (0, T], and Θ^ε(x,t) ≥ C^{-1} on the space-time cylinder. The energy functional M^ε(t) collects L^∞_t(H²) norms of (ρ^ε, q^ε), the L^∞_t(H¹) norm of u^ε, L^∞_t(H²) of θ^ε, the L^∞ bound on (ρ^ε)^{-1}, L^∞_t(L²) norms of (ρ^ε_t, u^ε_t, q^ε_t), L^∞_t(H¹) of θ^ε_t, and L²_t(H³) norms of (u^ε, ∇θ^ε) together with L²_t(H¹) of (u^ε_t, ∇θ^ε_t). The key point is that T does not shrink with ε, which is what makes the subsequent compactness argument possible.

Compactness passes the limit to the low Mach system

The convergence theorem takes a solution satisfying the uniform boundedness and lets ε tend to zero. The result is (ρ^ε, u^ε, θ^ε) ⇀ (r, ω, θ) weakly-* in L^∞(0,T; H²(Ω)), with r(1+θ) = 1. The proof records weak-* limits for ρ^ε, u^ε, θ^ε and their time derivatives in the appropriate spaces, and uses the Aubin-Lions-Simon lemma to upgrade (ρ^ε, θ^ε) to strong convergence in C([0,T], H^{2−δ}) and u^ε to C([0,T], H^{1−δ}) for any δ in (0,1).

Passing to the limit in the reformulated system gives the low Mach number Navier-Stokes equations: (1+θ)^{-1}(ω_t + ω·∇ω) + ∇π = μΔω + (μ+λ)∇div ω, together with (1+θ)^{-1}(θ_t + ω·∇θ) = (κ/2)Δθ and div ω = (κ/2)Δθ, with boundary conditions ω = 0 and ∂θ/∂n = 0. For the momentum equation, applying curl removes the singular term and yields curl[r(ω_t + ω·∇ω) − μΔω − (μ+λ)∇div ω] = 0, from which a pressure π with ∇π in L²(0,T; L²(Ω)) is recovered. The limit system is stated for well-prepared initial data, and the local existence theorem requires compatibility conditions u^ε_0 = u^ε_t(0) = u^ε_tt(0) = 0 and vanishing normal derivatives of Θ^ε_0, Θ^ε_t(0) and Θ^ε_tt(0) on the boundary.

The decisive difference is the boundary condition, not the Mach number

The paper's contribution is best read as a boundary-condition result rather than a small-Mach-number result. The Mach number is driven to zero in all these problems; what changes is whether the domain has a boundary and what the velocity does there. In whole-space or periodic settings, frequency localization handles the large-temperature-variation case. In bounded domains with slip-type conditions, the vorticity boundary relation closes the high-order estimates. Under no-slip, neither route is available, and the divergence-free Stokes lifting of the boundary trace of v_t is what replaces them.

The practical reading is therefore conditional. The uniform estimate and the convergence to the low Mach system are established for strong solutions in a three-dimensional smooth bounded domain, with well-prepared initial data, with β > 0, and on a time interval whose length is independent of ε but not asserted to be arbitrary. The limit system itself retains a nontrivial divergence constraint, so anyone comparing this result with the classical incompressible limit should check whether heat conduction is present and whether the limiting temperature is spatially constant; if either condition removes the right-hand side, the constraint collapses to div ω = 0 and the problem is a different one. The result also does not by itself cover ill-prepared initial data in this no-slip, large-temperature-variation setting, which the paper leaves as a distinct case handled elsewhere by different methods.

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