Spectral stability of noncharacteristic isentropic Navier–Stokes boundary layers

The small-amplitude results obtained in [19, 16, 23, 10] are of “general type”, making little use of the specific structure of the equations. Essentially, they all require that the difference between the boundary layer solution and its constant limit at $|x|=\infty$ be small in $L^{1}$.²²2Alternatively, as in [19, 23], the essentially equivalent condition that $x\hat{v}^{\prime}(x)$ be small in $L^{1}$. (For monotone profiles, $\int_{0}^{+\infty}|\hat{v}-v_{+}|dx=\pm\int_{0}^{+\infty}(\hat{v}-v_{+})dx=\mp\int_{0}^{+\infty}x\hat{v}^{\prime}dx$.) As pointed out in [10], this is the “gap lemma” regime in which standard asymptotic ODE estimates show that behavior is essentially governed by the limiting constant-coefficient equations at infinity, and thus stability may be concluded immediately from stability (computable by exact solution) of the constant layer identically equal to the limiting state. These methods do not suffice to treat either the (small-amplitude) characteristic limit or the large-amplitude case, which require more refined analyses. In particular, up to now, there was no analysis considering boundary layers approaching a full viscous shock profile, not even a profile of vanishingly small amplitude. Our analysis of this limit indicates why: the appearance of a small eigenvalue near zero prevents uniform estimates such as would be obtained by usual types of energy estimates.

By contrast, the large-amplitude results obtained here and (for expansive layers) in [19] make use of the specific form of the equations. In particular, both analyses make use of the advantageous structure in Lagrangian coordinates. The possibility to work in Lagrangian coordinates was first pointed out by Matsumura–Nishihara [19] in the inflow case, for which the stationary boundary transforms to a moving boundary with constant speed. Here we show how to convert the outflow problem also to Lagrangian coordinates, by converting the resulting variable-speed boundary problem to a constant-speed one with modified boundary condition. This trick seems of general use. In particular, it might be possible that the energy methods of [19] applied in this framework would yield unconditional stability of expansive boundary-layers, completing the analysis of the outflow case. Alternatively, this case could be attacked by the methods of the present paper. These are two further interesting direction for future investigation.

In the outflow case, a further transformation to the “balanced flux form” introduced in [22], in which the equations take the form of the integrated shock equations, allows us to establish stability in the characteristic limit by energy estimates like those of [18] in the shock case. The treatment of the characteristic inflow limit by the methods of [22, 7] seems to be another extremely interesting direction for future study.

Finally, we point to the extension of the present methods to full (nonisentropic) gas dynamics and multidimensions as the two outstanding open problems in this area.

New features of the present analysis as compared to the shock case considered in [3, 12] are the presence of two parameters, strength and displacement, indexing possible boundary layers, vs. the single parameter of strength in the shock case, and the fact that the limiting equations in several asymptotic regimes possess zero eigenvalues, making the limiting stability analysis much more delicate than in the shock case. The latter is seen, for example, in the limit as a compressive boundary layer approaches a full stationary shock solution, which we show to be spectrally equivalent to the situation of unintegrated shock equations on the whole line. As the equations on the line possess always a translational eigenvalue at $\lambda=0$, we may conclude existence of a zero at $\lambda=0$ for the limiting equations and thus a zero near $\lambda=0$ as we approach this limit, which could be stable or unstable. Similarly, the Evans function in the inflow case is shown to converge in the large-strength limit to a function with a zero at $\lambda=0$, with the same conclusions; see Section 3 for further details.

To deal with this latter circumstance, we find it necessary to make use also of topological information provided by the stability index of [21, 8, 25], a mod-two index counting the parity of the number of unstable eigenvalues. Together with the information that there is at most one unstable zero, the parity information provided by the stability index is sufficient to determine whether an unstable zero does or does not occur. Remarkably, in the isentropic case we are able to compute explicitly the stability index for all parameter values, recovering results obtained by indirect argument in [25], and thereby completing the stability analysis in the presence of a single possibly unstable zero.

The analyses of [12, 3] in the shock wave case were carried out in Lagrangian coordinates, which proved to be particularly convenient. Our first step, therefore, is to convert the Eulerian formulation (1) into Lagrangian coordinates similar to those of the shock case. However, standard Lagrangian coordinates in which the spatial variable $\tilde{x}$ is constant on particle paths are not appropriate for the boundary-value problem with inflow/outflow. We therefore introduce instead “psuedo-Lagrangian” coordinates

in which the physical boundary $x=0$ remains fixed at $\tilde{x}=0$.

Straightforward calculation reveals that in these coordinates (1) becomes

on $x>0$, where

so that $m(t)$ is the negative of the momentum at the boundary $x=\tilde{x}=0$. From now on, we drop the tilde, denoting $\tilde{x}$ simply as $x$.

Our next step is to rescale the equations in such a way that coefficients remain bounded in the strong boundary-layer limit. Consider the change of variables

where $\varepsilon$ is chosen so that

where $v_{+}$ is the limit as $x\to+\infty$ of the boundary layer (stationary solution) $(\hat{v},\hat{u})$ under consideration and $v_{-}$ is the limit as $x\to-\infty$ of its continuation into $x<0$ as a solution of the standing-wave ODE (discussed in more detail just below). Under the rescaling (12), (8) becomes

where $a=a_{0}\varepsilon^{-\gamma-1}s^{-2}$, $\sigma=-u(0,t)/v(0,t)+1$, on respective domains

Stationary boundary layers

of (14) satisfy

where (d) is imposed at $+\infty$ in the inflow case, $-\infty$ in the outflow case and (imposing $\sigma=0$) $u_{0}=v_{0}$. Using (15)(a) we can reduce this to the study of the scalar ODE,

with the same boundary conditions at $x=0$ and $x=\pm\infty$ as above. Taking the antiderivative of this equation yields

where $C$ is a constant of integration.

Noting that $\mathcal{H}_{C}$ is convex, we find that there are precisely two rest points of (17) whenever boundary-layer profiles exist, except at the single parameter value on the boundary between existence and nonexistence of solutions, for which there is a degenerate rest point (double root of $\mathcal{H}_{C}$). Ignoring this degenerate case, we see that boundary layers terminating at rest point $v_{+}$ as $x\to+\infty$ must either continue backward into $x<0$ to terminate at a second rest point $v_{-}$ as $x\to-\infty$, or else blow up to infinity as $x\to-\infty$. The first case we shall call compressive, the second expansive.

In the first case, the extended solution on the whole line may be recognized as a standing viscous shock wave; that is, for isentropic gas dynamics, compressive boundary layers are just restrictions to the half-line $x\geq 0$ [resp. $x\leq 0$] of standing shock waves. In the second case, as discussed in [19], the boundary layers are somewhat analogous to rarefaction waves on the whole line. From here on, we concentrate exclusively on the compressive case.

With the choice $v_{-}=1$, we may carry out the integration of (16) once more, this time as a definite integral from $-\infty$ to $x$, to obtain

where $a$ is found by letting $x\rightarrow+\infty$, yielding

in particular, $a\sim v_{+}^{\gamma}$ in the large boundary layer limit $v_{+}\to 0$. This is exactly the equation for viscous shock profiles considered in [12].

Linearizing (14) about $(\hat{v},\hat{u})$, we obtain

where $v_{0}=\hat{v}(0)$,

and $\tilde{v},\tilde{u}$ denote perturbations of $\hat{v},\hat{u}$.

The following estimates are established in Appendices A and B.

Setting $w:=\frac{u^{\prime}}{\hat{v}}+\frac{h(\hat{v})}{\hat{v}^{\gamma+1}}v-u$, we may express (22) as a first-order system

where

where

with $h$ as in (21) and $a$ as in (19), or, equivalently,

Eigenvalues of (22) correspond to nontrivial solutions $W$ for which the boundary conditions $W(\pm\infty)=0$ are satisfied. Because $A(x,\lambda)$ as a function of $\hat{v}$ is asymptotically constant in $x$, the behavior near $x=\pm\infty$ of solutions of (34) is governed by the limiting constant-coefficient systems

from which we readily find on the (nonstable) domain $\Re\lambda\geq 0$, $\lambda\neq 0$ of interest that there is a one-dimensional unstable manifold $W_{1}^{-}(x)$ of solutions decaying at $x=-\infty$ and a two-dimensional stable manifold $W_{2}^{+}(x)\wedge W_{3}^{+}(x)$ of solutions decaying at $x=+\infty$, analytic in $\lambda$, with asymptotic behavior

as $x\to\pm\infty$, where $\mu_{\pm}(\lambda)$ and $V_{j}^{\pm}(\lambda)$ are eigenvalues and associated analytically chosen eigenvectors of the limiting coefficient matrices $A_{\pm}(\lambda)$. A standard choice of eigenvectors $V_{j}^{\pm}$ [8, 5, 4, 13], uniquely specifying $W_{j}^{\pm}$ (up to constant factor) is obtained by Kato’s ODE [15], a linear, analytic ODE whose solution can be alternatively characterized by the property that there exist corresponding left eigenvectors $\tilde{V}_{j}^{\pm}$ such that

where “$\,\,\dot{}\,\,$” denotes $d/d\lambda$; for further discussion, see [15, 8, 13].

Taking a formal limit as $v_{+}\to 0$ of the rescaled equations (14) and recalling that $a\sim v_{+}^{\gamma}$, we obtain a limiting evolution equation

corresponding to a pressureless gas, or $\gamma=0$.

The associated limiting profile equation $v^{\prime}=v(v-1)$ has explicit solution

$\hat{v}^{0}(0)=\frac{1-\tanh(-\delta/2)}{2}=v_{0}$; the limiting eigenvalue system is $W^{\prime}=A^{0}(x,\lambda)W,$

where $f^{0}(\hat{v}^{0})=2\hat{v}^{0}-1=-\tanh\big{(}\frac{x+\delta}{2}\big{)}.$

Convergence of the profile and eigenvalue equations is uniform on any interval $\hat{v}^{0}\geq\epsilon>0$, or, equivalently, $x-\delta\leq L$, for $L$ any positive constant, where the sequence of coefficient matrices is therefore a regular perturbation of its limit. Following [12], we call $x\leq L+\delta$ the “regular region”. For $\hat{v}_{0}\to 0$ on the other hand, or $x\to\infty$, the limit is less well-behaved, as may be seen by the fact that $\partial f/\partial\hat{v}\sim\hat{v}^{-1}$ as $\hat{v}\to v_{+}$, a consequence of the appearance of $\big{(}\frac{v_{+}}{\hat{v}}\big{)}$ in the expression (36) for $f$. Similarly, $A(x,\lambda)$ does not converge to $A_{+}(\lambda)$ as $x\to+\infty$ with uniform exponential rate independent of $v_{+}$, $\gamma$, but rather as $C\hat{v}^{-1}e^{-x/2}$. As in the shock case, this makes problematic the treatment of $x\geq L+\delta$. Following [12] we call $x\geq L+\delta$ the “singular region”.

To put things in another way, the effects of pressure are not lost as $v_{+}\to 0$, but rather pushed to $x=+\infty$, where they must be studied by a careful boundary-layer analysis. (Note: this is not a boundary-layer in the same sense as the background solution, nor is it a singular perturbation in the usual sense, at least as we have framed the problem here.)

With the above definitions, we have the following main theorems characterizing the strong-layer limit $v_{+}\to 0$ as well as the limits $v_{0}\to 0,\,1$.

By similar computations, we obtain also the following direct result.

We have also the following parity information, obtained by stability-index computations as in [25].³³3Indeed, these may be deduced from the results of [25], taking account of the difference between Eulerian and Lagrangian coordinates.

Finally, we have the following auxiliary results established by energy estimates in Appendices C, D, E, and F.

Collecting information, we have the following analytical stability results.

Stability of inflow boundary layers in the characteristic limit $v_{+}\to 1$ is not treated here, but should be treatable analytically by the asymptotic ODE methods used in [22, 7] to study the small-amplitude (characteristic) shock limit. This would be an interesting direction for future investigation. The characteristic limit is not accessible numerically, since the exponential decay rate of the background profile decays to zero as $|1-v_{+}|$, so that the numerical domain of integration needed to resolve the eigenvalue ODE becomes infinitely large as $v_{+}\to 1$.

The asymptotic results of Section 3.2 reduce the problem of (uniformly noncharacteristic, $v_{+}$ bounded away from $v_{-}=1$) boundary layer stability to a bounded parameter range on which the Evans function may be efficiently computed numerically in a way that is uniformly well-conditioned; see [5]. Specifically, we may map a semicircle

enclosing $\Lambda$ for $\gamma\in[1,3]$ by $D^{0}_{\rm in}$, $D^{0}_{\rm out}$, $D_{\rm in}$, $D_{\rm out}$ and compute the winding number of its image about the origin to determine the number of zeroes of the various Evans functions within the semicircle, and thus within $\Lambda$. For details of the numerical algorithm, see [3, 5].

In all cases, we obtain results consistent with stability; that is, a winding number of zero or one, depending on the situation. In the case of a single nonzero root, we know from our limiting analysis that this root may be quite near $\lambda=0$, making delicate the direct determination of its stability; however, in this case we do not attempt to determine the stability numerically, but rely on the analytically computed stability index to conclude stability. See Section 6 for further details.

As in the shock case [3, 12], our results indicate unconditional stability of uniformly noncharacteristic boundary-layers for isentropic Navier–Stokes equations (and, for outflow layer, in the characteristic limit as well), despite the additional complexity of the boundary-layer case. However, two additional comments are in order, perhaps related. First, we point out that the apparent symmetry of Theorem 3.3 in the $v_{0}\to 0$ outflow and $v_{0}\to 1$ inflow limits is somewhat misleading. For, the limiting, shock Evans function possesses a single zero at $\lambda=0$, indicating that stability of inflow boundary layers is somewhat delicate as $v_{0}\to 1$: specifically, they have an eigenvalue near zero, which, though stable, is (since vanishingly small in the shock limit) not “very” stable. Likewise, the limiting Evans function $D^{0}_{\rm in}$ as $v_{+}\to 0$ possesses a zero at $\lambda=0$, with the same conclusions.

By contrast, the Evans functions of outflow boundary layers possess a spurious zero at $\lambda=0$, so that convergence to the shock or strong-layer limit in this case implies the absence of any eigenvalues near zero, or “uniform” stability as $v_{+}\to 0$. In this sense, strong outflow boundary layers appear to be more stable than inflow boundary layers. One may make interesting comparisons to physical attempts to stabilize laminar flow along an air- or hydro-foil by suction (outflow) along the boundary. See, for example, the interesting treatise [24].

Second, we point out the result of instability obtained in [25] for inflow boundary-layers of the full (nonisentropic) ideal-gas equations for appropriate ratio of the coefficients of viscosity and heat conduction. This suggests that the small eigenvalues of the strong inflow-layer limit may in some cases perturb to the unstable side. It would be very interesting to make these connections more precise, as we hope to do in future work.

We first block upper-triangularize by a static (constant) coordinate transformation the limiting matrix

at $x=+\infty$ using special block lower-triangular transformations

where $I$ denotes the $2\times 2$ identity matrix and $\theta_{+}\in\mathbb{C}^{1\times 2}$ is a $1\times 2$ row vector.