Compressible Boundary Layer Equations

• Compressible boundary layer equations are asymptotically reduced models that describe thin near-boundary regions with strong normal gradients and anisotropic diffusion.
• They encompass a range of formulations including Prandtl-type, viscous–thermal, and magneto-hydrodynamic extensions to capture complex compressible flow behavior.
• Analytical and numerical methods, such as matched asymptotics and Gevrey regularity, are employed to address challenges like tangential derivative loss and nonlocal coupling.

Compressible boundary layer equations are reduced equations for thin near-boundary or thin-shear-layer regions of compressible flow in which strong normal gradients coexist with comparatively slow tangential or streamwise variation. In the literature represented here, they appear as Prandtl-type systems for the zero-viscosity limit of compressible Navier–Stokes, as coupled viscous–thermal layer equations for non-isentropic flow, as magnetic and magneto-micropolar boundary layer systems, as thermal-layer equations when heat conduction remains but viscosity is negligible, and as boundary-region or wall-model reductions in porous, free-shear, and turbulent settings. This suggests that the topic is most accurately viewed as a family of asymptotic and reduced models rather than a single universal PDE (Wang et al., 2023, Wang et al., 24 Jul 2025).

1. Model class and defining reductions

A common structural feature is the thin-layer approximation: pressure is often constant in the wall-normal direction, viscous and thermal diffusion are retained primarily in the normal direction, and the outer flow supplies matching data at infinity. A representative compressible heat-conducting laminar boundary-layer system is

$$\frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial x} u + \rho \frac{\partial u}{\partial x} + \frac{\partial \rho}{\partial y} v + \rho \frac{\partial v}{\partial y} = 0,$$

$\frac{\partial p}{\partial y} =0,$

$$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$

$$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$

which was studied through a self-similar reduction in two dimensions (Barna et al., 2021).

A different canonical form, arising in the non-isentropic viscous–thermal setting, is

$$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$

with

$$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$

Here the velocity field is explicitly not divergence-free, since its divergence is determined by thermal diffusion and viscous dissipation (Wang et al., 24 Jul 2025).

Taken together, these formulations suggest several recurring defining reductions: elimination or simplification of normal pressure variation, anisotropic diffusion, ideal-gas or related closure, and nonlocal recovery of the normal velocity from continuity or a divergence constraint.

Variant	Representative structure	Reference
Heat-conducting laminar layer	$\partial_y p=0$, $p=R\rho T$, $y$-diffusion in momentum and temperature	(Barna et al., 2021)
Compressible Prandtl-type layer	outer Euler plus inner $u^p,v^p$ on $\frac{\partial p}{\partial y} =0,$0	(Wang et al., 2023)
Non-isentropic viscous–thermal layer	coupled $\frac{\partial p}{\partial y} =0,$1 with $\frac{\partial p}{\partial y} =0,$2	(Wang et al., 24 Jul 2025)
Compressible MHD layer	$\frac{\partial p}{\partial y} =0,$3, induction and magnetic divergence constraints	(Yongting et al., 2018)
Porous compressible layer	porosity-weighted continuity, Brinkman diffusion, Darcy–Forchheimer drag	(Fossà et al., 14 Aug 2025)

2. Inviscid limits, Prandtl scaling, and matched asymptotics

One central branch of the subject derives compressible boundary layer equations from vanishing dissipation limits. In the two-dimensional isentropic compressible Navier–Stokes system in the half-space with no-slip boundary condition, the asymptotic expansion is

$\frac{\partial p}{\partial y} =0,$4

where the outer flow solves compressible Euler and the inner correction solves a compressible Prandtl-type system (Wang et al., 2023).

The leading boundary-layer equations in that setting are

$\frac{\partial p}{\partial y} =0,$5

with $\frac{\partial p}{\partial y} =0,$6. The analytic theorem proves that, locally in time,

$\frac{\partial p}{\partial y} =0,$7

in $\frac{\partial p}{\partial y} =0,$8 (Wang et al., 2023).

A complementary one-dimensional tradition studies vanishing shear viscosity together with boundary-layer thickness and boundary-layer solution directly at the level of the full compressible MHD equations. For the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data, the vanishing shear viscosity limit is justified, convergence rates are obtained, boundary-layer thickness and boundary-layer solution are discussed, and the proofs make full use of the effective viscous flux, the material derivatives, and the structure of the one-dimensional equations (Ye et al., 2015).

These results indicate two rigorous asymptotic patterns within the subject: matched Euler–Prandtl expansions in multidimensional half-space problems, and direct vanishing-dissipation boundary-layer analysis in lower-dimensional compressible MHD settings.

3. Thermal layers and non-isentropic coupling

The non-isentropic case introduces a thermal layer that is not merely auxiliary to the viscous layer. In the two-dimensional compressible boundary layer system quoted above, the effective diffusion coefficient is $\frac{\partial p}{\partial y} =0,$9, and the temperature equation contains the viscous heating term $$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$0. The same system shows that the normal velocity is determined nonlocally through

$$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$1

using $$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$2. This is the source of the derivative loss emphasized in the analytic and Gevrey well-posedness theories (Wang et al., 24 Jul 2025).

A distinct thermal-layer model arises when heat conduction remains relevant but viscosity is absent or asymptotically negligible. In that case the principal system is

$$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$3

with $$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$4 and $$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$5 as $$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$6. The temperature equation is degenerate parabolic because the diffusion coefficient is proportional to $$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$7, and the paper gives a semi-explicit solution formula, proves convergence to the inviscid Prandtl equations when the initial temperature goes to a constant, and analyzes time-asymptotic stability of the linearized system around a shear flow (Liu et al., 2016).

Self-similar reductions provide another perspective on thermal coupling. In the two-dimensional compressible heat-conducting boundary layer studied with the Sedov-type ansatz

$$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$8

consistency forces

$$\rho \frac{\partial u}{\partial t} + \rho \left( u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} \right) = \mu \frac{\partial^2 u}{\partial y^2} - \frac{\partial p}{\partial x},$$9

and the system reduces to coupled ODEs for $$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$0, including a nonlinear third-order density equation. The paper states that, unlike the incompressible case, the compressible heated boundary layer does not admit explicit analytic similarity solutions and must be treated numerically (Barna et al., 2021).

A frequent misconception is that compressible boundary-layer theory is simply incompressible Prandtl theory with variable density. The cited thermal-layer models show that this is too narrow: temperature can determine parabolicity, divergence, and even the effective dimensional reduction.

4. Magnetic, magneto-micropolar, and steady subsonic extensions

Compressible MHD boundary layer equations add magnetic induction, total-pressure balance, and a second divergence constraint. In the two-dimensional non-isentropic compressible MHD boundary layer derived from simultaneous vanishing viscosity, heat conductivity, and magnetic diffusivity, the leading system includes

$$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$1

together with mass, tangential momentum, temperature, and induction equations. The normal constancy of total pressure yields

$$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$2

so density is eliminated algebraically through the ideal-gas law and magnetic field (Yongting et al., 2018).

The key structural assumption is nondegeneracy of the tangential magnetic field. Because

$$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$3

the transformation

$$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$4

is valid when $$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$5. This removes the derivative loss associated with recovering $$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$6 and $$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$7 from divergence constraints. Under this condition, local-in-time well-posedness is obtained in weighted Sobolev spaces for the compressible non-isentropic MHD boundary layer (Yongting et al., 2018).

The same magnetic stabilization mechanism also supports long-time theory. For two-dimensional compressible MHD boundary layer equations, when the initial data are a small perturbation of a steady solution with size $$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$8 and the far-field state is also a small perturbation around such a steady solution in Sobolev space, the lifespan of solutions is greater than $$c_p \rho \frac{\partial T}{\partial t} + c_p \rho \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = k \frac{\partial^2 T}{\partial y^2}, \qquad p = R \rho T,$$9 (Li et al., 2022). In a related 2025 result, local-in-time well-posedness is proved for the two-dimensional compressible magneto-micropolar boundary layer system in Sobolev spaces, using a nonlinear coordinate transformation and assuming that the initial tangential magnetic field is non-degenerate; the underlying boundary conditions are non-slip on velocity, Dirichlet on micro-rotational velocity, and perfectly conducting on magnetic field (Qin et al., 16 Apr 2025).

Steady compressible boundary layers admit a different reduction. For the two-dimensional steady isentropic compressible Navier–Stokes equations in the half-plane, the boundary layer is treated as a shear profile

$$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$0

and structural stability is established uniformly in the entire subsonic regime $$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$1. The paper states that, according to Prandtl’s theory, the pressure does not exhibit a leading order boundary layer, and consequently the density has no boundary layers at this order; the key reduced scalar object in the analysis is a compressible Orr–Sommerfeld-type equation involving

$$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$2

whose positivity encodes subsonicity (Li et al., 27 Jan 2025).

These magnetic and steady subsonic results suggest that compressible boundary-layer theory is strongly shaped by additional structure: a nondegenerate tangential magnetic field can replace velocity monotonicity, while steady isentropic subsonic problems may be governed more naturally by shear-profile stability and compressible Orr–Sommerfeld reductions than by a standalone Prandtl PDE.

5. Geometry, porosity, and free-shear generalizations

Compressible boundary layer equations are not confined to impermeable flat walls. Over an isotropic porous substrate, asymptotic reduction of the volume-averaged compressible Navier–Stokes equations yields

$$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$3

$$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$4

$$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$5

with $$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$6. Relative to the classical impermeable-wall case, continuity is porosity-weighted, viscous diffusion takes Brinkman form, Darcy and Forchheimer drag appear in the momentum equation, and thermal conduction is modified by surface porosity $$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$7 (Fossà et al., 14 Aug 2025).

For a special streamwise-growing permeability $$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$8, the porous problem admits a self-similar reduction using the Dorodnitsyn–Howarth transformation. The reduced ODE system is

$$\left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u=(\theta+\theta^E)\partial_y^2 u,\ &\partial_t \theta+u\partial_x \theta+v\partial_y \theta=(\theta+\theta^E)\partial_y^2 \theta+(\theta+\theta^E)(\partial_y u)^2,\ &\partial_x u+\partial_y v=\partial_y^2\theta+(\partial_y u)^2, \end{aligned} \right.$$9

$$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$0

and reduces above the interface to the classical compressible Blasius equations (Fossà et al., 14 Aug 2025).

A different generalization appears in curved free shear layers. There the compressible Navier–Stokes equations are reduced, in a high-Reynolds-number asymptotic framework, to nonlinear compressible boundary region equations (NCBRE) with slow streamwise evolution. The reduced system includes

$$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$1

$$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$2

$$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$3

so the curvature/Görtler term $$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$4 explicitly represents centrifugal effects. The resulting system is parabolic in the streamwise direction and is solved by streamwise marching (Es-SAhli et al., 2024).

Compressible boundary-region equations also arise in the asymptotic suction boundary layer. In the subsonic perfect-gas ASBL at large Reynolds number, the base compressible suction layer can be solved exactly after a Dorodnitsyn–Howarth transformation, and the subsequent boundary-region equations show that free-stream coherent structures generate both velocity and thermal streaks. The paper reports that increasing the free-stream Mach number enhances the thermal streaks, whereas varying the Prandtl number changes the location of the maximum amplitude of the thermal streak relative to the velocity streak (Johnstone et al., 2021).

6. Well-posedness frameworks and derivative loss

A central analytical obstacle is tangential derivative loss. In the non-isentropic compressible boundary layer system, the transport terms $$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$5, $$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$6, $$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$7, and $$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$8 are coupled with the nonlocal recovery of $$u|_{y=0}=v|_{y=0}=\partial_y\theta|_{y=0}=0,\qquad \lim_{y\to\infty}u=\lim_{y\to\infty}\theta=0.$$9, and direct Sobolev estimates are not available without additional structure. One route is analyticity in the tangential variable. For analytic initial data, local existence and uniqueness are proved in spaces that are analytic in the tangential variable and Sobolev in the normal variable, using Littlewood–Paley theory, weighted analytic norms, and a time-dependent analytic radius; the paper explicitly states that the system is a nonlinear coupled system of degenerate parabolic equations and an elliptic equation (Wang et al., 24 Jul 2025).

A second route is Gevrey regularity. In the Gevrey-2 theory for the same compressible, non-isentropic boundary layer system, the main novelty is the use of new auxiliary functions and a cancellation mechanism adapted to the strong interaction between viscous and thermal layers. The good unknowns are

$\partial_y p=0$0

and the paradifferential operator is

$\partial_y p=0$1

The resulting triangular structure

$\partial_y p=0$2

supports local existence and uniqueness in Gevrey-2 in the tangential variable and Sobolev regularity in the normal variable (Wang et al., 17 Apr 2026).

Analyticity also enters inviscid-limit justification. For the zero-viscosity limit of compressible Navier–Stokes in the half-space with no-slip boundary condition, analyticity is used to recover one lost derivative created by the boundary-layer terms, and the local-in-time Prandtl expansion is justified in analytic norms (Wang et al., 2023). In contrast, MHD systems can recover Sobolev well-posedness through magnetic nondegeneracy and coordinate transformation rather than analyticity alone (Yongting et al., 2018).

This suggests that there is no single preferred regularity class for compressible boundary layer equations. Instead, the choice of analytic, Gevrey, weighted Sobolev, or transformed Sobolev framework depends on which structural mechanism is available to offset derivative loss.

7. Turbulent inner-layer reductions and compressible wall laws

In turbulent compressible boundary layers, the phrase “boundary layer equations” often denotes reduced inner-layer models rather than full laminar Prandtl-type PDEs. A representative near-wall model starts from the standard one-dimensional equilibrium wall-model reduction

$\partial_y p=0$3

$\partial_y p=0$4

with the thin-boundary-layer assumption

$\partial_y p=0$5

The newer model replaces the direct compressible ODE treatment by the inverse of the Griffin–Fu–Moin velocity transformation plus an algebraic temperature–velocity relation, predicts mean temperature and velocity profiles across the inner layer, wall shear stress, and wall heat flux, and solves one ODE instead of two (Griffin et al., 2023).

The velocity transformation itself has been reformulated using a generalized total stress. In that framework, the generalized total stress is

$\partial_y p=0$6

with

$\partial_y p=0$7

and the transformed mean shear is

$\partial_y p=0$8

The paper argues that density and viscosity fluctuation effects are dynamically significant and that the generalized total stress and its viscous/turbulent partition are approximately Mach-invariant in the inner layer (Lee et al., 2021).

These turbulent reductions do not solve the full compressible boundary-layer PDE across the entire layer. They instead model the inner layer under equilibrium or near-equilibrium assumptions. Even so, they belong to the same research landscape because they preserve the core compressible boundary-layer themes: variable density and viscosity, wall-normal balance, heat transfer, and asymptotic reduction of thin-shear-layer dynamics into lower-dimensional equations.