Long-time stability of multi-dimensional noncharacteristic viscous boundary layers
aa r X i v : . [ m a t h - ph ] A ug LONG-TIME STABILITY OF MULTI-DIMENSIONALNONCHARACTERISTIC VISCOUS BOUNDARY LAYERS
TOAN NGUYEN AND KEVIN ZUMBRUN
Abstract.
We establish long-time stability of multi-dimensional noncharacteristic bound-ary layers of a class of hyperbolic–parabolic systems including the compressible Navier–Stokes equations with inflow [outflow] boundary conditions, under the assumption of strongspectral, or uniform Evans, stability. Evans stabiity has been verified for small-amplitudelayers by Gu`es, M´etivier, Williams, and Zumbrun. For large-amplitude layers, it maybe efficiently checked numerically, as done in the one-dimensional case by Costanzino,Humpherys, Nguyen, and Zumbrun.
Contents
1. Introduction 21.1. Equations and assumptions 21.2. The Evans condition and strong spectral stability 51.3. Main results 61.4. Discussion and open problems 72. Resolvent kernel: construction and low-frequency bounds 92.1. Construction 92.2. Pointwise low-frequency bounds 123. Linearized estimates 133.1. Resolvent bounds 133.2. Estimates on homogeneous solution operators 153.3. Boundary estimates 193.4. Duhamel formula 213.5. Proof of linearized stability 224. Nonlinear stability 224.1. Auxiliary energy estimates 224.2. Proof of nonlinear stability 33Appendix A. Physical discussion in the isentropic case 39A.1. Existence 39A.2. Stability 40A.3. Discussion 41References 41
Date : November 4, 2018.This work was supported in part by the National Science Foundation award number DMS-0300487. Introduction
We consider a boundary layer, or stationary solution,(1.1) ˜ U = ¯ U ( x ) , lim z → + ∞ ¯ U ( z ) = U + , ¯ U (0) = ¯ U of a system of conservation laws on the quarter-space(1.2) ˜ U t + X j F j ( ˜ U ) x j = X jk ( B jk ( ˜ U ) ˜ U x k ) x j , x ∈ R d + = { x > } , t > , ˜ U , F j ∈ R n , B jk ∈ R n × n , with initial data ˜ U ( x,
0) = ˜ U ( x ) and Dirichlet type boundaryconditions specified in (1.5), (1.6) below. A fundamental question connected to the physicalmotivations from aerodynamics is whether or not such boundary layer solutions are stable in the sense of PDE, i.e., whether or not a sufficiently small perturbation of ¯ U remainsclose to ¯ U , or converges time-asymptotically to ¯ U , under the evolution of (1.2). That is thequestion we address here.1.1. Equations and assumptions.
We consider the general hyperbolic-parabolic systemof conservation laws (1.2) in conserved variable ˜ U , with˜ U = (cid:18) ˜ u ˜ v (cid:19) , B = (cid:18) b jk b jk (cid:19) , ˜ u ∈ R n − r , and ˜ v ∈ R r , where ℜ σ X jk b jk ξ j ξ k ≥ θ | ξ | > , ∀ ξ ∈ R n \{ } . Following [MaZ4, Z3, Z4], we assume that equations (1.2) can be written, alternatively,after a triangular change of coordinates(1.3) ˜ W := ˜ W ( ˜ U ) = (cid:18) ˜ w I (˜ u )˜ w II (˜ u, ˜ v ) (cid:19) , in the quasilinear, partially symmetric hyperbolic-parabolic form (1.4) ˜ A ˜ W t + X j ˜ A j ˜ W x j = X jk ( ˜ B jk ˜ W x k ) x j + ˜ G, where, defining ˜ W + := ˜ W ( U + ),(A1) ˜ A j ( ˜ W + ) , ˜ A , ˜ A are symmetric, ˜ A block diagonal, ˜ A ≥ θ > ξ ∈ R d \ { } , no eigenvector of P j ξ j ˜ A j ( ˜ A ) − ( ˜ W + ) lies in the kernel of P jk ξ j ξ k ˜ B jk ( ˜ A ) − ( ˜ W + ),(A3) ˜ B jk = (cid:18) b jk (cid:19) , P ˜ b jk ξ j ξ k ≥ θ | ξ | , and ˜ G = (cid:18) g (cid:19) with ˜ g ( ˜ W x , ˜ W x ) = O ( | ˜ W x | ) . Along with the above structural assumptions, we make the following technical hypotheses:(H0) F j , B jk , ˜ A , ˜ A j , ˜ B jk , ˜ W ( · ) , ˜ g ( · , · ) ∈ C s , with s ≥ [( d − /
2] + 5 in our analysis oflinearized stability, and s ≥ s ( d ) := [( d − /
2] + 7 in our analysis of nonlinear stability.
TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 3 (H1) ˜ A is either strictly positive or strictly negative, that is, either ˜ A ≥ θ > , or˜ A ≤ − θ <
0. (We shall call these cases the inflow case or outflow case , correspondingly.)(H2) The eigenvalues of dF ( U + ) are distinct and nonzero.(H3) The eigenvalues of P j dF j ± ξ j have constant multiplicity with respect to ξ ∈ R d , ξ = 0.(H4) The set of branch points of the eigenvalues of ( ˜ A ) − ( iτ ˜ A + P j =1 iξ j ˜ A j ) ± , τ ∈ R ,˜ ξ ∈ R d − is the (possibly intersecting) union of finitely many smooth curves τ = η ± q ( ˜ ξ ), onwhich the branching eigenvalue has constant multiplicity s q (by definition ≥ U + of the associated first-order hyperbolic system obtainedby dropping second-order terms. The assumptions (A1)-(A3) and (H0)-(H2) are satisfiedfor gas dynamics and MHD with van der Waals equation of state under inflow or outflowconditions; see discussions in [MaZ4, CHNZ, GMWZ5, GMWZ6]. Condition (H3) holdsalways for gas dynamics, but fails always for MHD in dimension d ≥
2. Condition (H4) is atechnical requirement of the analysis introduced in [Z2]. It is satisfied always in dimension d = 2 or for rotationally invariant systems in dimensions d ≥
2, for which it serves only todefine notation; in particular, it holds always for gas dynamics.We also assume:(B) Dirichlet boundary conditions in ˜ W -coordinates:(1.5) ( ˜ w I , ˜ w II )(0 , ˜ x, t ) = ˜ h (˜ x, t ) := (˜ h , ˜ h )(˜ x, t )for the inflow case, and(1.6) ˜ w II (0 , ˜ x, t ) = ˜ h (˜ x, t )for the outflow case, with x = ( x , ˜ x ) ∈ R d .This is sufficient for the main physical applications; the situation of more general, Neu-mann and mixed-type boundary conditions on the parabolic variable v can be treated asdiscussed in [GMWZ5, GMWZ6]. Example 1.1.
The main example we have in mind consists of laminar solutions ( ρ, u, e )( x , t )of the compressible Navier–Stokes equations(1.7) ∂ t ρ + div( ρu ) = 0 ∂ t ( ρu ) + div( ρu t u ) + ∇ p = εµ ∆ u + ε ( µ + η ) ∇ div u∂ t ( ρE ) + div (cid:0) ( ρE + p ) u (cid:1) = εκ ∆ T + εµ div (cid:0) ( u · ∇ ) u (cid:1) + ε ( µ + η ) ∇ ( u · div u ) ,x ∈ R d , on a half-space x >
0, where ρ denotes density, u ∈ R d velocity, e specific internalenergy, E = e + | u | specific total energy, p = p ( ρ, e ) pressure, T = T ( ρ, e ) temperature, µ > | η | ≤ µ first and second coefficients of viscosity, κ > T. NGUYEN AND K. ZUMBRUN conduction, and ε > suction-type boundary conditions on the velocity, u j (0 , x , . . . , x d ) = 0 , j = 1 and u (0 , x , . . . , x d ) = V ( x ) < , and prescribed temperature, T (0 , x , . . . , x d ) = T wall (˜ x ) . Under the standard assumptions p ρ , T e >
0, this can be seen to satisfy all of the hypotheses (A1)–(A3), (H0)–(H4), (B) inthe outflow case (1.6); indeed these are satisfied also under much weaker van der Waals gasassumptions [MaZ4, Z3, CHNZ, GMWZ5, GMWZ6]. In particular, boundary-layer solutionsare of noncharacteristic type, scaling as ( ρ, u, e ) = (¯ ρ, ¯ u, ¯ e )( x /ε ), with layer thickness ∼ ε as compared to the ∼ √ ε thickness of the characteristic type found for an impermeableboundary.This corresponds to the situation of an airfoil with microscopic holes through whichgas is pumped from the surrounding flow, the microscopic suction imposing a fixed normalvelocity while the macroscopic surface imposes standard temperature conditions as in flowpast a (nonporous) plate. This configuration was suggested by Prandtl and tested experi-mentally by G.I. Taylor as a means to reduce drag by stabilizing laminar flow; see [S, Bra].It was implemented in the NASA F-16XL experimental aircraft program in the 1990’s withreported 25% reduction in drag at supersonic speeds [Bra]. Possible mechanisms for thisreduction are smaller thickness ∼ ε << √ ε of noncharacteristic boundary layers as com-pared to characteristic type, and greater stability, delaying the transition from laminar toturbulent flow. In particular, stability properties appear to be quite important for theunderstanding of this phenomenon. For further discussion, including the related issuesof matched asymptotic expansion, multi-dimensional effects, and more general boundaryconfigurations, see [GMWZ5]. Example 1.2.
Alternatively, we may consider the compressible Navier–Stokes equations(1.7) with blowing-type boundary conditions u j (0 , x , . . . , x d ) = 0 , j = 1 and u (0 , x , . . . , x d ) = V ( x ) > , and prescribed temperature and pressure T (0 , x , . . . , x d ) = T wall (˜ x ) , p (0 , x , . . . , x d ) = p wall (˜ x )(equivalently, prescribed temperature and density). Under the standard assumptions p ρ , T e > inflow case (1.5). Lemma 1.3 ([MaZ3, Z3, GMWZ5, NZ]) . Given (A1)-(A3) and (H0)-(H2), a standing wavesolution (1.1) of (1.2) , (B) satisfies (1.8) (cid:12)(cid:12)(cid:12) ( d/dx ) k ( ¯ U − U + ) (cid:12)(cid:12)(cid:12) ≤ Ce − θx , ≤ k ≤ s + 1 , as x → + ∞ , s as in (H0). Moreover, a solution, if it exists, is in the inflow or strictlyparabolic case unique; in the outflow case it is locally unique.Proof. See Lemma 1.3, [NZ]. (cid:3) TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 5
The Evans condition and strong spectral stability.
The linearized equations of(1.2), (B) about ¯ U are(1.9) U t = LU := X j,k ( B jk U x k ) x j − X j ( A j U ) x j with initial data U (0) = U and boundary conditions in (linearized) ˜ W -coordinates of W (0 , ˜ x, t ) := ( w I , w II ) T (0 , ˜ x, t ) = h for the inflow case, and w II (0 , ˜ x, t ) = h for the outflow case, with x = ( x , ˜ x ) ∈ R d , where W := ( ∂ ˜ W /∂U )( ¯ U ) U .A necessary condition for linearized stability is weak spectral stability, defined as nonexis-tence of unstable spectra ℜ λ > L about the wave. As describedin Section 2.1.1, this is equivalent to nonvanishing for all ˜ ξ ∈ R d − , ℜ λ > Evansfunction D L ( ˜ ξ, λ )(defined in (2.8)), a Wronskian associated with the Fourier-transformed eigenvalue ODE. Definition 1.4.
We define strong spectral stability as uniform Evans stability :(D) | D L ( ˜ ξ, λ ) | ≥ θ ( C ) > ξ, λ ) on bounded subsets C ⊂ { ˜ ξ ∈ R d − , ℜ λ ≥ } \ { } .For the class of equations we consider, this is equivalent to the uniform Evans condition of[GMWZ5, GMWZ6], which includes an additional high-frequency condition that for theseequations is always satisfied (see Proposition 3.8, [GMWZ5]). A fundamental result provedin [GMWZ5] is that small-amplitude noncharacteristic boundary-layers are always stronglyspectrally stable. Proposition 1.5 ([GMWZ5]) . Assuming (A1)-(A3), (H0)-(H3), (B) for some fixed end-state (or compact set of endstates) U + , boundary layers with amplitude k ¯ U − U + k L ∞ [0 , + ∞ ] sufficiently small satisfy the strong spectral stability condition (D). As demonstrated in [SZ], stability of large-amplitude boundary layers may fail for theclass of equations considered here, even in a single space dimension, so there is no suchgeneral theorem in the large-amplitude case. Stability of large-amplitude boundary-layersmay be checked efficiently by numerical Evans computations as in [BDG, Br1, Br2, BrZ,HuZ, BHRZ, HLZ, CHNZ, HLyZ1, HLyZ2]. The result of [GMWZ5] applies also to more general types of boundary conditions and in some situationsto systems with variable multiplicity characteristics, including, in some parameter ranges, MHD.
T. NGUYEN AND K. ZUMBRUN
Main results.
Our main results are as follows.
Theorem 1.6 (Linearized stability) . Assuming (A1)-(A3), (H0)-(H4), (B), and strongspectral stability (D), we obtain asymptotic L ∩ H [( d − / → L p stability of (1.9) indimension d ≥ , and any ≤ p ≤ ∞ , with rate of decay (1.10) | U ( t ) | L ≤ C (1 + t ) − d − ( | U | L ∩ H + E ) , | U ( t ) | L p ≤ C (1 + t ) − d (1 − /p )+1 / p ( | U | L ∩ H [( d − / + E ) , provided that the initial perturbations U are in L ∩ H for p = 2 , or in L ∩ H [( d − / for p > , and boundary perturbations h satisfy (1.11) | h ( t ) | L x ≤ E (1 + t ) − ( d +1) / , | h ( t ) | L ∞ ˜ x ≤ E (1 + t ) − d/ |D h ( t ) | L x ∩ H [( d − / x ≤ E (1 + t ) − d/ − ǫ , where D h ( t ) := | h t | + | h ˜ x | + | h ˜ x ˜ x | , E is some positive constant, and ǫ > is arbitrary smallfor the case d = 2 and ǫ = 0 for d ≥ . Theorem 1.7 (Nonlinear stability) . Assuming (A1)-(A3), (H0)-(H4), (B), and strongspectral stability (D), we obtain asymptotic L ∩ H s → L p ∩ H s stability of ¯ U as a solutionof (1.2) in dimension d ≥ , for s ≥ s ( d ) as defined in (H0), and any ≤ p ≤ ∞ , with rateof decay (1.12) | ˜ U ( t ) − ¯ U | L p ≤ C (1 + t ) − d (1 − /p )+1 / p ( | U | L ∩ H s + E ) | ˜ U ( t ) − ¯ U | H s ≤ C (1 + t ) − d − ( | U | L ∩ H s + E ) , provided that the initial perturbations U := ˜ U − ¯ U are sufficiently small in L ∩ H s andboundary perturbations h ( t ) := ˜ h ( t ) − W ( ¯ U ) satisfy (1.11) and (1.13) B h ( t ) ≤ E (1 + t ) − d − , with sufficiently small E , where the boundary measure B h is defined as (1.14) B h ( t ) := | h | H s (˜ x ) + [( s +1) / X i =0 | ∂ it h | L (˜ x ) for the outflow case, and similarly (1.15) B h ( t ) := | h | H s (˜ x ) + [( s +1) / X i =0 | ∂ it h | L (˜ x ) + s X i =0 | ∂ it h | L (˜ x ) for the inflow case. Combining Theorem 1.7 and Proposition 1.5, we obtain the following small-amplitudestability result, applying in particular to the motivating situation of Example 1.1.
TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 7
Corollary 1.8.
Assuming (A1)-(A3), (H0)-(H4), (B) for some fixed endstate (or compactset of endstates) U + , boundary layers with amplitude k ¯ U − U + k L ∞ [0 , + ∞ ] sufficiently small are linearly and nonlinearly stable in the sense of Theorems 1.6 and 1.7. Remark 1.9.
The obtained rate of decay in L may be recognized as that of a ( d − L ∞ as that of a d -dimensionalheat kernel. We believe that the sharp rate of decay in L is rather that of a d -dimensionalheat kernel and the sharp rate of decay in L ∞ dependent on the characteristic structure ofthe associated inviscid equations, as in the constant-coefficient case [HoZ1, HoZ2]. Remark 1.10.
In one dimension, strong spectral stability is necessary for linearized as-ymptotic stability; see Theorem 1.6, [NZ]. However, in multi-dimensions, it appears likelythat, as in the shock case [Z3], there are intermediate possibilities between strong and weakspectral stability for which linearized stability might hold with degraded rates of decay. Inany case, the gap between the necessary weak spectral and the sufficient strong spectralstability conditions concerns only pure imaginary spectra ℜ λ = 0 on the boundary betweenstrictly stable and unstable half-planes, so this should not interfere with investigation ofphysical stability regions.1.4. Discussion and open problems.
Asymptotic stability, without rates of decay, hasbeen shown for small amplitude noncharacteristic “normal” boundary layers of the isen-tropic compressible Navier–Stokes equations with outflow boundary conditions and vanish-ing transverse velocity in [KK], using energy estimates. Corollary 1.8 recovers this existingresult and extends it to the general arbitrary transverse velocity, outflow or inflow, andisentropic or nonisentropic (full compressible Navier–Stokes) case, in addition giving as-ymptotic rates of decay. Moreover, we treat perturbations of boundary as well as initialdata, as previous time-asymptotic investigations (with the exception of direct predecessors[YZ, NZ]) do not. As discussed in Appendix A, the type of boundary layer relevant to thedrag-reduction strategy discussed in Examples 1.1–1.2 is a noncharacteristic “transverse”type with constant normal velocity, complementary to the normal type considered in [KK].The large-amplitude asymptotic stability result of Theorem 1.7 extends to multi dimen-sions corresponding one-dimensional results of [YZ, NZ], reducing the problem of stability toverification of a numerically checkable Evans condition. See also the related, but technicallyrather different, work on the small viscosity limit in [MZ, GMWZ5, GMWZ6]. By a combi-nation of numerical Evans function computations and asymptotic ODE estimates, spectralstability has been checked for arbitrary amplitude noncharacteristic boundary layers of theone-dimensional isentropic compressible Navier–Stokes equations in [CHNZ]. Extensions tothe nonisentropic and multi-dimensional case should be possible by the methods used in[HLyZ1] and [HLyZ2] respectively to treat the related shock stability problem.This (investigation of large-amplitude spectral stability) would be a very interesting di-rection for further investigation. In particular, note that it is large-amplitude stability thatis relevant to drag-reduction at flight speeds, since the transverse relative velocity (i.e.,velocity parallel to the airfoil) is zero at the wing surface and flight speed outside a thinboundary layer, so that variation across the boundary layer is substantial. We discuss thisproblem further in Appendix A for the model isentropic case.
T. NGUYEN AND K. ZUMBRUN
Our method of analysis follows the basic approach introduced in [Z2, Z3, Z4] for thestudy of multi-dimensional shock stability and we are able to make use of much of thatanalysis without modification. However, there are some new difficulties to be overcome inthe boundary-layer case.The main new difficulty is that the boundary-layer case is analogous to the undercom-pressive shock case rather than the more favorable
Lax shock case emphasized in [Z3], inthat G y t − / G as in the Lax shock case but rather G y ∼ ( e − θ | y | + t − / ) G , θ >
0, as inthe undercompressive case. This is a significant difficulty; indeed, for this reason, the un-dercompressive shock analysis was carried out in [Z3] only in nonphysical dimensions d ≥ G is somewhat better in the boundary layer than in the shockcase.Thus, the difficulty of the present problem is roughly intermediate to that of the Laxand undercompressive shock cases. Though the undercompressive shock case is still open inmulti-dimensions for d ≤
3, the slight advantage afforded by lack of pole terms allows us toclose the argument in the boundary-layer case. Specifically, thanks to the absence of poleterms, we are able to get a slightly improved rate of decay in L ∞ ( x ) norms, though our L ( x ) estimates remain the same as in the shock case. By keeping track of these improvedsup norm bounds throughout the proof, we are able to close the argument without usingdetailed pointwise bounds as in the one-dimensional analyses of [HZ, RZ].Other difficulties include the appearance of boundary terms in integrations by parts,which makes the auxiliary energy estimates by which we control high-frequency effectsconsiderably more difficult in the boundary-layer than in the shock-layer case, and thetreatment of boundary perturbations. In terms of the homogeneous Green function G ,boundary perturbations lead by a standard duality argument to contributions consisting ofintegrals on the boundary of perturbations against various derivatives of G , and these area bit too singular as time goes to zero to be absolutely integrable. Following the strategyintroduced in [YZ, NZ], we instead use duality to convert these to less singular integralsover the whole space, that are absolutely integrable in time. However, we make a keyimprovement here over the treatment in [YZ, NZ], integrating against an exponentiallydecaying test function to obtain terms of exactly the same form already treated for thehomogeneous problem. This is necessary for us in the multi-dimensional case, for whichwe have insufficient information about individual parts of the solution operator to estimatethem separately as in [YZ, NZ], but makes things much more transparent also in the one-dimensional case.Among physical systems, our hypotheses appear to apply to and essentially only to thecase of compressible Navier–Stokes equations with inflow or outflow boundary conditions.However, the method of analysis should apply, with suitable modifications, to more generalsituations such as MHD; see for example the recent results on the related small-viscosityproblem in [GMWZ5, GMWZ6]. The extension to MHD is a very interesting open problem.Finally, as pointed out in Remark 1.10, the strong spectral stability condition does notappear to be necessary for asymptotic stability. It would be interesting to develop a refinedstability condition similarly as was done in [SZ, Z2, Z3, Z4] for the shock case. TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 9 Resolvent kernel: construction and low-frequency bounds
In this section, we briefly recal the construction of resolvent kernel and then establishthe pointwise low-frequency bounds on G ˜ ξ,λ , by appropriately modifying the proof in [Z3]in the boundary layer context [YZ, NZ].2.1. Construction.
We construct a representation for the family of elliptic Green distri-butions G ˜ ξ,λ ( x , y ),(2.1) G ˜ ξ,λ ( · , y ) := ( L ˜ ξ − λ ) − δ y ( · ) , associated with the ordinary differential operators ( L ˜ ξ − λ ), i.e. the resolvent kernel ofthe Fourier transform L ˜ ξ of the linearized operator L of (1.9). To do so, we study thehomogeneous eigenvalue equation ( L ˜ ξ − λ ) U = 0, or(2.2) L U z }| { ( B U ′ ) ′ − ( A U ) ′ − i X j =1 A j ξ j U + i X j =1 B j ξ j U ′ + i X k =1 ( B k ξ k U ) ′ − X j,k =1 B jk ξ j ξ k U − λU = 0 , with boundary conditions (translated from those in W -coordinates)(2.3) (cid:18) A − A ( b ) − b b b (cid:19) U (0) ≡ (cid:18) ∗ (cid:19) where ∗ = 0 for the inflow case and is arbitrary for the outflow case.Define Λ ˜ ξ := n \ j =1 Λ + j ( ˜ ξ )where Λ + j ( ˜ ξ ) denote the open sets bounded on the left by the algebraic curves λ + j ( ξ , ˜ ξ )determined by the eigenvalues of the symbols − ξ B + − iξA + of the limiting constant-coefficient operators L ˜ ξ + w := B + w ′′ − A + w ′ as ξ is varied along the real axis, with ˜ ξ held fixed. The curves λ + j ( · , ˜ ξ ) comprise theessential spectrum of operators L ˜ ξ + . Let Λ denote the set of ( ˜ ξ, λ ) such that λ ∈ Λ ˜ ξ .For ( ˜ ξ, λ ) ∈ Λ ˜ ξ , introduce locally analytically chosen (in ˜ ξ , λ ) matrices(2.4) Φ + = ( φ +1 , · · · , φ + k ) , Φ = ( φ k +1 , · · · , φ n + r ) , and(2.5) Φ = (Φ + , Φ ) , whose columns span the subspaces of solutions of (2.2) that, respectively, decay at x = + ∞ and satisfy the prescribed boundary conditions at x = 0, and locally analytically chosenmatrices(2.6) Ψ = ( ψ , · · · , ψ k ) , Ψ + = ( ψ + k +1 , · · · , ψ + n + r ) and(2.7) Ψ = (Ψ , Ψ + ) . whose columns span complementary subspaces. The existence of such matrices is guaranteedby the general Evans function framework of [AGJ, GZ, MaZ3]; see in particular [Z3, NZ].That dimensions sum to n + r follows by a general result of [GMWZ5]; see also [SZ].2.1.1. The Evans function.
Following [AGJ, GZ, SZ], we define on Λ the
Evans function (2.8) D L ( ˜ ξ, λ ) := det(Φ , Φ + ) | x =0 . Evidently, eigenfunctions decaying at + ∞ and satisfying the prescribed boundary conditionsat x = 0 occur precisely when the subspaces span Φ and span Φ + intersect, i.e., at zerosof the Evans function D L ( ˜ ξ, λ ) = 0 . The Evans function as constructed here is locally analytic in ( ˜ ξ, λ ), which is all that weneed for our analysis; we prescribe different versions of the Evans function as needed ondifferent neighborhoods of Λ. Note that Λ includes all of { ˜ ξ ∈ R d − , ℜ λ ≥ } \ { } , so thatDefinition 1.4 is well-defined and equivalent to simple nonvanishing, away from the origin( ˜ ξ, λ ) = (0 , j in (2.8) remain uniformly bounded , a condition that can always be achieved bylimiting the neighborhood of definition.For the class of equations we consider, the Evans function may in fact be extendedcontinuously along rays through the origin [R2, MZ, GMWZ5, GMWZ6].2.1.2. Basic representation formulae.
Define the solution operator from y to x of ODE( L ˜ ξ − λ ) U = 0, denoted by F y → x , as F y → x = Φ( x , λ )Φ − ( y , λ )and the projections Π y , Π + y on the stable manifolds at 0 , + ∞ asΠ + y = (cid:0) Φ + ( y ) 0 (cid:1) Φ − ( y ) , Π y = (cid:0) ( y ) (cid:1) Φ − ( y ) . We define also the dual subspaces of solutions of ( L ∗ ˜ ξ − λ ∗ ) ˜ W = 0. We denote growingsolutions(2.9) ˜Φ = ( ˜ φ , · · · , ˜ φ k ) , ˜Φ + = ( ˜ φ + k +1 , · · · , ˜ φ + n + r ) , ˜Φ := ( ˜Φ , ˜Φ + ) and decaying solutions(2.10) ˜Ψ = ( ˜ ψ , · · · , ˜ ψ + k ) , ˜Ψ + = ( ˜ ψ + k +1 , · · · , ˜ ψ + n + r ) , and ˜Ψ := ( ˜Ψ , ˜Ψ + ), satisfying the relations(2.11) (cid:0) ˜Ψ ˜Φ (cid:1) ∗ , + ¯ S ˜ ξ (cid:0) Ψ Φ (cid:1) , + ≡ I, where(2.12) ¯ S ˜ ξ = − A + iB ξ + iB ˜ ξ (cid:18) I r (cid:19)(cid:0) − ( b ) − b I − I r (cid:1) . TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 11
With these preparations, the construction of the Resolvent kernel goes exactly as in theconstruction performed in [ZH, MaZ3, Z3] on the whole line and [YZ, NZ] on the half line,yielding the following basic representation formulae; for a proof, see [MaZ3, NZ].
Proposition 2.1.
We have the following representation (2.13) G ˜ ξ,λ ( x , y ) = ( ( I n , F y → x Π + y ( ¯ S ˜ ξ ) − ( y )( I n , tr , f or x > y , − ( I n , F y → x Π y ( ¯ S ˜ ξ ) − ( y )( I n , tr , f or x < y . Proposition 2.2.
The resolvent kernel may alternatively be expressed as G ˜ ξ,λ ( x , y ) = ( ( I n , + ( x ; λ ) M + ( λ ) ˜Ψ ∗ ( y ; λ )( I n , tr x > y , − ( I n , ( x ; λ ) M ( λ ) ˜Ψ + ∗ ( y ; λ )( I n , tr x < y , where (2.14) M ( λ ) := diag( M + ( λ ) , M ( λ )) = Φ − ( z ; λ )( ¯ S ˜ ξ ) − ( z ) ˜Ψ − ∗ ( z ; λ ) . Scattering decomposition.
From Propositions 2.1 and 2.2, we obtain the followingscattering decomposition, generalizing the Fourier transform representation in the constant-coefficient case, from which we will obtain pointwise bounds in the low-frequency regime.
Corollary 2.3. On Λ ˜ ξ ∩ ρ ( L ˜ ξ ) , (2.15) G ˜ ξ,λ ( x , y ) = X j,k d + jk φ + j ( x ; λ ) ˜ ψ + k ( y ; λ ) ∗ + X k φ + k ( x ; λ ) ˜ φ + k ( y ; λ ) ∗ for ≤ y ≤ x , and (2.16) G ˜ ξ,λ ( x , y ) = X j,k d jk φ + j ( x ; λ ) ˜ ψ + k ( y ; λ ) ∗ − X k ψ + k ( x ; λ ) ˜ ψ + k ( y ; λ ) ∗ for ≤ x ≤ y , where (2.17) d , + jk ( λ ) = ( I, (cid:0) Φ + Φ (cid:1) − Ψ + . Proof.
For 0 ≤ x ≤ y , we obtain the preliminary representation G ˜ ξ,λ ( x , y ) = X j,k d jk ( λ ) φ + j ( x ; λ ) ˜ ψ + k ( y ; λ ) ∗ + X jk e jk ψ + j ( x ; λ ) ˜ ψ + k ( y ; λ ) ∗ from which, together with duality (2.11), representation (2.13), and the fact that Π = I − Π + , we have(2.18) (cid:18) d e (cid:19) = − (cid:0) ˜Φ + ˜Ψ + (cid:1) ∗ A Π Ψ + = − (cid:0) Φ + Ψ + (cid:1) − h I − (cid:0) Φ + (cid:1) (cid:0) Φ + Φ (cid:1) − i Ψ + = (cid:18) − I k (cid:19) + (cid:18) I n − k
00 0 (cid:19) (cid:0) Φ + Φ (cid:1) − Ψ + . Similarly, for 0 ≤ y ≤ x , we obtain the preliminary representation G ˜ ξ,λ ( x , y ) = X j,k d + jk ( λ ) φ + j ( x ; λ ) ˜ ψ + k ( y ; λ ) ∗ + X jk e + jk φ + j ( x ; λ ) ˜ φ + k ( y ; λ ) ∗ from which, together with duality (2.11) and representation (2.13), we have(2.19) (cid:18) d + e + (cid:19) = ˜Φ + ∗ A Π + (cid:0) Ψ + Φ + (cid:1) = (Φ + ) − (cid:0) Φ + (cid:1) (cid:0) Φ + Φ (cid:1) − (cid:0) Ψ + Φ + (cid:1) = (cid:0) I (cid:1) (cid:0) Φ + Φ (cid:1) − (cid:0) Ψ + Φ + (cid:1) = (cid:18) I n − k
00 0 (cid:19) (cid:0) Φ + Φ (cid:1) − Ψ + + (cid:18) I k (cid:19) (cid:18) I k (cid:19) . (cid:3) Remark 2.4.
In the constant-coefficient case, with a choice of common bases Ψ , + = Φ + , at 0 , + ∞ , the above representation reduces to the simple formula(2.20) G ˜ ξ,λ ( x , y ) = (P Nj = k +1 φ + j ( x ; λ ) ˜ φ + ∗ j ( y ; λ ) x > y , − P kj =1 ψ + j ( x ; λ ) ˜ ψ + ∗ j ( y ; λ ) x < y . Pointwise low-frequency bounds.
We obtain pointwise low-frequency bounds onthe resolvent kernel G ˜ ξ,λ ( x , y ) by appealing to the detailed analysis of [Z2, Z3, GMWZ1]in the viscous shock case. Restrict attention to the surface(2.21) Γ ˜ ξ := { λ : ℜ eλ = − θ ( | ˜ ξ | + |ℑ mλ | ) } , for θ > Proposition 2.5 ([Z3]) . Under the hypotheses of Theorem 1.7, for λ ∈ Γ ˜ ξ and ρ := | ( ˜ ξ, λ ) | , θ > , and θ > sufficiently small, there hold: (2.22) | G ˜ ξ,λ ( x , y ) | ≤ Cγ e − θρ | x − y | . and (2.23) | ∂ βy G ˜ ξ,λ ( x , y ) | ≤ Cγ ( ρ β + βe − θy ) e − θρ | x − y | where (2.24) γ := 1 + X j h ρ − |ℑ mλ − η + j ( ˜ ξ ) | + ρ i /s j − , and s j , η + j ( ˜ ξ ) are as defined in (H4).Proof. This follows by a simplified version of the analysis of [Z3], Section 5 in the viscousshock case, replacing Φ − , Ψ − with Φ , Ψ , omitting the refined derivative bounds of Lem-mas 5.23 and 5.27 describing special properties of the Lax and overcompressive shock case(not relevant here), and setting ℓ = 0, or ˜ γ ≡ ℓ is the mul-tiplicity to which the Evans function vanishes at the origin, ( ˜ ξ, λ ) = (0 , + , Ψ + at plus spatial infinity are the same for theboundary-layer as for the shock case.This leads to the pointwise bounds (5.37)–(5.38) given in Proposition 5.10 of [Z3] in case α = 1, γ ≡ TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 13 without the first O ( ρ − ), or “pole”, terms appearing on the righthand side, which derivefrom cases ˜ γ ∼ ρ − not arising here. But, these are exactly the claimed bounds (2.22)–(2.24).We omit the (substantial) details of this computation, referring the reader to [Z3]. How-ever, the basic idea is, starting with the scattering decomposition of Corollary 2.1.3, to note,first, that the normal modes Φ j , Ψ j , ˜Φ j , ˜Ψ j can be approximated up to an exponentiallytrivial coordinate change by solutions of the constant-coefficient limiting system at x → + ∞ (the conjugation lemma of [MZ]) and, second, that the coefficients M jk , d jk may be well-estimated through formulae (2.14) and (2.17) using Kramer’s rule and the assumed lowerbound on th Evans function | D | appearing in the denominator. This is relatively straight-forward away from the branch points ℑ λ = η j ( ˜ ξ ) or “glancing set” of hyperbolic theory;the treatment near these points involves some delicate matrix perturbation theory appliedto the limiting constant-coefficient system at x → + ∞ followed by careful bookkeeping inthe application of Kramer’s rule. (cid:3) Linearized estimates
We next establish estimates on the linearized inhomogeneous problem(3.1) U t − LU = f with initial data U (0) = U and Dirichlet boundary conditions as usual in ˜ W -coordinates:(3.2) W (0 , ˜ x, t ) := ( w I , w II ) T (0 , ˜ x, t ) = h for the inflow case, and(3.3) w II (0 , ˜ x, t ) = h for the outflow case, with x = ( x , ˜ x ) ∈ R d .3.1. Resolvent bounds.
Our first step is to estimate solutions of the resolvent equationwith homogeneous boundary data ˆ h ≡ Proposition 3.1 (High-frequency bounds) . Given (A1)-(A2), (H0)-(H2), and homoge-neous boundary conditions (B), for some
R, C sufficiently large and θ > sufficiently small, (3.4) | ( L ˜ ξ − λ ) − ˆ f | ˆ H ( x ) ≤ C | ˆ f | ˆ H ( x ) , and (3.5) | ( L ˜ ξ − λ ) − ˆ f | L ( x ) ≤ C | λ | / | ˆ f | ˆ H ( x ) , for all | ( ˜ ξ, λ ) | ≥ R and ℜ eλ ≥ − θ , where ˆ f is the Fourier transform of f in variable ˜ x and | ˆ f | ˆ H ( x ) := | (1 + | ∂ x | + | ˜ ξ | ) ˆ f | L ( x ) .Proof. First observe that a Laplace-Fourier transformed version with respect to variables( λ, ˜ x ) of the nonlinear energy estimate in Section 4.1 with s = 1, carried out on the linearizedequations written in W -coordinates, yields(3.6) ( ℜ eλ + θ ) | (1 + | ˜ ξ | + | ∂ x | ) W | ≤ C (cid:16) | W | + (1 + | ˜ ξ | ) | W || ˆ f | + | ∂ x W || ∂ x ˆ f | (cid:17) for some C big and θ > | . | denotes | . | L ( x ) . Applying Young’sinequality, we obtain(3.7) ( ℜ eλ + θ ) | (1 + | ˜ ξ | + | ∂ x | ) W | ≤ C | W | + C | (1 + | ˜ ξ | + | ∂ x | ) ˆ f | . On the other hand, taking the imaginary part of the L inner product of U against λU = f + LU , we have also the standard estimate(3.8) |ℑ mλ || U | L ≤ C | U | H + C | f | L , and thus, taking the Fourier transform in ˜ x , we obtain(3.9) |ℑ mλ || W | ≤ C | ˆ f | + C | (1 + | ˜ ξ | + | ∂ x | ) W | . Therefore, taking θ = θ /
2, we obtain from (3.7) and (3.9)(3.10) | (1 + | λ | / + | ˜ ξ | + | ∂ x | ) W | ≤ C | W | + C | (1 + | ˜ ξ | + | ∂ x | ) ˆ f | , for any ℜ eλ ≥ − θ . Now take R sufficiently large such that | W | on the right hand sideof the above can be absorbed into the left hand side, and thus, for all | ( ˜ ξ, λ ) | ≥ R and ℜ eλ ≥ − θ ,(3.11) | (1 + | λ | / + | ˜ ξ | + | ∂ x | ) W | ≤ C | (1 + | ˜ ξ | + | ∂ x | ) ˆ f | , for some large C >
0, which gives the result. (cid:3)
We next have the following:
Proposition 3.2 (Mid-frequency bounds) . Given (A1)-(A2), (H0)-(H2), and strong spec-tral stability (D), (3.12) | ( L ˜ ξ − λ ) − | ˆ H ( x ) ≤ C, for R − ≤ | ( ˜ ξ, λ ) | ≤ R and ℜ eλ ≥ − θ, for any R and C = C ( R ) sufficiently large and θ = θ ( R ) > sufficiently small, where | ˆ f | ˆ H ( x ) is defined as in Proposition 3.1.Proof. Immediate, by compactness of the set of frequencies under consideration togetherwith the fact that the resolvent ( λ − L ˜ ξ ) − is analytic with respect to H in ( ˜ ξ, λ ); seeProposition 4.8, [Z4]. (cid:3) We next obtain the following resolvent bound for low-frequency regions as a direct con-sequence of pointwise bounds on the resolvent kernel, obtained in Proposition 2.5.
Proposition 3.3 (Low-frequency bounds) . Under the hypotheses of Theorem 1.7, for λ ∈ Γ ˜ ξ and ρ := | ( ˜ ξ, λ ) | , θ sufficiently small, there holds the resolvent bound (3.13) | ( L ˜ ξ − λ ) − ∂ βx ˆ f | L p ( x ) ≤ Cγ ρ − /p h ρ β | ˆ f | L ( x ) + β | ˆ f | L ∞ ( x ) i for all ≤ p ≤ ∞ , β = 0 , , where γ is as defined in (2.24) .Proof. Using the convolution inequality | g ∗ h | L p ≤ | g | L p | h | L and noticing that | ∂ βy G ˜ ξ,λ ( x , y ) | ≤ Cγ ( ρ β + βe − θy ) e − θρ | x − y | , TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 15 we obtain(3.14) | ( L ˜ ξ − λ ) − ∂ βx ˆ f | L p ( x ) = (cid:12)(cid:12)(cid:12) Z ∂ βy G ˜ ξ,λ ( x , y ) ˆ f ( y , ˜ ξ ) dy (cid:12)(cid:12)(cid:12) L p ( x ) ≤ (cid:12)(cid:12)(cid:12) Z Cγ ( ρ β + βe − θy ) e − θρ | x − y | | ˆ f ( y , ˜ ξ ) | dy (cid:12)(cid:12)(cid:12) L p ≤ Cγ ρ − /p h ρ β | ˆ f | L ( x ) + β | ˆ f | L ∞ ( x ) i as claimed. (cid:3) Remark 3.4.
The above L p bounds may alternatively be obtained directly by the argu-ment of Section 12, [GMWZ1], using quite different Kreiss symmetrizer techniques, againomitting pole terms arising from vanishing of the Evans function at the origin, and also theauxiliary problem construction of Section 12.6 used to obtain sharpened bounds in the Laxor overcompressive shock case (not relevant here).3.2. Estimates on homogeneous solution operators.
Define low- and high-frequencyparts of the linearized solution operator S ( t ) of the linearized problem with homogeneousboundary and forcing data, f , h ≡
0, as(3.15) S ( t ) := 1(2 πi ) d Z | ˜ ξ |≤ r I Γ ˜ ξ ∩{| λ |≤ r } e λt + i ˜ ξ · ˜ x ( L ˜ ξ − λ ) − dλd ˜ ξ and(3.16) S ( t ) := e Lt − S ( t ) . Then we obtain the following:
Proposition 3.5 (Low-frequency estimate) . Under the hypotheses of Theorem 1.7, for β = ( β , β ′ ) with β = 0 , , (3.17) |S ( t ) ∂ βx f | L x ≤ C (1 + t ) − ( d − / −| β | / | f | L x + Cβ (1 + t ) − ( d − / | f | L , ∞ ˜ x,x , |S ( t ) ∂ βx f | L , ∞ ˜ x,x ≤ C (1 + t ) − ( d +1) / −| β | / | f | L x + Cβ (1 + t ) − ( d +1) / | f | L , ∞ ˜ x,x , |S ( t ) ∂ βx f | L ∞ ˜ x,x ≤ C (1 + t ) − d/ −| β | / | f | L x + Cβ (1 + t ) − d/ | f | L , ∞ ˜ x,x , where | · | L p,q ˜ x,x denotes the norm in L p (˜ x ; L q ( x )) .Proof. The proof will follow closely the treatment of the shock case in [Z3]. Let ˆ u ( x , ˜ ξ, λ )denote the solution of ( L ˜ ξ − λ )ˆ u = ˆ f , where ˆ f ( x , ˜ ξ ) denotes Fourier transform of f , and u ( x, t ) := S ( t ) f = 1(2 πi ) d Z | ˜ ξ |≤ r I Γ ˜ ξ ∩{| λ |≤ r } e λt + i ˜ ξ · ˜ x ( L ˜ ξ − λ ) − ˆ f ( x , ˜ ξ ) dλd ˜ ξ. Recalling the resolvent estimates in Proposition 3.3, we have | ˆ u ( x , ˜ ξ, λ ) | L p ( x ) ≤ Cγ ρ − /p | ˆ f | L ( x ) ≤ Cγ ρ − /p | f | L ( x ) where γ is as defined in (2.24). Therefore, using Parseval’s identity, Fubini’s theorem, and the triangle inequality, wemay estimate | u | L ( x , ˜ x ) ( t ) = 1(2 π ) d Z x Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e λt ˆ u ( x , ˜ ξ, λ ) dλ (cid:12)(cid:12)(cid:12) d ˜ ξdx = 1(2 π ) d Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e λt ˆ u ( x , ˜ ξ, λ ) dλ (cid:12)(cid:12)(cid:12) L ( x ) d ˜ ξ ≤ π ) d Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e ℜ eλt | ˆ u ( x , ˜ ξ, λ ) | L ( x ) dλ (cid:12)(cid:12)(cid:12) d ˜ ξ ≤ C | f | L ( x ) Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e ℜ eλt γ ρ − dλ (cid:12)(cid:12)(cid:12) d ˜ ξ. Specifically, parametrizing Γ ˜ ξ by λ ( ˜ ξ, k ) = ik − θ ( k + | ˜ ξ | ) , k ∈ R , and observing that by (2.24),(3.18) γ ρ − ≤ ( | k | + | ˜ ξ | ) − h X j (cid:16) | k − τ j ( ˜ ξ ) | ρ (cid:17) /s j − i ≤ ( | k | + | ˜ ξ | ) − h X j (cid:16) | k − τ j ( ˜ ξ ) | ρ (cid:17) ǫ − i , where ǫ := j s j (0 < ǫ < Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e ℜ eλt γ ρ − dλ (cid:12)(cid:12)(cid:12) d ˜ ξ ≤ Z ˜ ξ (cid:12)(cid:12)(cid:12) Z R e − θ ( k + | ˜ ξ | ) t γ ρ − dk (cid:12)(cid:12)(cid:12) d ˜ ξ ≤ Z ˜ ξ e − θ | ˜ ξ | t | ˜ ξ | − ǫ (cid:12)(cid:12)(cid:12) Z R e − θ k t | k | ǫ − dk (cid:12)(cid:12)(cid:12) d ˜ ξ + X j Z ˜ ξ e − θ | ˜ ξ | t | ˜ ξ | − ǫ (cid:12)(cid:12)(cid:12) Z R e − θ k t | k − τ j ( ˜ ξ ) | ǫ − dk (cid:12)(cid:12)(cid:12) d ˜ ξ ≤ Z ˜ ξ e − θ | ˜ ξ | t | ˜ ξ | − ǫ (cid:12)(cid:12)(cid:12) Z R e − θ k t | k | ǫ − dk (cid:12)(cid:12)(cid:12) d ˜ ξ ≤ Ct − ( d − / . Likewise, we have | u | L , ∞ ˜ x,x ( t ) = 1(2 π ) d Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e λt ˆ u ( x , ˜ ξ, λ ) dλ (cid:12)(cid:12)(cid:12) L ∞ ( x ) d ˜ ξ ≤ π ) d Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e ℜ eλt | ˆ u ( x , ˜ ξ, λ ) | L ∞ ( x ) dλ (cid:12)(cid:12)(cid:12) d ˜ ξ ≤ C | f | L ( x ) Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e ℜ eλt γ dλ (cid:12)(cid:12)(cid:12) d ˜ ξ TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 17 where Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e ℜ eλt γ dλ (cid:12)(cid:12)(cid:12) d ˜ ξ ≤ Z ˜ ξ e − θ | ˜ ξ | t (cid:12)(cid:12)(cid:12) Z R e − θ k t dk (cid:12)(cid:12)(cid:12) d ˜ ξ + X j Z ˜ ξ e − θ | ˜ ξ | t | ˜ ξ | − ǫ (cid:12)(cid:12)(cid:12) Z R e − θ k t | k − τ j ( ˜ ξ ) | ǫ − dk (cid:12)(cid:12)(cid:12) d ˜ ξ ≤ Ct − ( d +1) / + C Z ˜ ξ e − θ | ˜ ξ | t | ˜ ξ | − ǫ (cid:12)(cid:12)(cid:12) Z R e − θ k t | k | ǫ − dk (cid:12)(cid:12)(cid:12) d ˜ ξ ≤ Ct − ( d +1) / . Similarly, we estimate | u | L ∞ ˜ x,x ( t ) ≤ π ) d Z ˜ ξ (cid:12)(cid:12)(cid:12) I Γ ˜ ξ ∩{| λ |≤ r } e λt ˆ u ( x , ˜ ξ, λ ) dλ (cid:12)(cid:12)(cid:12) L ∞ ( x ) d ˜ ξ ≤ π ) d Z ˜ ξ I Γ ˜ ξ ∩{| λ |≤ r } e ℜ eλt | ˆ u ( x , ˜ ξ, λ ) | L ∞ ( x ) dλd ˜ ξ ≤ C | f | L ( x ) Z ˜ ξ I Γ ˜ ξ ∩{| λ |≤ r } e ℜ eλt γ dλd ˜ ξ where as above we have Z ˜ ξ I Γ ˜ ξ ∩{| λ |≤ r } e ℜ eλt γ dλd ˜ ξ ≤ Z ˜ ξ e − θ | ˜ ξ | t Z R e − θ k t dkd ˜ ξ + X j Z ˜ ξ e − θ | ˜ ξ | t | ˜ ξ | − ǫ Z R e − θ k t | k − τ j ( ˜ ξ ) | ǫ − dkd ˜ ξ ≤ Ct − d/ + C Z ˜ ξ e − θ | ˜ ξ | t | ˜ ξ | − ǫ Z R e − θ k t | k | ǫ − dkd ˜ ξ ≤ Ct − d/ . The x -derivative bounds follow similarly by using the resolvent bounds in Proposition3.3 with β = 1. The ˜ x -derivative bounds are straightforward by the fact that d ∂ ˜ β ˜ x f = ( i ˜ ξ ) ˜ β ˆ f .Finally, each of the above integrals is bounded by C | f | L ( x ) as the product of | f | L ( x ) times the integral quantities γ ρ − , γ over a bounded domain, hence we may replace t by(1 + t ) in the above estimates. (cid:3) Next, we obtain estimates on the high-frequency part S ( t ) of the linearized solutionoperator. Recall that S ( t ) = S ( t ) − S ( t ), where S ( t ) = 1(2 πi ) d Z R d − e i ˜ ξ · ˜ x e L ˜ ξ t d ˜ ξ and S ( t ) = 1(2 πi ) d Z | ˜ ξ |≤ r I Γ ˜ ξ ∩{| λ |≤ r } e λt + i ˜ ξ · ˜ x ( L ˜ ξ − λ ) − dλd ˜ ξ. Then according to [Z4, Corollary 4.11], we can write(3.19) S ( t ) f = 1(2 πi ) d P.V. Z − θ + i ∞− θ − i ∞ Z R d − χ | ˜ ξ | + |ℑ mλ | ≥ θ + θ × e i ˜ ξ · ˜ x + λt ( λ − L ˜ ξ ) − ˆ f ( x , ˜ ξ ) d ˜ ξdλ. Proposition 3.6 (High-frequency estimate) . Given (A1)-(A2), (H0)-(H2), (D), and ho-mogeneous boundary conditions (B), for ≤ | α | ≤ s − , s as in (H0), (3.20) |S ( t ) f | L x ≤ Ce − θ t | f | H x , | ∂ αx S ( t ) f | L x ≤ Ce − θ t | f | H | α | +3 x . Proof.
The proof starts with the following resolvent identity, using analyticity on the resol-vent set ρ ( L ˜ ξ ) of the resolvent ( λ − L ˜ ξ ) − , for all f ∈ D ( L ˜ ξ ),(3.21) ( λ − L ˜ ξ ) − f = λ − ( λ − L ˜ ξ ) − L ˜ ξ f + λ − f. Using this identity and (3.19), we estimate(3.22) S ( t ) f = 1(2 πi ) d P.V. Z − θ + i ∞− θ − i ∞ Z R d − χ | ˜ ξ | + |ℑ mλ | ≥ θ + θ × e i ˜ ξ · ˜ x + λt λ − ( λ − L ˜ ξ ) − L ˜ ξ ˆ f ( x , ˜ ξ ) d ˜ ξdλ + 1(2 πi ) d P.V. Z − θ + i ∞− θ − i ∞ Z R d − χ | ˜ ξ | + |ℑ mλ | ≥ θ + θ × e i ˜ ξ · ˜ x + λt λ − ˆ f ( x , ˜ ξ ) d ˜ ξdλ =: S + S , where, by Plancherel’s identity and Propositions 3.6 and 3.2, we have | S | L (˜ x,x ) ≤ C Z − θ + i ∞− θ − i ∞ | λ | − | e λt || ( λ − L ˜ ξ ) − L ˜ ξ ˆ f | L (˜ ξ,x ) | dλ |≤ Ce − θ t Z − θ + i ∞− θ − i ∞ | λ | − / (cid:12)(cid:12)(cid:12) (1 + | ˜ ξ | ) | L ˜ ξ ˆ f | H ( x ) (cid:12)(cid:12)(cid:12) L (˜ ξ ) | dλ |≤ Ce − θ t | f | H x and(3.23) | S | L x ≤ π ) d (cid:12)(cid:12)(cid:12) P.V. Z − θ + i ∞− θ − i ∞ λ − e λt dλ Z R d − e i ˜ x · ˜ ξ ˆ f ( x , ˜ ξ ) d ˜ ξ (cid:12)(cid:12)(cid:12) L + 1(2 π ) d (cid:12)(cid:12)(cid:12) P.V. Z − θ + ir − θ − ir λ − e λt dλ Z R d − e i ˜ x · ˜ ξ ˆ f ( x , ˜ ξ ) d ˜ ξ (cid:12)(cid:12)(cid:12) L ≤ Ce − θ t | f | L x , by direct computations, noting that the integral in λ in the first term is identically zero.This completes the proof of the first inequality stated in the proposition. Derivative boundsfollow similarly. (cid:3) TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 19
Remark 3.7.
Here, we have used the λ / improvement in (3.5) over (3.4) together withmodifications introduced in [KZ] to greatly simplify the original high-frequency argumentgiven in [Z3] for the shock case.3.3. Boundary estimates.
For the purpose of studying the nonzero boundary perturba-tion, we need the following proposition. For h := h (˜ x, t ), define(3.24) D h ( t ) := ( | h t | + | h ˜ x | + | h ˜ x ˜ x | )( t ) , and(3.25) Γ h ( t ) := Z t Z R d − (cid:16) X k G y k B k + GA (cid:17) ( x, t − s ; 0 , ˜ y ) h (˜ y, s ) d ˜ yds, where G ( x, t ; y ) is the Green function of ∂ t − L . This boundary term will appear when wewrite down the Duhamel formulas for the linearized and nonlinear equations (see (3.37) and(4.55)). Noting that for the outflow case, the fact that G ( x, t ; 0 , ˜ y ) ≡ h to(3.26) Γ h ( t ) = Z t Z R d − G y ( x, t − s ; 0 , ˜ y ) B h d ˜ yds. Therefore when dealing with the outflow case, instead of putting assumptions on h itselfas in the inflow case, we make assumptions on B h , matching with the hypotheses on W -coordinates. Proposition 3.8.
Assume that h = h (˜ x, t ) satisfies (3.27) | h ( t ) | L x ≤ E (1 + t ) − ( d +1) / , | h ( t ) | L ∞ ˜ x ≤ E (1 + t ) − d/ |D h ( t ) | L x ∩ H | γ | +3˜ x ≤ E (1 + t ) − d/ − ǫ , for some positive constant E ; here | γ | = [( d − /
2] + 2 , and ǫ > is arbitrary small for d = 2 and ǫ = 0 for d ≥ . For the outflow case, we replace these assumptions on h bythose on B h . Then we obtain (3.28) | Γ h ( t ) | L ≤ CE (1 + t ) − ( d − / , | Γ h ( t ) | L , ∞ ˜ x,x ≤ CE (1 + t ) − ( d +1) / , | Γ h ( t ) | L ∞ ≤ CE (1 + t ) − d/ , and derivative bounds (3.29) | ∂ x Γ h ( t ) | L , ∞ ˜ x,x ≤ CE (1 + t ) − ( d +1) / , | ∂ x Γ h ( t ) | L , ∞ ˜ x,x ≤ CE (1 + t ) − ( d +1) / , for all t ≥ .Proof. We first recall that G ( x, t − s ; y ) is a solution of ( ∂ s − L y ) ∗ G ∗ = 0, that is,(3.30) − G s − X j ( GA j ) y j + X j GA jy j = X jk ( G y k B kj ) y j . Integrating this on R d + × [0 , t ] against(3.31) g ( y , ˜ y, s ) := e − y h (˜ y, s ) , and integrating by parts twice, we obtainΓ h = − Z t Z R d + (cid:16) X jk G y k B kj + X j GA j (cid:17) g y j dyds − Z t Z R d + (cid:16) − G s + X j GA jy j (cid:17) g ( y, s ) dyds where, recalling that S ( t ) f ( x ) = Z R d + G ( x, t ; y ) f ( y ) dy, we get − Z t Z R d + X jk (cid:16) G y k B kj + X j GA j (cid:17) g y j dyds = − Z t S ( t − s ) (cid:16) − X jk ( B kj g x j ) x k + X j A j g x j (cid:17) ds and − Z t Z R d + (cid:16) − G s + X j GA jy j (cid:17) g ( y, s ) dyds = − Z t S ( t − s ) (cid:16) g s + X j A jx j g (cid:17) ds + g ( x, t ) − S ( t ) g ( x, . Therefore combining all these estimates yields(3.32) Γ h = g ( x, t ) − S ( t ) g − Z t S ( t − s )( g s − L x g ( x, s )) ds with g ( x ) := g ( x,
0) and L x g = − P j ( A j g ) x j + P jk ( B jk g x k ) x j . Now we are ready to employ estimates obtained in the previous section on the solutionoperator S ( t ) = S ( t ) + S ( t ). Noting that | g | L px ≤ C | h | L p ˜ x , TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 21 we estimate | Γ h | L ≤ | g | L + |S ( t ) g | L + |S ( t ) g | L + Z t |S ( t − s )( g s − Lg ) | L + |S ( t − s )( g s − Lg ) | L ds ≤ | h ( t ) | L x + C (1 + t ) − d − | g | L + Ce − ηt | g | H + Z t (1 + t − s ) − ( d − / ( | g s | + | Lg | ) L + e − θ ( t − s ) ( | g s | + | Lg | ) H ds ≤ | h ( t ) | L x + C (1 + t ) − d − | h | L x ∩ H x + Z t (1 + t − s ) − ( d − / |D h ( s ) | L x + e − θ ( t − s ) |D h ( s ) | H x ds ≤ CE (1 + t ) − d − and similarly we also obtain(3.33) | Γ h | L , ∞ ˜ x,x ≤ | h ( t ) | L x + C (1 + t ) − d +14 | h | L x ∩ H x + C Z t (1 + t − s ) − ( d +1) / |D h ( s ) | L x + e − θ ( t − s ) |D h ( s ) | H x ds ≤ CE (1 + t ) − d +14 and(3.34) | Γ h | L ∞ ≤ | h ( t ) | L ∞ ˜ x + C (1 + t ) − d | h | L x ∩ H | γ | +3˜ x + C Z t (1 + t − s ) − d/ |D h ( s ) | L x + e − θ ( t − s ) |D h ( s ) | H | γ | +3˜ x ds ≤ CE (1 + t ) − d . Similar bounds hold for derivatives.This completes the proof of the proposition. (cid:3)
Duhamel formula.
The following integral representation formula expresses the solu-tion of the inhomogeneous equation (3.1) in terms of the homogeneous solution operator S for f , h ≡ Lemma 3.9 (Integral formulation) . Solutions U of (3.1) may be expressed as (3.35) U ( x, t ) = S ( t ) U + Z t S ( t − s ) f ( · , s ) + Γ U (0 , ˜ x, t ) where U ( x,
0) = U ( x ) , (3.36) Γ U (0 , ˜ x, t ) := Z t Z R d − ( X j G y j B j + GA )( x, t − s ; 0 , ˜ y ) U (0 , ˜ y, s ) d ˜ yds, and G ( · , t ; y ) = S ( t ) δ y ( · ) is the Green function of ∂ t − L . Proof.
Integrating on R d + the linearized equations( ∂ s − L y ) U = f against G ( x, t − s ; y ) and using the fact that by duality( ∂ s − L y ) ∗ G ∗ ( x, t − s ; y ) ≡ , we easily obtain the lemma as in the one-dimensional case (see [YZ, NZ]), recalling that S ( t ) f = Z R d + G ( x, t ; y ) f ( y ) dy. (cid:3) Proof of linearized stability.
Proof of Theorem 1.6.
Writing the Duhamel formula for the linearized equations(3.37) U ( x, t ) = S ( t ) U + Γ h (˜ x, t ) , with Γ h defined in (3.25), where U ( x,
0) = U ( x ) and U (0 , ˜ x, t ) = h (˜ x, t ), and applyingestimates on low- and high-frequency operators S ( t ) and S ( t ), we obtain(3.38) | U ( t ) | L ≤ |S ( t ) U | L + |S ( t ) U | L + | Γ h ( t ) | L ≤ C (1 + t ) − d − | U | L + Ce − ηt | U | H + CE (1 + t ) − ( d − / ≤ C (1 + t ) − d − ( | U | L ∩ H + E )and(3.39) | U ( t ) | L ∞ ≤ |S ( t ) U | L ∞ + |S ( t ) U | L ∞ + | Γ h ( t ) | L ∞ ≤ C (1 + t ) − d | U | L + C |S ( t ) U | H [( d − / + CE (1 + t ) − d/ ≤ C (1 + t ) − d | U | L + Ce − ηt | U | H [( d − / + CE (1 + t ) − d/ ≤ C (1 + t ) − d ( | U | L ∩ H [( d − / + E ) . These prove the bounds as stated in the theorem for p = 2 and p = ∞ . For 2 < p < ∞ , weuse the interpolation inequality between L and L ∞ . (cid:3) Nonlinear stability
Auxiliary energy estimates.
For the analysis of nonlinear stability, we need thefollowing energy estimate adapted from [MaZ4, NZ, Z4]. Define the nonlinear perturbationvariables U = ( u, v ) by(4.1) U ( x, t ) := ˜ U ( x, t ) − ¯ U ( x ) . Proposition 4.1.
Under the hypotheses of Theorem 1.7, let U ∈ H s and U = ( u, v ) T be asolution of (1.2) and (4.1) . Suppose that, for ≤ t ≤ T , the W , ∞ x norm of the solution U remains bounded by a sufficiently small constant ζ > . Then | U ( t ) | H s ≤ Ce − θt | U | H s + C Z t e − θ ( t − τ ) (cid:16) | U ( τ ) | L + |B h ( τ ) | (cid:17) dτ (4.2) for all ≤ t ≤ T , where the boundary term B h is defined as in Theorem 1.7. TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 23
Proof.
Observe that a straightforward calculation shows that | U | H r ∼ | W | H r , (4.3) W = ˜ W − ¯ W := W ( ˜ U ) − W ( ¯ U ) , for 0 ≤ r ≤ s , provided | U | W , ∞ remains bounded, hence it is sufficient to prove a corre-sponding bound in the special variable W . We first carry out a complete proof in the morestraightforward case with conditions (A1)-(A3) replaced by the following global versions,indicating afterward by a few brief remarks the changes needed to carry out the proof inthe general case.(A1’) ˜ A j ( ˜ W ) , ˜ A , ˜ A are symmetric, ˜ A ≥ θ > W = (cid:18) ˜ w I ˜ w II (cid:19) , ˜ B jk = ˜ B kj = (cid:18) b jk (cid:19) , P ξ j ξ k ˜ b jk ≥ θ | ξ | , and ˜ G ≡ A W t + X j A j W x j = X jk ( B jk W x k ) x j + M ¯ W x + X j ( M j ¯ W x ) x j (4.4)where A := A ( W + ¯ W ) is symmetric positive definite, A j := A j ( W + ¯ W ) are symmetric, M = A ( W + ¯ W ) − A ( ¯ W ) = (cid:16) Z dA ( ¯ W + θW ) dθ (cid:17) W,M j = B j ( W + ¯ W ) − B j ( ¯ W ) = (cid:18) R db j ( ¯ W + θW ) dθ ) W (cid:19) . As shown in [MaZ4], we have bounds | A | ≤ C, | A t | ≤ C | W t | ≤ C ( | W x | + | w IIxx | ) ≤ Cζ, (4.5) | ∂ x A | + | ∂ x A | ≤ C ( X k =1 | ∂ kx W | + | ¯ W x | ) ≤ C ( ζ + | ¯ W x | ) . (4.6)We have the same bounds for A j , B jk , and also due to the form of M , M , | M | , | M | ≤ C ( ζ + | ¯ W x | ) | W | . (4.7)Note that thanks to Lemma 1.3 we have the bound on the profile: | ¯ W x | ≤ Ce − θ | x | , as x → + ∞ .The following results assert that hyperbolic effects can compensate for degenerate vis-cosity B , as revealed by the existence of a compensating matrix K . Lemma 4.2 ([KSh]) . Assuming (A1’), condition (A2’) is equivalent to the following:(K1) There exist smooth skew-symmetric “compensating matrices” K ( ξ ) , homogeneousdegree one in ξ , such that (4.8) ℜ e (cid:16) X j,k ξ j ξ k B jk − K ( ξ )( A ) − X k ξ k A k (cid:17) ( W + ) ≥ θ | ξ | > for all ξ ∈ R d \ { } . Define α by the ODE(4.9) α x = − sign( A ) c ∗ | ¯ W x | α, α (0) = 1where c ∗ > α x /α ) A ≤ − c ∗ θ | ¯ W x | =: − ω ( x )(4.10)and | α x /α | ≤ c ∗ | ¯ W x | = θ − ω ( x ) . (4.11)In what follows, we shall use h· , ·i as the α -weighted L inner product defined as h f, g i = h αf, g i L ( R d + ) and k f k s = s X i =0 X | α | = i D ∂ αx f, ∂ αx f E / as the norm in weighted H s space. Note that for any symmetric operator S , h Sf x j , f i = − h S x j f, f i , j = 1 h Sf x , f i = − h ( S x + ( α x /α ) S ) f, f i − h Sf, f i , where h· , ·i denotes the integration on R d := { x = 0 } × R d − . Also we define k f k ,s = k f k H s ( R d ) = s X i =0 X | α | = i D ∂ α ˜ x f, ∂ α ˜ x f E / . Note that in what follows, we shall pay attention to keeping track of c ∗ . For constantsindependent of c ∗ , we simply write them as C . Also, for simplicity, the sum symbol willsometimes be dropped where it is no confusion. We write k f x k = P j k f x j k and k ∂ kx f k = P | α | = k k ∂ αx f k .4.1.1. Zeroth order “Friedrichs-type” estimate.
First, by integration by parts and estimates(4.5), (4.6), and then (4.10), we obtain for j = 1, −h A j W x j , W i = 12 h A jx j W, W i ≤ C h ( ζ + | ¯ W x | ) w I , w I i + C k w II k and for j = 1, −h A W x , W i = 12 h ( A x + ( α x /α ) A ) W, W i + 12 h A W, W i ≤ h ( α x /α ) A w I , w I i + C h ( ζ + | ¯ W x | ) | W | + ω ( x ) | w II | , | W |i + J b ≤ − h ω ( x ) w I , w I i + C h ( ζ + | ¯ W x | ) w I , w I i + C ( c ∗ ) k w II k + J b , TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 25 where J b denotes the boundary term h A W, W i . The term h| ¯ W x | w I , w I i may be easilyabsorbed into the first term of the right-hand side, since for c ∗ sufficiently large,(4.12) h| ¯ W x | w I , w I i ≤ ( c ∗ θ ) − h ω ( x ) w I , w I i ≤ C h ω ( x ) w I , w I i . Also, integration by parts yields h ( B jk W x k ) x j , W i = −h B jk W x k , W x j i − h ( α x /α ) B k W x k , W i − h B k W x k , W i ≤ − θ k w IIx k + C h ω ( x ) w IIx , w II i − h b k w IIx k , w II i ≤ − θ k w IIx k + C ( c ∗ ) k w II k − h b k w IIx k , w II i . where we used the fact that B jk W x · W = b jk w IIx · w II , noting that B has block-diagonal formwith the first block identical to zero. Similarly, recalling that M j = B j ( W + ¯ W ) − B j ( ¯ W ),we have h ( M j ¯ W x ) x j , W i = −h M j ¯ W x , W x j i − h ( α x /α ) M ¯ W x , W i − h M ¯ W x , W i ≤ C h| ¯ W x || W | , | w IIx |i + C h ω ( x ) | W | , w II i − h m ¯ W x , w II i ≤ ξ k w IIx k + C (cid:16) ǫ h ω ( x ) w I , w I i + C ( c ∗ ) k w II k (cid:17) − h m ¯ W x , w II i for any small ξ, ǫ . Note that C is independent of c ∗ . Therefore, for ξ = θ/ c ∗ sufficientlylarge, combining all above estimates, we obtain(4.13)12 ddt h A W, W i = h A W t , W i + 12 h A t W, W i = h− A j W x j + ( B jk W x k ) x j + M ¯ W x + ( M j ¯ W x ) x j , W i + 12 h A t W, W i≤ −
14 [ h ω ( x ) w I , w I i + θ k w IIx k ] + Cζ k w I k + C ( c ∗ ) k w II k + I b where the boundary term(4.14) I b := 12 h A W, W i − h b k w IIx k , w II i − h m ¯ W x , w II i which, in the outflow case (thanks to the negative definiteness of A ), is estimated as(4.15) I b ≤ − θ k w I k , + C ( k w II k , + k w IIx k , k w II k , ) , and similarly in the inflow case, estimated as(4.16) I b ≤ C ( k W k , + k w IIx k , k w II k , ) . Here we recall that k · k ,s := k · k H s ( R d ) . First order “Friedrichs-type” estimate.
Similarly as above, we need the following keyestimate, computing by the use of integration by parts, (4.12), and c ∗ being sufficientlylarge,(4.17) − X j h W x i , A j W x i x j i = 12 X j h W x i , A jx j W x i i + 12 h W x i , ( α x /α ) A W x i i + 12 h W x i , A W x i i ≤ − h ω ( x ) w Ix , w Ix i + Cζ k w Ix k + Cc ∗ k w IIx k + 12 h W x i , A W x i i . We deal with the boundary term later. Now let us compute12 ddt h A W x i , W x i i = h W x i , ( A W t ) x i i − h W x i , A x i W t i + 12 h A t W x i , W x i i . (4.18)We control each term in turn. By (4.5) and (4.6), we first have h A t W x i , W x i i ≤ Cζ k W x k and by multiplying ( A ) − into (4.4), |h W x i , A x i W t i| ≤ C h ( ζ + | ¯ W x | ) | W x | , ( | W x | + | w IIxx | + | W | ) i≤ ξ k w IIxx k + C h ( ζ + | ¯ W x | ) w Ix , w Ix i + C h ( ζ + | ¯ W x | ) w I , w I i + C k w II k , where the term h| ¯ W x | w Ix , w Ix i may be treated in the same way as was h| ¯ W x | w I , w I i in(4.12). Using (4.4), we write the first term in the right-hand side of (4.18) as h W x i , ( A W t ) x i i = h W x i , [ − A j W x j + ( B jk W x k ) x j + M ¯ W x + ( M j ¯ W x ) x j ] x i i = − h W x i , A j W x i x j i + h W x i , − A jx i W x j + ( M ¯ W x ) x i i− h W x i x j , [( B jk W x k ) x i + ( M j ¯ W x ) x i ] i− h ( α x /α ) W x i , [( B k W x k ) x i + ( M ¯ W x ) x i ] i− h W x i , [( B k W x k ) x i + ( M ¯ W x ) x i ] i ≤ − h h ω ( x ) w Ix , w Ix i + θ k w IIxx k i + C h ζ k w I k + C ( c ∗ ) k w IIx k + h| ¯ W x | w I , w I i i + I b where I b denotes the boundary terms(4.19) I b : = 12 h W x i , A W x i i − h W x i , [( B k W x k ) x i + ( M ¯ W x ) x i ] i = 12 h W x i , A W x i i − h w IIx i , [( b k w IIx k ) x i + ( m ¯ W x ) x i ] i , and we have used (A3) for each fixed i and ξ j = ( W x i ) x j to get(4.20) X jk h W x i x j , B jk W x k x i i ≥ θ X j k W x i x j k , TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 27 and estimates (4.17),(4.12) for w I , w Ix , and Young’s inequality to obtain: h W x , − A jx W x + ( M ¯ W x ) x i ≤ C h ( ζ + | ¯ W x | ) | W x | , | W x | + | W |i . −h W xx + ( α x /α ) W x , ( B jk W x ) x i ≤− θ k w IIxx k + C h| w IIxx | + ω ( x ) | w IIx | , ( ζ + | ¯ W x | ) | w IIx |i−h W xx + ( α x /α ) W x , ( M j ¯ W x ) x i ≤ C h| w IIxx | + ω ( x ) | w IIx | , ( ζ + | ¯ W x | )( | W x | + | W | ) i . Putting these estimates together into (4.18), we have obtained12 ddt h A W x , W x i + 14 θ k w IIxx k + 14 h ω ( x ) w Ix , w Ix i≤ C h ζ k w I k + h| ¯ W x | w I , w I i + C ( c ∗ ) k w II k i + I b . (4.21)Let us now treat the boundary term. First observe that using the parabolic equations,noting that A is the diagonal-block form, we can estimate( b jk w IIx k ) x j (0 , ˜ x, t ) ≤ C (cid:16) | w IIt | + | W x j | + | W | (cid:17) (0 , ˜ x, t )(4.22)and thus for i = 1(4.23) h w IIx i , [( b k w IIx k ) x i + ( m ¯ W x ) x i ] i ≤ Z R d | w IIx i x i | (cid:0) | W | + | w IIx k | (cid:17) ≤ C Z R d (cid:16) | W | + | w IIx | + | w II ˜ x ˜ x | (cid:17) and for i = 1, using b k = b k , (4.22), and recalling here that we always use the sumconvention,(4.24) X k ( b k w IIx k ) x = 12 (cid:16) ( b k w IIx k ) x + ( b j w IIx ) x j + b kx w IIx k − b j x j w IIx (cid:17) = 12 (cid:16) ( b jk w IIx k ) x j + b kx w IIx k − b j x j w IIx − X j =1; k =1 ( b jk w IIx k ) x j (cid:17) ≤ C (cid:0) | w IIt | + | W | + | W x j | + | w II ˜ x ˜ x | (cid:17) . Therefore h w IIx , [( b k w IIx k ) x + ( m ¯ W x ) x ] i ≤ ǫ Z R d | w Ix | + C Z R d (cid:16) | w IIt | + | W | + | w IIx | + | w II ˜ x ˜ x | (cid:17) For the first term in I b , we consider each inflow/outflow case separately. For the outflowcase, since A ≤ − θ <
0, we get A W x · W x ≤ − θ | w Ix | + C | w IIx | . Therefore I b ≤ − θ Z R d | w Ix | + Z R d (cid:16) | W | + | w IIx | + | w IIt | + | w II ˜ x ˜ x | (cid:17) . (4.25)Meanwhile, for the inflow case, since A ≥ θ >
0, we have | A W x · W x | ≤ C | W x | . In this case, the invertibility of A allows us to use the hyperbolic equation to derive | w Ix | ≤ C ( | w It | + | w IIx | + | w I ˜ x | ) . Therefore we get I b ≤ Z R d (cid:16) | W | + | W t | + | w I ˜ x | + | w IIx | + | w II ˜ x ˜ x | (cid:17) . (4.26)Now apply the standard Sobolev inequality(4.27) | w (0) | ≤ C k w k L ( R ) ( k w x k L ( R ) + k w k L ( R ) )to control the term | w IIx (0) | in I b in both cases. We get(4.28) Z R d | w IIx | ≤ ǫ ′ k w IIxx k + C k w IIx k . Using this with ǫ ′ = θ/
8, (4.19), and (4.25), the estimate (4.21) reads(4.29) ddt h A W x , W x i + k w IIxx k + h ω ( x ) w Ix , w Ix i≤ C (cid:16) ζ k w I k + h| ¯ W x | w I , w I i + C ( c ∗ ) k w II k (cid:17) + I b where the (new) boundary term I b satisfies(4.30) I b ≤ − θ Z R d | w Ix | + C Z R d (cid:16) | W | + | w II ˜ x | + | w IIt | + | w II ˜ x ˜ x | (cid:17) for the outflow case, and(4.31) I b ≤ Z R d (cid:16) | W | + | W t | + | W ˜ x | + | w II ˜ x ˜ x | (cid:17) for the inflow case.4.1.3. Higher order “Friedrichs-type” estimate.
For any fixed multi-index α = ( α x , · · · , α x d ), α = 0 , | α | = k = 2 , ..., s , by computing ddt h A ∂ αx W, ∂ αx W i and following the same spiritas the above subsection, we easily obtain(4.32) ddt h A ∂ αx W, ∂ αx W i + θ k ∂ α +1 x w II k + h ω ( x ) ∂ αx w I , ∂ αx w I i≤ C (cid:16) C ( c ∗ ) k w II k k + ζ k w I k k + k − X i =1 h| ¯ W x | ∂ ix w I , ∂ ix w I i (cid:17) + I αb TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 29 where ∂ αx : = ∂ α x · · · ∂ α d x d , ∂ α +1 x := X j ∂ α x · · · ∂ α d x d ∂ x j , ∂ ix = X | β | = i ∂ β x · · · ∂ β d x d and the boundary term I αb satisfies(4.33) I αb ≤ − θ Z R d | ∂ αx w I | + C Z R d (cid:16) [( k +1) / X i =1 | ∂ it w II | + k − X i =0 | ∂ ix w I | + k X i =0 | ∂ i ˜ x w II | (cid:17) for the outflow case, and(4.34) I αb ≤ Z R d (cid:16) k X i =0 | ∂ it w I | + [( k +1) / X i =1 | ∂ it w II | + k X i =0 | ∂ i ˜ x W | (cid:17) for the inflow case.Now for α with α = 2 , ..., s we observe that the estimate (4.32) still holds. Indeed, usingintegration by parts and computing ddt h A ∂ αx W, ∂ αx W i as above leaves the boundary termsas(4.35) I αb := 12 h ∂ αx W, A ∂ αx W i − h ∂ αx w II , ∂ αx [( b k w IIx k ) + ( m ¯ W x )] i . Then we can use the parabolic equations to solve w IIx x = ( b ) − (cid:16) A w IIt + A j W x j − ( b jk w IIx k ) x j − b x w IIx − M ¯ W x − ( m j ¯ W x ) x j (cid:17) . Using this we can reduce the order of derivative with respect to x in ∂ αx to one, with thesame spirit as (4.23) and (4.24). Finally we use the Sobolev embedding similar to (4.28) toobtain the estimate for the normal derivative ∂ x , and get the estimate for I αb as claimed in(4.33) and (4.34).We recall next the following Kawashima-type estimate, presented in [Z3], to bound theterm k w I k k appearing on the left hand side of (4.32).4.1.4. “Kawashima-type” estimate. Let K ( ξ ) be the skew-symmetry in (4.8). Using Plancherel’sidentity and the equations (4.4), we compute(4.36) 12 ddt h K ( ∂ x ) ∂ rx W, ∂ rx i = 12 ddt h iK ( ξ )( iξ ) r ˆ W , ( iξ ) r ˆ W i = h iK ( ξ )( iξ ) r ˆ W , ( iξ ) r ˆ W t i = h ( iξ ) r ˆ W , − K ( ξ )( A ) − X j ξ j A j + ( iξ ) r ˆ W i + h iK ( ξ )( iξ ) r ˆ W , ( iξ ) r ˆ H i , where(4.37) H := X j (cid:16) ( A ) − A j + − ( A ) − A j (cid:17) W x j + ( A ) − (cid:16) X jk ( B jk W x k ) x j + M ¯ W x + X j ( M j ¯ W x ) x j (cid:17) . By using the fact that | ( A ) − A j + − ( A ) − A j | = O ( ζ + | ¯ W x | ), we can easily obtain k ∂ rx H k ≤ C k w II k r +2 + C r +1 X k =0 h ( ζ + | ¯ W x | ) ∂ kx w I , ∂ kx w I i . Meanwhile, applying (4.8) into the first term of the last line in (4.36), we get h ( iξ ) r ˆ W , − K ( ξ )( A ) − X j ξ j A j + ( iξ ) r ˆ W i≥ θ k| ξ | r +1 ˆ W k − C k| ξ | r +1 ˆ w II k = θ k ∂ r +1 x w I k − C k ∂ r +1 x w II k . Putting these estimates together into (4.36), we have obtained the high order “Kawashima-type” estimate:(4.38) ddt h K ( ∂ x ) ∂ rx W, ∂ rx W i ≤ − θ k ∂ r +1 x w I k + C k w II k r +2 + C r +1 X i =0 h ( ζ + | ¯ W x | ) ∂ ix w I , ∂ ix w I i Final estimates.
We are ready to conclude our result. First combining the estimate(4.29) with (4.13), we easily obtain12 ddt (cid:16) h A W x , W x i + M h A W, W i (cid:17) ≤ − (cid:16) θ k w IIxx k + 14 h ω ( x ) w Ix , w Ix i (cid:17) + C (cid:16) ζ k w I k + h| ¯ W x | w I , w I i + C ( c ∗ ) k w II k (cid:17) + I b − M (cid:16) h ω ( x ) w I , w I i + θ k w IIx k (cid:17) + CM ζ k w I k + M C ( c ∗ ) k w II k + M I b By choosing M sufficiently large such that M θ ≫ C ( c ∗ ), and noting that c ∗ θ | ¯ W x | ≤ ω ( x ), we get(4.39) 12 ddt (cid:16) h A W x , W x i + M h A W, W i (cid:17) ≤ − (cid:16) θ k w II k + h ω ( x ) w I , w I i + h ω ( x ) w Ix , w Ix i (cid:17) + C (cid:16) ζ k w I k + C ( c ∗ ) k w II k (cid:17) + I b + M I b . We shall treat the boundary terms later. Now we use the estimate (4.38) (for r = 0) toabsorb the term k ∂ x w I k into the left hand side. Indeed, fixing c ∗ large as above, adding(4.39) with (4.38) times ǫ , and choosing ǫ, ζ sufficiently small such that ǫC ( c ∗ ) ≪ θ, ǫ ≪ ζ ≪ ǫθ , we obtain TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 31 ddt (cid:16) h A W x , W x i + M h A W, W i + ǫ h KW x , W i (cid:17) ≤ − (cid:16) θ k w II k + h ω ( x ) w I , w I i + h ω ( x ) w Ix , w Ix i (cid:17) + C (cid:16) ζ k w I k + C ( c ∗ ) k w II k (cid:17) − θ ǫ k w Ix k + Cǫ (cid:16) k w II k + ζ k w I k + h ω ( x ) w I , w I i + h ω ( x ) w Ix , w Ix i (cid:17) + I b + M I b ≤ − (cid:16) θ k w II k + θ ǫ k w Ix k (cid:17) + C ( c ∗ ) k W k + I b where I b := I b + M I b .In view of boundary terms I b and I b , we treat the term I b in each inflow/outflow caseseparately. Recall the inequality (4.28), k w IIx k , ≤ C k w II k . Thus, using this, for theinflow case we have(4.40) I b ≤ C ( k W k , + k w IIx k , k w II k , ) ≤ C ( k W k , + k w II ˜ x k , + ǫ k w II k )and for the outflow case,(4.41) I b ≤ − θ k w I k , + C ( k w II k , + k w IIx k , k w II k , ) ≤ − θ k w I k , + C ( k w II k , + k w II ˜ x k , + ǫ k w II k ) . Therefore these together with (4.30) and (4.31), using the good estimate of k w IIxx k , yield(4.42) I b ≤ − θ Z R d ( | w I | + | w Ix | ) + C Z R d (cid:16) | w II | + | w II ˜ x | + | w IIt | + | w II ˜ x ˜ x | (cid:17) for the outflow case, and(4.43) I b ≤ Z R d (cid:16) | W | + | W t | + | W ˜ x | + | w II ˜ x ˜ x | (cid:17) for the inflow case.Now by Cauchy-Schwarz’s inequality, | K ( ξ ) | ≤ C | ξ | , and positive definiteness of A , it iseasy to see that(4.44) E : = h A W x , W x i + M h A W, W i + ǫ h K ( ∂ x ) W, W i ∼ k W k H α ∼ k W k H . The last equivalence is due to the fact that α is bounded above and below away from zero.Thus the above yields ddt E ( W )( t ) ≤ − θ E ( W )( t ) + C ( c ∗ ) (cid:16) k W ( t ) k L + |B ( t ) | (cid:17) , for some positive constant θ , which by the Gronwall inequality implies(4.45) k W ( t ) k H ≤ Ce − θt k W k H + C ( c ∗ ) Z t e − θ ( t − τ ) (cid:16) k W ( τ ) k L + |B ( τ ) | (cid:17) dτ, where W ( x,
0) = W ( x ) and(4.46) |B ( τ ) | := Z R d (cid:16) | W | + | W t | + | W ˜ x | + | w II ˜ x ˜ x | (cid:17) for the inflow case, and(4.47) |B ( τ ) | := Z R d (cid:16) | w II | + | w II ˜ x | + | w IIt | + | w II ˜ x ˜ x | (cid:17) for the outflow case.Similarly, by induction, we can derive the same estimates for W in H s . To do that, letus define E ( W ) := h A W x , W x i + M h A W, W i + ǫ h KW x , W iE k ( W ) := h A ∂ kx W, ∂ kx W i + M E k − ( W ) + ǫ h K∂ kx W, ∂ k − x W i , k ≤ s. Then similarly by the Cauchy-Schwarz inequality, E s ( W ) ∼ k W k H s , and by induction,we obtain ddt E s ( W )( t ) ≤ − θ E s ( W )( t ) + C ( c ∗ )( k W ( t ) k L + |B h ( t ) | ) , for some positive constant θ , which by the Gronwall inequality yields(4.48) k W ( t ) k H s ≤ Ce − θt k W k H s + C ( c ∗ ) Z t e − θ ( t − τ ) ( k W ( τ ) k L + |B h ( τ ) | ) dτ, where W ( x,
0) = W ( x ), and B h are defined as in (1.14) and (1.15).4.1.6. The general case.
Following [MaZ4, Z3], the general case that hypotheses (A1)-(A3)hold can easily be covered via following simple observations. First, we may express matrix A in (4.4) as(4.49) A j ( W + ¯ W ) = ˆ A j + ( ζ + | ¯ W x | ) (cid:18) O (1) O (1) O (1) (cid:19) , where ˆ A j is a symmetric matrix obeying the same derivative bounds as described for A j ,ˆ A identical to A in the 11 block and obtained in other blocks kl by(4.50) A kl ( W + ¯ W ) = A kl ( ¯ W ) + A kl ( W + ¯ W ) − A kl ( ¯ W )= A kl ( W + ) + O ( | W x | + | ¯ W x | )= A kl ( W + ) + O ( ζ + | ¯ W x | )and meanwhile, ˆ A j , j = 1, obtained by A j = A j ( W + ) + O ( ζ + | ¯ W x | ), similarly as in (4.50).Replacing A j by ˆ A j in the k th order Friedrichs-type bounds above, we find that theresulting error terms may be expressed as h ∂ kx O ( ζ + | ¯ W x | ) | W | , | ∂ k +1 x w II |i , plus lower order terms, easily absorbed using Young’s inequality, and boundary terms O ( k X i =0 | ∂ ix w II (0) || ∂ kx w I (0) | ) TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 33 resulting from the use of integration by parts as we deal with the 12-block. However theseboundary terms were already treated somewhere as before. Hence we can recover the sameFriedrichs-type estimates obtained above. Thus we may relax ( A ′ ) to ( A A ′ ) to ( A B jk = B kj isnot necessary. Indeed, by writing X jk ( B jk W x k ) x j = X jk (cid:16)
12 ( B jk + B kj ) W x k (cid:17) x j + 12 X jk ( B jk − B kj ) x j W x k , we can just replace B jk by ˜ B jk := ( B jk + B kj ), satisfying the same ( A ′ ), and thus stillobtain the energy estimates as before, with a harmless error term (last term in the aboveidentity). Next notice that the term g ( ˜ W x ) − g ( ¯ W x ) in the perturbation equation may beTaylor expanded as (cid:18) g ( ˜ W x , ¯ W x ) + g ( ¯ W x , ˜ W x ) (cid:19) + (cid:18) O ( | W x | ) (cid:19) The first term, since it vanishes in the first component and since | ¯ W x | decays at plus spatialinfinity, yields by Young’s inequality the estimate D (cid:18) g ( ˜ W x , ¯ W x ) + g ( ¯ W x , ˜ W x ) (cid:19) , (cid:18) w Ix w IIx (cid:19) E ≤ C (cid:16) h ( ζ + | ¯ W x | ) w Ix , w Ix i + k w IIx k (cid:17) which can be treated in the Friedrichs-type estimates. The (0 , O ( | W x | ) T nonlinear termmay be treated as other source terms in the energy estimates. Specifically, the worst-caseterm D ∂ kx W,∂ kx (cid:18) O ( | W x | ) (cid:19) E = −h ∂ k +1 x w II , ∂ k − x O ( | W x | ) i − ∂ kx w II (0) ∂ k − x O ( | W x | )(0)may be bounded by k ∂ k +1 x w II k L k W k W , ∞ k W k H k − ∂ kx w II (0) ∂ k − x O ( | W x | )(0) . The boundary term will contribute to energy estimates in the form (4.35) of I αb , and thuswe may use the parabolic equations to get rid of this term as we did in (4.23), (4.24). Thus,we may relax ( A ′ ) to ( A A − ( A
3) and theproposition. (cid:3)
Proof of nonlinear stability.
Defining the perturbation variable U := ˜ U − ¯ U , weobtain the nonlinear perturbation equations(4.51) U t − LU = X j Q j ( U, U x ) x j , where(4.52) Q j ( U, U x ) = O ( | U || U x | + | U | ) Q j ( U, U x ) x j = O ( | U || U x | + | U || U xx | + | U x | ) Q j ( U, U x ) x j x k = O ( | U || U xx | + | U x || U xx | + | U x | + | U || U xxx | ) so long as | U | remains bounded.For boundary conditions written in U -coordinates, (B) gives(4.53) h = ˜ h − ¯ h = ( ˜ W ( U + ¯ U ) − ˜ W ( ¯ U ))(0 , ˜ x, t )= ( ∂ ˜ W /∂ ˜ U )( ¯ U ) U (0 , ˜ x, t ) + O ( | U (0 , ˜ x, t ) | ) . in inflow case, where ( ∂ ˜ W /∂ ˜ U )( ¯ U ) is constant and invertible, and(4.54) h = ˜ h − ¯ h = ( ˜ w II ( U + ¯ U ) − ˜ w II ( ¯ U ))(0 , ˜ x, t )= ( ∂ ˜ w II /∂ ˜ U )( ¯ U ) U (0 , ˜ x, t ) + O ( | U (0 , ˜ x, t ) | )= m (cid:0) ¯ b ¯ b (cid:1) ( ¯ U ) U (0 , ˜ x, t ) + O ( | U (0 , ˜ x, t ) | )= mB ( ¯ U ) U (0 , ˜ x, t ) + O ( | U (0 , ˜ x, t ) | )for some invertible constant matrix m .Applying Lemma 3.9 to (4.51), we obtain(4.55) U ( x, t ) = S ( t ) U + Z t S ( t − s ) X j ∂ x j Q j ( U, U x ) ds + Γ U (0 , ˜ x, t )where U ( x,
0) = U ( x ) , (4.56) Γ U (0 , ˜ x, t ) := Z t Z R d − ( X j G y j B j + GA )( x, t − s ; 0 , ˜ y ) U (0 , ˜ y, s ) d ˜ yds, and G is the Green function of ∂ t − L . Proof of Theorem 1.7.
Define(4.57) ζ ( t ) := sup s (cid:16) | U ( s ) | L x (1 + s ) d − + | U ( s ) | L ∞ x (1 + s ) d + ( | U ( s ) | + | U x ( s ) | + | ∂ x U ( s ) | ) L , ∞ ˜ x,x (1 + s ) d +14 (cid:17) . We shall prove here that for all t ≥ ζ ( t ) uniformlybounded by some fixed, sufficiently small constant, there holds(4.58) ζ ( t ) ≤ C ( | U | L ∩ H s + E + ζ ( t ) ) . This bound together with continuity of ζ ( t ) implies that(4.59) ζ ( t ) ≤ C ( | U | L ∩ H s + E )for t ≥
0, provided that | U | L ∩ H s + E < / C . This would complete the proof of thebounds as claimed in the theorem, and thus give the main theorem.By standard short-time theory/local well-posedness in H s , and the standard principleof continuation, there exists a solution U ∈ H s on the open time-interval for which | U | H s remains bounded, and on this interval ζ ( t ) is well-defined and continuous. Now, let [0 , T )be the maximal interval on which | U | H s remains strictly bounded by some fixed, sufficiently TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 35 small constant δ >
0. By Proposition 4.1, and the Sobolev embeding inequality | U | W , ∞ ≤ C | U | H s , we have(4.60) | U ( t ) | H s ≤ Ce − θt | U | H s + C Z t e − θ ( t − τ ) (cid:16) | U ( τ ) | L + |B h ( τ ) | (cid:17) dτ ≤ C ( | U | H s + E + ζ ( t ) )(1 + t ) − ( d − / . and so the solution continues so long as ζ remains small, with bound (4.59), yielding exis-tence and the claimed bounds.Thus, it remains to prove the claim (4.58). First by (4.55), we obtain(4.61) | U ( t ) | L ≤|S ( t ) U | L + Z t |S ( t − s ) ∂ x j Q j ( s ) | L ds + Z t |S ( t − s ) ∂ x j Q j ( s ) | L ds + | Γ U (0 , ˜ x, t ) | L ≤ I + I + I + | Γ U (0 , ˜ x, t ) | L where I : = |S ( t ) U | L ≤ C (1 + t ) − d − | U | L ∩ H ,I : = Z t |S ( t − s ) ∂ x j Q j ( s ) | L ds ≤ C Z t (1 + t − s ) − d − − | Q j ( s ) | L + (1 + s ) − d − | Q j ( s ) | L , ∞ ˜ x,x ds ≤ C Z t (1 + t − s ) − d − − | U | H + (1 + t − s ) − d − (cid:16) | U | L , ∞ ˜ x,x + | U x | L , ∞ ˜ x,x (cid:17) ds ≤ C ( | U | H s + ζ ( t ) ) Z t h (1 + t − s ) − d − − (1 + s ) − d − + (1 + t − s ) − d − (1 + s ) − d +12 i ds ≤ C (1 + t ) − d − ( | U | H s + ζ ( t ) ) and I : = Z t |S ( t − s ) ∂ x j Q j ( s ) | L ds ≤ Z t e − θ ( t − s ) | ∂ x j Q j ( s ) | H ds ≤ C Z t e − θ ( t − s ) ( | U | L ∞ + | U x | L ∞ ) | U | H ds ≤ C Z t e − θ ( t − s ) | U | H s ds ≤ C ( | U | H s + ζ ( t ) ) Z t e − θ ( t − s ) (1 + s ) − d − ds ≤ C (1 + t ) − d − ( | U | H s + ζ ( t ) ) . Meanwhile, for the boundary term | Γ U (0 , ˜ x, t ) | L , we treat two cases separately. First forthe inflow case, then by (4.53) we have | U (0 , ˜ x, t ) | ≤ C | h (˜ x, t ) | + O ( | U (0 , ˜ x, t ) | ) , and thus | U (0 , ˜ x, t ) | ≤ C | h (˜ x, t ) | , provided that | h | is sufficiently small. Therefore under thehypotheses on h in Theorem 1.7, Proposition 3.8 yields | Γ U (0 , · , · ) | L x ≤ CE (1 + t ) − d − . Now for the outflow case, recall that in this case G ( x, t ; 0 , ˜ y ) ≡
0. Thus (4.56) simplifiesto(4.62) Γ U (0 , ˜ x, t ) = Z t Z R d − G y ( x, t − s ; 0 , ˜ y ) B U (0 , ˜ y, s ) d ˜ yds. To deal with this term, we shall use Proposition 3.8 as in the inflow case. In view of (4.54), | B U (0 , ˜ y, s ) | ≤ C | h (˜ y, t ) | + O ( | U (0 , ˜ y, s ) | ) , and assumptions on h are imposed as in Theorem 1.6, so that (3.27) is satisfied. To checkthe last term O ( | U (0) | ) , using the definition (4.57) of ζ ( t ), we have |O ( | U (0 , ˜ y, s ) | ) | L ≤ C | U | L ∞ | U | L , ∞ ˜ x,x ≤ Cζ ( t )(1 + s ) − d − d +14 |O ( | U (0 , ˜ y, s ) | ) | L ∞ ≤ C | U | L ∞ ≤ Cζ ( t )(1 + s ) − d and for the term D h with h replaced by O ( | U (0 , ˜ y, s ) | ), using the standard H¨older inequalityto get |D h | L x ≤ C ( | U | L , ∞ + | U x | L , ∞ + | U ˜ x ˜ x | L , ∞ ) ≤ Cζ ( t )(1 + s ) − d +12 |D h | H [( d − / x ≤ C | U | L ∞ | U | H s ≤ Cζ ( t )(1 + s ) − d/ − ( d − / . We remark here that Sobolev bounds (4.60) are not good enough for estimates of D h in L ,requiring a decay at rate (1 + t ) − d/ − ǫ for the two-dimensional case (see Proposition 3.8).This is exactly why we have to keep track of U ˜ x ˜ x in L , ∞ norm in ζ ( t ) as well, to gain abound as above for D h . TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 37
Therefore applying Proposition 3.8, we also obtain (4.62) for the outflow case. Combiningthese above estimates yields(4.63) | U ( t ) | L (1 + t ) d − ≤ C ( | U | L ∩ H s + E + ζ ( t ) ) . Next, we estimate(4.64) | U ( t ) | L , ∞ ˜ x,x ≤|S ( t ) U | L , ∞ ˜ x,x + Z t |S ( t − s ) ∂ x j Q j ( s ) | L , ∞ ˜ x,x ds + Z t |S ( t − s ) ∂ x j Q j ( s ) | L , ∞ ˜ x,x ds + | Γ U (0 , ˜ x, t ) | L , ∞ ˜ x,x ≤ J + J + J + | Γ U (0 , ˜ x, t ) | L , ∞ ˜ x,x where J : = |S ( t ) U | L , ∞ ˜ x,x ≤ C (1 + t ) − d +14 | U | L ∩ H J : = Z t |S ( t − s ) ∂ x j Q j ( s ) | L , ∞ ˜ x,x ds ≤ C Z t (1 + t − s ) − d +14 − | Q j ( s ) | L + (1 + s ) − d +14 | Q j ( s ) | L , ∞ ˜ x,x ds ≤ C Z t (1 + t − s ) − d +14 − | U | H + (1 + t − s ) − d +14 (cid:16) | U | L , ∞ ˜ x,x + | U x | L , ∞ ˜ x,x (cid:17) ds ≤ C ( | U | H s + ζ ( t ) ) Z t (1 + t − s ) − d +14 − (1 + s ) − d − + (1 + t − s ) − d +14 (1 + s ) − d +12 ds ≤ C (1 + t ) − d +14 ( | U | H s + ζ ( t ) )and (by Moser’s inequality) J : = Z t |S ( t − s ) ∂ x j Q j ( s ) | L , ∞ ˜ x,x ds ≤ C Z t e − θ ( t − s ) | ∂ x j Q j ( s ) | H ds ≤ C Z t e − θ ( t − s ) | U | L ∞ x | U | H ds ≤ C ( | U | H s + ζ ( t ) ) Z t e − θ ( t − s ) (1 + s ) − d (1 + s ) − d − ds ≤ C (1 + t ) − d +14 ( | U | H s + ζ ( t ) ) . These estimates together with similar treatment for the boundary term yield(4.65) | U ( t ) | L , ∞ ˜ x,x (1 + t ) d +14 ≤ C ( | U | L ∩ H s + E + ζ ( t ) ) . Similarly, we have the same estimate for | U x ( t ) | L , ∞ ˜ x,x . Indeed, we have(4.66) | U x ( t ) | L , ∞ ˜ x,x ≤| ∂ x S ( t ) U | L , ∞ ˜ x,x + Z t | ∂ x S ( t − s ) ∂ x j Q j ( s ) | L , ∞ ˜ x,x ds + Z t | ∂ x S ( t − s ) ∂ x j Q j ( s ) | L , ∞ ˜ x,x ds + | ∂ x Γ U (0 , ˜ x, t ) | L , ∞ ˜ x,x ≤ K + K + K + | ∂ x Γ U (0 , ˜ x, t ) | L , ∞ ˜ x,x where K and K are treated exactly in the same way as the treatment of J , J , yet inthe first term of K it is a bit better by a factor t − / . Similar bounds hold for | U ˜ x ˜ x | in L , ∞ , noting that there are no higher derivatives in x involved and thus similar to thosein (4.64).Finally, we estimate the L ∞ norm of U . By Duhamel’s formula (4.55), we obtain(4.67) | U ( t ) | L ∞ ≤|S ( t ) U | L ∞ + Z t |S ( t − s ) ∂ x j Q j ( s ) | L ∞ ds + Z t |S ( t − s ) ∂ x j Q j ( s ) | L ∞ ds + | Γ U (0 , ˜ x, t ) | L ∞ ≤ L + L + L + | Γ U (0 , ˜ x, t ) | L ∞ where the boundary term is treated in the same way as above, and for | γ | = [( d − /
2] + 2, L : = |S ( t ) U | L ∞ ≤ C (1 + t ) − d | U | L ∩ H | γ | +3 ,L : = Z t |S ( t − s ) ∂ x j Q j ( s ) | L ∞ ds ≤ C Z t (1 + t − s ) − d − | Q j ( s ) | L + (1 + s ) − d | Q j ( s ) | L , ∞ ˜ x,x ds ≤ C Z t (1 + t − s ) − d − | U | H + (1 + t − s ) − d (cid:16) | U | L , ∞ ˜ x,x + | U x | L , ∞ ˜ x,x (cid:17) ds ≤ C ( | U | H s + ζ ( t ) ) Z t h (1 + t − s ) − d − (1 + s ) − d − + (1 + t − s ) − d (1 + s ) − d +12 i ds ≤ C (1 + t ) − d ( | U | H s + ζ ( t ) ) TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 39 and (again by Moser’s inequality), L : = Z t |S ( t − s ) ∂ x j Q j ( s ) | L ∞ ds ≤ Z t |S ( t − s ) ∂ x j Q j ( s ) | H | γ | ds ≤ Z t e − θ ( t − s ) | ∂ x Q j ( s ) | H | γ | +3 ds ≤ C Z t e − θ ( t − s ) | U | L ∞ | U | H | γ | +5 ds ≤ C ( | U | H s + ζ ( t ) ) Z t e − θ ( t − s ) (1 + s ) − d (1 + s ) − d − ds ≤ C (1 + t ) − d ( | U | H s + ζ ( t ) ) . Therefore we have obtained(4.68) | U ( t ) | L ∞ x (1 + t ) d ≤ C ( | U | L ∩ H s + E + ζ ( t ) )and thus completed the proof of claim (4.58), and the main theorem. (cid:3) Appendix A. Physical discussion in the isentropic case
In this appendix, we revisit in slightly more detail the drag-reduction problem sketched inExamples 1.1–1.2, in the simplified context of the two-dimensional isentropic case. Followingthe notation of [GMWZ5], consider the two-dimensional isentropic compressible Navier–Stokes equations ρ t + ( ρu ) x + ( ρv ) y = 0 , (A.1) ( ρu ) t + ( ρu ) x + ( ρuv ) y + p x = (2 µ + η ) u xx + µu yy + ( µ + η ) v xy , (A.2) ( ρv ) t + ( ρuv ) x + ( ρv ) y + p y = µv xx + (2 µ + η ) v yy + ( µ + η ) u yx (A.3)on the half-space y >
0, where ρ is density, u and v are velocities in x and y directions,and p = p ( ρ ) is pressure, and µ > | η | ≥ p ′ ( ρ ) > x -axis, with constant imposed normal velocity v (0) = V and zero transverse relative velocity u (0) = 0 imposed at the airfoil surface, andseek a laminar boundary-layer flow ( ρ, u, v )( y ) with transverse relative velocity u ∞ a shortdistance away the airfoil, with | V | much less than the sound speed c ∞ and | u ∞ | of an orderroughly comparable to c ∞ .A.1. Existence.
The possible boundary-layer solutions have been completely categorizedin this case in Section 5.1 of [GMWZ5]. We here cite the relevant conclusions, referring to[GMWZ5] for the (straightforward) justifying computations.
A.1.1.
Outflow case (
V < ). In the outflow case, the scenario described above correspondsto case (5.15) of [GMWZ5], in which it is found that the only solutions are purely transverse flows(A.4) ( ρ, v ) ≡ ( ρ , V ) , u ( y ) = u ∞ (1 − e ρ V y/µ ) , varying only in the tranverse velocity u . The drag force per unit length at the airfoil, byNewton’s law of viscosity, is(A.5) µ ¯ u y | y =0 = u ∞ ρ ∞ | V | , since momentum m := ρ V = ρ ∞ V is constant throughout the layer, so that ( ρ ∞ , u ∞ beingimposed by ambient conditions away from the wing) drag is proportional to the speed | V | of the imposed normal velocity .A.1.2. Inflow case (
V > ). Consulting again [GMWZ5] (p. 61), we find for
V > ρ, u, v )(0) of the orders described above, the only solutions are purely normal flows,(A.6) u ≡ u (0) , ( ρ, v ) = ( ρ, v )( y ) , varying only in the normal velocity v . Thus, it is not possible to reconcile the velocity u (0)at the airfoil with the velocity u ∞ >> c some distance away.As discussed in [MN], the expected behavior in such a case consists rather of a combi-nation of a boundary-layer at y = 0 and one or more elementary planar shock, rarefaction,or contact waves moving away from y = 0: in this case a shear wave moving with normalfluid velocity V into the half-space, across which the transverse velocity changes from zeroto u ∞ . That is, a characteristic layer analogous to the solid-boundary case detaches fromthe airfoil and travels outward into the flow field. In this case, one would not expect dragreduction compared to the solid-boundary case, but rather some increase.A.2. Stability.
If we consider one-dimensional stability, or stability with respect to per-turbations depending only on y , we find that the linearized eigenvalue equations decoupleinto the constant-coefficient linearized eigenvalue equations for ( ρ, v ) about a constant layer( ρ, v ) ≡ ( ρ , V ), and the scalar linearized eigenvalue equation(A.7) λ ¯ ρu + mu y = µu yy associated with the constant-coefficient convection-diffusion equation ¯ ρu t + mu = µu yy ,m := ¯ ρ ¯ v ≡ ρ V , ¯ ρ ≡ ρ . As the constant layer ( ρ , V ) is stable by Corollary 1.5 or directcalculation (Fourier transform), and (A.7) is stable by direct calculation, we may thusconclude that purely transverse layers are one-dimensionally stable .Considered with respect to general perturbations, the equations do not decouple, nor dothey reduce to constant-coefficient form, but to a second order system whose coefficients arequadratic polynomials in e ρ V y . It would be very interesting to try to resolve the questionof spectral stability by direct solution using this special form, or, alternatively, to performa numerical study as done in [HLyZ2] for the multi-dimensional shock wave case.
Remark A.1.
For general laminar boundary layers (¯ ρ, ¯ u, ¯ v )( y ), the one-dimensional sta-bility problem, now variable-coefficient, does not completely decouple, but has triangular TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 41 form, breaking into a system in ( ρ, v ) alone and an equation in u forced by ( ρ, v ). Stabil-ity with respect to general perturbations, therefore, is equivalent to stability with respectto perturbations of form ( ρ, , v ) or (0 , u, ρ, u, v ) = (0 , u, u equation again becomes (A.7), with µ , m still constant, but ¯ ρ varying in y . Taking the realpart of the complex L inner product of u against (A.7) gives ℜ λ k u k L + k u y k L = 0 , hence for ℜ λ ≥ u ≡ constant = 0. Thus, the layer is one-dimensionally stable if and onlyif the normal part (¯ ρ, ¯ v ) is stable with respect to perturbations ( ρ, v ). Stability of normallayers was studied in [CHNZ] for a γ -law gas p ( ρ ) = aρ γ , 1 ≤ γ ≤
3, with the conclusionthat all layers are one-dimensionally stable , independent of amplitude, in the general inflowand compressive outflow cases. Hence, we can make the same conclusion for full layers(¯ ρ, ¯ u, ¯ v ). In the present context, this includes all cases except for suction with supersonicvelocity | V | > c ∞ , which in the notation of [CHNZ] is of expansive outflow type (expectedalso to be stable, but not considered in [CHNZ]), since | ¯ v | is decreasing with y , so thatdensity ¯ ρ (since m = ¯ ρ ¯ v ≡ constant) is increasing.A.3. Discussion.
Note that we do not achieve by subsonic boundary suction an exactlaminar flow connecting the values ( u, v ) = (0 , V ) at the wing to the values ( u ∞ ,
0) of theambient flow at infinity, but rather to an intermediate value ( u ∞ , V ). That is, we tradea large variation u ∞ in shear for a possibly small variation V in normal velocity, whichappears now as a boundary condition for the outer, approximately Euler flow away fromthe boundary layer. Whether the full solution is stable appears to be a question concerningalso nonstationary Euler flow. It is not clear either what is the optimal outflux velocity V . From (A.5) and the discussion just above, it appears desirable to minimize | V | , sincethis minimizes both drag and the imbalance between flow v ∞ just outside the boundarylayer and the ambient flow at infinity. On the other hand, we expect that stability becomesmore delicate in the characteristic limit V → − , in the sense that the size of the basin ofattraction of the boundary layer shrinks to zero (recall, we have ignored throughout ouranalysis the size of the basin of attraction, taking perturbations as small as needed withoutkeeping track of constants). These would be quite interesting issues for further investigation. References [AGJ] J. Alexander, R. Gardner, and C. Jones.
A topological invariant arising in the stability analysisof travelling waves,
J. Reine Angew. Math., 410:167–212, 1990.[BHRZ] B. Barker, J. Humpherys, K. Rudd, and K. Zumbrun.
Stability of viscous shocks in isentropicgas dynamics, to appear, Comm. Math. Phys.[Bra] Braslow, A.L.,
A history of suction-type laminar-flow control with emphasis on flight research ,NSA History Division, Monographs in aerospace history, number 13 (1999).[BDG] T. J. Bridges, G. Derks, and G. Gottwald,
Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework,
Phys. D, 172(1-4):190–216, 2002.[Br1] L. Q. Brin.
Numerical testing of the stability of viscous shock waves,
PhD thesis, Indiana Univer-sity, Bloomington, 1998.[Br2] L. Q. Brin.
Numerical testing of the stability of viscous shock waves,
Math. Comp., 70(235):1071–1088, 2001. [BrZ] L. Q. Brin and K. Zumbrun.
Analytically varying eigenvectors and the stability of viscous shockwaves,
Mat. Contemp., 22:19–32, 2002, Seventh Workshop on Partial Differential Equations, PartI (Rio de Janeiro, 2001).[CHNZ] N. Costanzino, J. Humpherys, T. Nguyen, and K. Zumbrun,
Spectral stability of noncharacteristicboundary layers of isentropic Navier–Stokes equations, to appear, Arch. Ration. Mech. Anal.[GZ] R. A. Gardner and K. Zumbrun.
The gap lemma and geometric criteria for instability of viscousshock profiles.
Comm. Pure Appl. Math., 51(7):797–855, 1998.[GR] Grenier, E. and Rousset, F.,
Stability of one dimensional boundary layers by using Green’s func-tions , Comm. Pure Appl. Math. 54 (2001), 1343-1385.[GMWZ1] C. M. I. O. Gu`es, G. M´etivier, M. Williams, and K. Zumbrun.
Multidimensional viscous shocksI: degenerate symmetrizers and long time stability,
J. Amer. Math. Soc. 18 (2005), no. 1, 61–120.[GMWZ5] C. M. I. O. Gu`es, G. M´etivier, M. Williams, and K. Zumbrun.
Existence and stability of non-characteristic hyperbolic-parabolic boundary-layers.
Preprint, 2008.[GMWZ6] C. M. I. O. Gu`es, G. M´etivier, M. Williams, and K. Zumbrun.
Viscous boundary value problemsfor symmetric systems with variable multiplicities
J. Differential Equations 244 (2008) 309–387.[HZ] P. Howard and K. Zumbrun,
Stability of undercompressive viscous shock waves , in press, J.Differential Equations 225 (2006), no. 1, 308–360.[HLZ] J. Humpherys, O. Lafitte, and K. Zumbrun.
Stability of viscous shock profiles in the high Machnumber limit, (Preprint, 2007).[HLyZ1] Humpherys, J., Lyng, G., and Zumbrun, K.,
Spectral stability of ideal-gas shock layers , Preprint(2007).[HLyZ2] Humpherys, J., Lyng, G., and Zumbrun, K.,
Multidimensional spectral stability of large-amplitudeNavier-Stokes shocks , in preparation.[HoZ1] D. Hoff and K. Zumbrun,
Multi-dimensional diffusion waves for the Navier-Stokes equations ofcompressible flow,
Indiana Univ. Math. J. 44 (1995), no. 2, 603–676.[HoZ2] D. Hoff and K. Zumbrun,
Pointwise decay estimates for multidimensional Navier-Stokes diffusionwaves,
Z. Angew. Math. Phys. 48 (1997), no. 4, 597–614.[HuZ] J. Humpherys and K. Zumbrun.
An efficient shooting algorithm for evans function calculationsin large systems,
Physica D, 220(2):116–126, 2006.[KK] Y. Kagei and S. Kawashima
Stability of planar stationary solutions to the compressible Navier-Stokes equations in the half space,
Comm. Math. Phys. 266 (2006), 401-430.[KNZ] S. Kawashima, S. Nishibata, and P. Zhu,
Asymptotic stability of the stationary solution to thecompressible Navier-Stokes equations in the half space,
Comm. Math. Phys. 240 (2003), no. 3,483–500.[KSh] S. Kawashima and Y. Shizuta.
Systems of equations of hyperbolic-parabolic type with applicationsto the discrete Boltzmann equation. Hokkaido Math. J. , 14(2):249–275, 1985.[KZ] B. Kwon and K. Zumbrun,
Asymptotic Behavior of Multidimensional scalar Relaxation Shocks ,Preprint, 2008[MaZ3] C. Mascia and K. Zumbrun.
Pointwise Green function bounds for shock profiles of systems withreal viscosity.
Arch. Ration. Mech. Anal., 169(3):177–263, 2003.[MaZ4] C. Mascia and K. Zumbrun.
Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems.
Arch. Ration. Mech. Anal., 172(1):93–131, 2004.[MN] Matsumura, A. and Nishihara, K.,
Large-time behaviors of solutions to an inflow problem in thehalf space for a one-dimensional system of compressible viscous gas,
Comm. Math. Phys., 222(2001), no. 3, 449–474.[MZ] M´etivier, G. and Zumbrun, K.,
Viscous Boundary Layers for Noncharacteristic Nonlinear Hy-perbolic Problems , Memoirs AMS, 826 (2005).[NZ] T. Nguyen and K. Zumbrun,
Long-time stability of large-amplitude noncharacteristic boundarylayers for hyperbolic-parabolic systems , Preprint, 2008[PW] R. L. Pego and M. I. Weinstein.
Eigenvalues, and instabilities of solitary waves.
Philos. Trans.Roy. Soc. London Ser. A, 340(1656):47–94, 1992.
TABILITY OF MULTI-DIMENSIONAL BOUNDARY LAYERS 43 [RZ] M. Raoofi and K. Zumbrun,
Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems
Preprint, 2007.[R2] F. Rousset,
Inviscid boundary conditions and stability of viscous boundary layers . Asymptot.Anal. 26 (2001), no. 3-4, 285–306.[R3] Rousset, F.,
Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems,
Trans. Amer. Math. Soc. 355 (2003), no. 7, 2991–3008.[S] H. Schlichting,
Boundary layer theory , Translated by J. Kestin. 4th ed. McGraw-Hill Series inMechanical Engineering. McGraw-Hill Book Co., Inc., New York, 1960.[SZ] Serre, D. and Zumbrun, K.,
Boundary layer stability in real vanishing-viscosity limit , Comm.Math. Phys. 221 (2001), no. 2, 267–292.[YZ] S. Yarahmadian and K. Zumbrun,
Pointwise Green function bounds and long-time stability oflarge-amplitude noncharacteristic boundary layers , Preprint (2008).[Z2] K. Zumbrun. Multidimensional stability of planar viscous shock waves. In
Advances in the the-ory of shock waves , volume 47 of
Progr. Nonlinear Differential Equations Appl. , pages 307–516.Birkh¨auser Boston, Boston, MA, 2001.[Z3] K. Zumbrun. Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In
Handbook of mathematical fluid dynamics. Vol. III , pages 311–533. North-Holland, Amsterdam,2004. With an appendix by Helge Kristian Jenssen and Gregory Lyng.[Z4] K. Zumbrun. Planar stability criteria for viscous shock waves of systems with real viscosity.In
Hyperbolic systems of balance laws , volume 1911 of
Lecture Notes in Math. , pages 229–326.Springer, Berlin, 2007.[ZH] K. Zumbrun and P. Howard.
Pointwise semigroup methods and stability of viscous shock waves.
Indiana Univ. Math. J., 47(3):741–871, 1998.
Department of Mathematics, Indiana University, Bloomington, IN 47402
E-mail address : [email protected] Department of Mathematics, Indiana University, Bloomington, IN 47402
E-mail address ::