Long time stability of large amplitude noncharacteristic boundary layers for hyperbolic–parabolic systems

arXiv:0804.1345v1 [math.AP] Apr 2008 LONG-TIME STABILITY OF LARGE-AMPLITUDE NONCHARACTERISTIC BOUNDARY LAYERS FOR HYPERBOLIC–PARABOLIC SYSTEMS TOAN NGUYEN AND KEVIN ZUMBRUN Abstract Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic-parabolic systems including the Navier–Stokes equations of compressible gas- and magnetohydrodynamics, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index γ ≥ Together with these previous results, our results thus give nonlinear stability of largeamplitude isentropic boundary layers, the first such result for compressive (“shock-type”) layers in other than the nearly-constant case The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime Contents Introduction 1.1 Equations and assumptions 1.2 Main results 1.3 Discussion and open problems Pointwise bounds on resolvent kernel Gλ 2.1 Evans function framework 2.2 Construction of the resolvent kernel 2.3 High frequency estimates 2.4 Low frequency estimates Pointwise bounds on Green function G(x, t; y) Energy estimates 4.1 Energy estimate I 4.2 Energy estimate II 11 11 15 18 27 29 35 35 50 Date: Last Updated: April 5, 2008 This work was supported in part by the National Science Foundation award number DMS-0300487 T NGUYEN AND K ZUMBRUN Stability analysis 5.1 Integral formulation 5.2 Convolution estimates 5.3 Linearized stability 5.4 Nonlinear argument References 50 51 53 57 58 61 Introduction In this paper, we study the stability of boundary layers assuming that the boundary layer solution is noncharacteristic, which means, roughly, that signals are transmitted into or out of but not along the boundary In the context of gas dynamics or magnetohydrodynamics (MHD), this corresponds to the situation of a porous boundary with prescribed inflow or outflow conditions accomplished by suction or blowing, a scenario that has been suggested as a means to reduce drag along an airfoil by stabilizing laminar flow; see Example 1.1 below We consider a boundary layer, or stationary solution, ˜ =U ¯ (x), ¯ (z) = U+ , U ¯ (0) = U ¯0 (1) U lim U z→+∞ of a system of conservation laws on the quarter-plane ˜t + F (U˜ )x = (B(U ˜ )U˜x )x , x, t > 0, (2) U ˜ , F ∈ Rn , B ∈ Rn×n , with initial data U ˜ (x, 0) = U ˜0 (x) and Dirichlet type U boundary conditions specified in (5), (6) below A fundamental question connected to the physical motivations from aerodynamics is whether or not such boundary layer solutions are stable in the sense of PDE, i.e., whether ¯ remains close to U ¯ , or converges or not a sufficiently small perturbation of U ¯ , under the evolution of (2) That is the question time-asymptotically to U we address here Our main result, in the general spirit of [ZH, MaZ3, MaZ4, Z3, HZ, YZ], is to reduce the questions of linear and nonlinear stability to verification of a simple and numerically well-posed Evans function, or generalized spectral stability, condition, which can then be checked either numerically or by the variety of methods available for study of eigenvalue ODE; see, for example, [Br1, Br2, BrZ, BDG, HuZ2, PZ, FS, BHRZ, HLZ, HLyZ1, HLyZ2, CHNZ] Together with the results of [CHNZ], this yields in particular nonlinear stability of sufficiently large-amplitude boundary-layers of the compressible Navier–Stokes equations of isentropic ideal gas dynamics, with adiabatic index γ ≥ 1, the first such result for a large compressive, or “shock-type”, boundary layers The main new difficulty beyond the strictly parabolic case of [YZ] is to treat the more singular, hyperbolic behavior in the highfrequency regime, both in obtaining pointwise Green function bounds, and in deriving energy estimates by which the nonlinear analysis is closed STABILITY OF BOUNDARY LAYERS 1.1 Equations and assumptions We consider the general hyperbolic˜ , with parabolic system of conservation laws (2) in conserved variable U ˜ = U u ˜ , v˜ B= 0 , b1 b2 σ(b2 ) ≥ θ > 0, u ˜ ∈ R, and v˜ ∈ Rn−1 , where, here and elsewhere, σ denotes spectrum of a linearized operator or matrix Here for simplicity, we have restricted to the case (as in standard gas dynamics and MHD) that the hyperbolic part (equation for u ˜) consists of a single scalar equation As in [MaZ3], the results extend in straightforward fashion to the case u˜ ∈ Rk , k > 1, with σ(A11 ) strictly positive or strictly negative Following [MaZ4, Z3], we assume that equations (2) can be written, alternatively, after a triangular change of coordinates (3) ˜ := W ˜ (U ˜) = W w ˜I (˜ u) , II w ˜ (˜ u, v˜) in the quasilinear, partially symmetric hyperbolic-parabolic form (4) ˜ t + A˜W ˜ x = (B ˜W ˜ x )x + G, ˜ A˜0 W ˜ + := W ˜ (U+ ), where, defining W ˜W ˜ + ), A˜0 , A˜11 are symmetric, A0 block diagonal, A˜0 ≥ θ0 > 0, (A1) A( ˜ A˜0 )−1 (W ˜ + ) lies in the kernel of B( ˜ A˜0 )−1 (W ˜ + ), (A2) no eigenvector of A( 0 ˜ ˜ = with g˜(W ˜ x, W ˜ x ) = O(|W ˜ x |2 ) , b ≥ θ > 0, and G ˜ g˜ b Along with the above structural assumptions, we make the following technical hypotheses: ˜= (A3) B ˜ B, ˜ W ˜ (·), g˜(·, ·) ∈ C (H0) F, B, A˜0 , A, (H1) A˜11 (scalar) is either strictly positive or strictly negative, that is, either A˜11 ≥ θ1 > 0, or A˜11 ≤ −θ1 < (We shall call these cases the inflow case or the outflow case, correspondingly.) (H2) The eigenvalues of dF 11 (U+ ) are real, distinct, and nonzero ¯ is unique (H3) Solution U Condition (H1) corresponds to noncharacteristicity, while (H2) is the condition for the hyperbolicity of U+ The assumptions (A1)-(A3) and (H0)-(H3) are satisfied for gas dynamics and MHD with van der Waals equation of state under inflow or outflow conditions; see discussions in [MaZ4, CHNZ, GMWZ5, GMWZ6] We also assume: ˜ -coordinates: (B) Dirichlet boundary conditions in W (5) ˜ := (h ˜1, h ˜ )(t) (w ˜I , w ˜II )(0, t) = h(t) T NGUYEN AND K ZUMBRUN for the inflow case, and ˜ w ˜II (0, t) = h(t) (6) for the outflow case This is sufficient for the main physical applications; the situation of more general, Neumann- and mixed-type boundary conditions on the parabolic variable v can be treated as discussed in [GMWZ5, GMWZ6] Example 1.1 The main example we have in mind consists of laminar solutions (ρ, u, e)(x1 , t) of the compressible Navier–Stokes equations  ∂t ρ + div(ρu) =      ∂t (ρu) + div(ρut u) + ∇p = εµ∆u + ε(µ + η)∇divu (7)  ∂t (ρE) + div (ρE + p)u = εκ∆T + εµdiv (u · ∇)u     + ε(µ + η)∇(u · divu), x ∈ Rd , on a half-space x1 > 0, where ρ denotes density, u ∈ Rd velocity, e specific internal energy, E = e + |u|2 specific total energy, p = p(ρ, e) pressure, T = T (ρ, e) temperature, µ > and |η| ≤ µ first and second coefficients of viscosity, κ > the coefficient of heat conduction, and ε > (typically small) the reciprocal of the Reynolds number, with no-slip suctiontype boundary conditions on the velocity, uj (0, x2 , , xd ) = 0, j = and u1 (0, x2 , , xd ) = V (x) < 0, and prescribed temperature, T (0, x2 , , xd ) = Twall (x) Under the standard assumptions pρ , Te > 0, this can be seen to satisfy all of the hypotheses (A1)–(A3), (H0)–(H3); indeed these are satisfied also under much weaker van der Waals gas assumptions [MaZ4, Z3, CHNZ, GMWZ5, GMWZ6] In particular, boundary-layer solutions are of noncharacteristic type, scaling√as (ρ, u, e) = (¯ ρ, u ¯, e¯)(x1 /ε), with layer thickness ∼ ε as compared to the ∼ ε thickness of the characteristic type found for an impermeable boundary This corresponds to the situation of an airfoil with microscopic holes through which gas is pumped from the surrounding flow, the microscopic suction imposing a fixed normal velocity while the macroscopic surface imposes standard temperature conditions as in flow past a (nonporous) plate This configuration was suggested by Prandtl and tested experimentally 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 with reported 25% reduction in drag at supersonic speeds [Bra] √ Possible mechanisms for this reduction are smaller thickness ∼ ε 0, the problem (10) with initial data U0 in X and homogeneous boundary data h ≡ has a unique global solution U (·, t) such that |U (·, t)|Y ≤ C|U0 |X for all t; it is said to be linearly asymptotically X → Y stable if also |U (·, t)|Y → as t → ∞ We define the following stability criterion, where D(λ) described below, denotes the Evans function associated with the linearized operator L about the layer, an analytic function analogous to the characteristic polynomial of a finite-dimensional operator, whose zeroes away from the essential spectrum agree in location and multiplicity with the eigenvalues of L: (D) There exist no zeroes of D(·) in the nonstable half-plane Reλ ≥ As discussed, e.g., in [R2, MZ1, GMWZ5, GMWZ6], under assumptions (H0)-(H3), this is equivalent to strong spectral stability, σ(L) ⊂ {Reλ < ¯ as a solution of the connection problem in the 0}, (ii) transversality of U associated standing-wave ODE, and hyperbolic stability of an associated boundary value problem obtained by formal matched asymptotics See [GMWZ5, GMWZ6] for further discussions ¯ is said to be nonlinearly X → Y Definition 1.5 The boundary layer U ˜0 sufficiently stable if, for each ε > 0, the problem (2) with initial data U ¯ ˜ close to the profile U in | · |X has a unique global solution U (·, t) such that ˜ (·, t) − U ¯ (·)|Y < ε for all t; it is said to be nonlinearly asymptotically |U ˜ (·, t) − U ¯ (·)|Y → as t → ∞ We shall sometimes X → Y stable if also |U not explicitly define the norm X, speaking instead of stability or asymptotic stability in Y under perturbations satisfying specified smallness conditions Our first main result is as follows Theorem 1.6 (Linearized stability) Assume (A1)-(A3), (H0)-(H3), and ¯ be a boundary layer Then linearized (B) with |h(t)| ≤ E0 (1 + t)−1 Let U p p L ∩ L → L ∩ L stability, ≤ p ≤ ∞, is equivalent to (D) In the case of stability, there holds also linearized asymptotic L1 ∩ Lp → Lp stability, p > 1, with rate (14) 1 |U (·, t)|Lp ≤ C(1 + t)− (1−1/p) |U0 |L1 ∩Lp + CE0 (1 + t)− (1−1/p) To state the pointwise nonlinear stability result, we need some notations Denoting by + + a+ < a2 < · · · < an (15) the eigenvalues of of the limiting convection matrix A+ := dF (U+ ), define (16) θ(x, t) := + /M t (1 + t)−1/2 e−|x−aj t| a+ j >0 , STABILITY OF BOUNDARY LAYERS (17) ψ1 (x, t) := χ(x, t) a+ j >0 −1/2 (1 + |x| + t)−1/2 (1 + |x − a+ , j t|) and (18) 1/2 −3/2 ψ2 (x, t) := (1 − χ(x, t))(1 + |x − a+ ) , n t| + t where χ(x, t) = for x ∈ [0, a+ n t] and χ(x, t) = otherwise and M > is a sufficiently large constant For simplicity, we measure the boundary data by function (19) Bh (t) := r=0 |(d/dt)r h|2 for the outflow case, and (20) Bh (t) := r=0 |(d/dt)r h1 |2 + r=0 |(d/dt)r h2 |2 for the inflow case Then, our next result is as follows Theorem 1.7 (Nonlinear stability) Assuming (A1)-(A3), (H0)-(H3), (B), ¯ is nonlinearly asympand the linear stability condition (D), the profile U p totically stable in L ∩ H , p > 1, with respect to perturbations U0 ∈ H , h ∈ C in initial and boundary data satisfying (1 + |x|2 )3/4 U0 H4 ≤ E0 and |Bh (t)| ≤ E0 (1 + |t|)−1/2 for E0 sufficiently small More precisely, ˜ (x, t) − U ¯ (x)| ≤ CE0 (θ + ψ1 + ψ2 )(x, t), |U ˜x (x, t) − U ¯x (x)| ≤ CE0 (θ + ψ1 + ψ2 )(x, t), |U (21) ˜ (x, t) denotes the solution of (2) with initial and boundary data where U ˜ ¯ (x) + U0 (x) and U ˜ (0, t) = U ¯0 + h(t), yielding the sharp rates U (x, 0) = U (22) (23) ˜ (x, t) − U ¯ (x) U ˜ (x, t) − U ¯ (x) U 1 Lp ≤ CE0 (1 + t)− (1− p ) , L1 ∩H − 14 ≤ CE0 (1 + t) ≤ p ≤ ∞, Remark 1.8 By the one dimensional Sobolev embedding, from the hypothesis on U0 , we automatically assume that U0 H4 ≤ E0 , |U0 (x)| + |U0′ (x)| ≤ E0 (1 + |x|)−3/2 A crucial step in establishing Theorems 1.6 and 1.7 is to obtain pointwise bounds on the Green function G(x, t; y) of the linearized evolution equations (10) (more properly speaking, a distribution), which we now describe Let + + a+ j , j = 1, , n denote the eigenvalues of A(+∞), and lj , rj associated left T NGUYEN AND K ZUMBRUN and right eigenvectors, respectively, normalized so that lj+ rk+ = δjk Eigenvalues aj (x), and eigenvectors lj (x), rj (x) correspond to large-time convection rates and modes of propagation of the linearized model (10) Define time-asymptotic, scalar diffusion rates (24) βj+ := (lj Brj )+ , j = 1, , n, and local dissipation coefficient (25) η∗ := −D∗ (x) where −1 −1 D∗ (x) := A12 b−1 A21 − A22 b−1 2 b1 + b2 b1 A∗ + b2 ∂x (b2 b1 ) (x) is an effective dissipation analogous to the effective diffusion predicted by formal, Chapman-Enskog expansion in the (dual) relaxation case, A∗ := A11 − A12 b−1 b1 Note that as a consequence of dissipativity, (A2), we obtain (26) η∗+ > 0, βj+ > 0, for all j We also define modes of propagation for the reduced, hyperbolic part of system (10) as (27) L∗ = 0n−1 , R∗ = −b−1 b1 We define the Green function G(x, t; y) of the linearized evolution equations (10) with homogeneous boundary conditions (more properly speaking, a distribution), by (i) (∂t − Lx )G = in the distributional sense, for all x, y, t > 0; (ii) G(x, t; y) → δ(x − y) as t → 0; A¯ ∗ (iii) for all y, t > 0, ¯ ∗ ¯ G(0, t; y) = where ∗ = for the inflow b1 b2 case A¯∗ > and ∗ is arbitrary for the outflow case A¯∗ < 0, noting that no boundary condition is needed to be prescribed on the hyperbolic part By standard arguments as in [MaZ3], we have the spectral resolution, or inverse Laplace transform formulae (28) eLt f = P.V 2πi η+i∞ η−i∞ eλt (λ − L)−1 f dλ and (29) G(x, t; y) = P.V 2πi η+i∞ eλt Gλ (x, y) dλ η−i∞ for any large positive η We prove the following pointwise bounds on the Green function G(x, t; y) STABILITY OF BOUNDARY LAYERS Proposition 1.9 Under assumptions (A1)-(A3), (H0)-(H3), (B), and (D), we obtain ˜ t; y), (30) G(x, t; y) = H(x, t; y) + G(x, where Rx A∗ (x)−1 A∗ (y)δx−¯a∗ t (y)e− y (η∗ /A∗ )(z)dz R∗ Ltr ∗ 2π = O(e−η0 t )δx−¯a∗ t (y)R∗ Ltr ∗ , H(x, t; y) = (31) and (32) ˜ t; y)| ≤ Ce−η(|x−y|+t) |∂xγ ∂yα G(x, + n C(t−(|α|+|γ|)/2 + |α|e−η|y| + |γ|e−η|x| ) + χ{|a+ t|≥|y|} t + a+ k 0 /M t k=1 + + −1/2 −(x−aj (t−|y/ak |))2 /M t k + t−1/2 e−(x−y−ak t) e , ≤ |α|, |γ| ≤ 1, for some η, C, M > 0, where indicator function χ{|a+ t|≥|y|} k is for |a+ k t| ≥ |y| and otherwise Here, the averaged convection rate a ¯∗ (x, t) in (31) denotes the time-averages over [0, t] of A∗ (z) along backward characteristic paths z∗ = z∗ (x, t) defined by dz∗ = A∗ (z∗ (x, t)), z∗ (t) = x (33) dt In all equations, a+ j , A∗ , L∗ , R∗ are as defined just above 1.3 Discussion and open problems The stability of noncharacteristic boundary layers in gas dynamics has been treated using energy estimates in, e.g., [MN, KNZ, R3], for both “compressive” boundary layers including the truncated shock-solutions (8), and for “expansive” solutions analogous to rarefaction waves However, in the case of compressive waves, these and most subsequent analyses were restricted to the small-amplitude case (34) u ¯ − u+ L1 (R+ ) sufficiently small Examining this condition even for the special class (8) of truncated shock solutions, we find that it is extremely restrictive For, consider the one-parameter family u ¯x0 (x) = u ¯(x − x0 ) of boundarylayers associated with a standing shock u ¯ of amplitude δ := |u+ − u− | 0) boundary condition v(0) = or outflow (A11 < 0) condition v(0) arbitrary, is by inspection exactly the whole-line STABILITY OF BOUNDARY LAYERS 49 Replacing A by Aˆ in the kth order Friedrichs-type bounds above, we find that the resulting error terms may be expressed as ¯ x |)|W |, |∂ k+1 wII | , ∂xk O(ζ + |W x plus lower order terms, easily absorbed using Young’s inequality, and boundary terms k O( i=0 |∂xi wII (0)||∂xk wI (0)|) resulting from the use of integration by parts as we deal with the 12−block However these boundary terms were already treated somewhere as before (see (207)) Hence we can recover the same Friedrichs-type estimates obtained above Thus we may relax (A1′ ) to (A1) The second observation is that, because of the favorable terms ¯ x |∂xk wI , ∂xk wI c∗ θ1 |W occurring in the lefthand sides of the Friedrichs-type estimates (214), we need the Kawashima-type bound only to control the contribution to |∂xk wI |2 coming from x near +∞; more precisely, we require from this estimate only a favorable term ¯ x |))∂xk wI , ∂xk wI −θ2 (1 − O(ζ + |W rather than θ2 ∂xk wI 20 as in (218) But, this may easily be obtained by ˆ := K(W+ ), substituting for K a skew-symmetric matrix-valued function K and using the fact that ℜe(K(A0 )−1 A + B)(W+ ) ≥ θ2 > 0, ˆ + O(ζ + |W ¯ x |), we have and same as (227), K = K ¯ x |)) > ℜe(K(A0 )−1 A + B)(W ) ≥ θ2 (1 − O(ζ + |W Thus we may relax (A2′ ) to (A2) ˜ x ) − g(W ¯ x ) in the perturbation equation Finally, notice that the term g(W may be Taylor expanded as 0 + ˜ ¯ ¯ ˜ O(|Wx |2 ) g1 (Wx , Wx ) + g1 (Wx , Wx ) The first, since linear term on the righthand side decays at plus spatial infinity and vanishes in the 1-1 block, it may be treated as follows by Young’s inequality ˜ x, W ¯ x ) + g1 (W ¯ x, W ˜ x) g1 (W wxI wxII ≤C ¯ x |)wxI , wxI + wxII (ζ + |W which can be treated in the Friedrichs-type estimates The (0, O(|Wx |2 ) nonlinear term may be treated as other source terms in the energy estimates 50 T NGUYEN AND K ZUMBRUN Specifically, the worst-case term O(|Wx |2 ) ∂xk W, ∂xk = − ∂xk+1 wII , ∂xk−1 O(|Wx |2 ) −∂xk wII (0)∂xk−1 O(|Wx |2 )(0) may be bounded by ∂xk+1 wII L2 W W 2,∞ W Hk − ∂xk wII (0)∂xk−1 O(|Wx |2 )(0) The boundary term will be contributed into the form (205) of Ib , and hence using the parabolic equations to get rid of this term as treating in (206) Thus, we may relax (A3′ ) to (A3), completing the proof of the general case (A1) − (A3) and the proposition 4.2 Energy estimate II We require also the following estimate: Lemma 4.3 ([HR]) Under the hypotheses of Theorem 1.7, let E0 := (1 + |x|2 )3/4 U0 H , and suppose that, for ≤ t ≤ T , the W 2,∞ norm of the solution U of (231) remains bounded by some constant C > Then, for all ≤ t ≤ T , (228) (1 + |x|2 )3/4 U (x, t) H4 ≤ M E0 eM t Proof This follows by standard Friedrichs symmetrizer estimates carried out in the weighted H norm Remark 4.4 An immediate consequence of Lemma 4.3, by Sobolev embedding: W 3,∞ ⊂ H and equation (231), is that, if E0 and U H are uniformly bounded on [0, T ], then (229) (1 + |x|)3/2 |U | + |Ut | + |Ux | + |Uxt | (x, t) is uniformly bounded on [0, T ] as well Stability analysis In this section, we shall prove Theorems 1.6 and 1.7 Following [HZ, MaZ3], define the nonlinear perturbation U = (u, v) by (230) we obtain ˜ (x, t) − U ¯ (x), U (x, t) := U (231) where linearized operator (232) where Ut − LU = Q(U, Ux )x , LU := −(AU )x + (BUx )x ¯x , B = B(U ¯) AU := dF (U¯ )U − (dB(U¯ )U )U and the second-order Taylor remainder: ¯ + U ) − F (U ¯ ) + A(U ¯ )U + (B(U ¯ + U ) − B(U))U ¯ Q(U, Ux ) = F (U x STABILITY OF BOUNDARY LAYERS 51 satisfying |Q(U, Ux )| ≤ C(|U ||Ux | + |U |2 ) (233) |Π1 Q(U, Ux )x | ≤ C(|U ||Ux | + |U |2 ) |Q(U, Ux )x | ≤ C(|U ||Uxx | + |Ux |2 + |U ||Ux |) |Q(U, Ux )xx | ≤ C(|U ||Uxx | + |U ||Uxxx | + |Ux ||Uxx | + |Ux |2 ) so long as |U | remains bounded For boundary conditions written in U −coordinates, (B) gives ˜ −h ¯ = (W ˜ (U + U ¯) − W ˜ (U ¯ ))(0, t) h(t) = h(t) (234) ˜ /∂ U˜ )(U ¯0 )U (0, t) + O(|U (0, t)|2 ) = (∂ W in inflow case and (235) ¯ = (w ¯) − w ¯ ))(0, t) h(t) = ˜ h(t) − h ˜II (U + U ˜ II (U ¯0 )U (0, t) + O(|U (0, t)|2 ) = (∂ w ˜II /∂ U˜ )(U ¯0 )U (0, t) + O(|U (0, t)|2 ) = m ¯b1 ¯b2 (U ¯0 )U (0, t) + O(|U (0, t)|2 ) = mB(U 5.1 Integral formulation We obtain the following: Lemma 5.1 (Integral formulation) We have ∞ G(x, t; y)U0 (y) dy U (x, t) = t + (236) ˜ y (x, t − s; 0)BU (0, s) + G(x, t − s; 0)AU (0, s) ds G t ∞ + t − H(x, t − s; y)Π1 Q(U, Uy )y (y, s) dy ds 0 ∞ ˜ y (x, t − s; y)Π2 Q(U, Uy )(y, s) dy ds G where U (y, 0) = U0 (y) Proof From the duality (see [ZH, Lemma 4.3]), we find that G(x, t − s; y) considered as a function of y, s satisfies the adjoint equation (237) or (238) (∂s − Ly )∗ G∗ (x, t − s; y) = 0, − Gs − (GA)y + GAy = (Gy B)y in the distributional sense, for all x, y, t > s > 0, where the adjoint operator of Ly is defined by (239) with V ∗ = V tr L∗y V := Vy∗ A + (Vy∗ B)y , 52 T NGUYEN AND K ZUMBRUN Likewise, for boundary conditions, we have, by duality (iii’) for all x, t > 0, G(x, t; 0) ≡ in the outflow case A¯∗ < 0; and G(x, t; 0)B = in the inflow case A¯∗ > 0, noting that no boundary condition need be applied on the hyperbolic part for the adjoint equations in the inflow case Thus, integrating G against (231), we obtain for any classical solution that t ∞ G(x, t − s; y)Q(U, Uy )y (y, s) dy ds = t (240) ∞ G(x, t − s; y)(∂s − Ly )U (y, s) dy ds = : I1 + I2 Integrating by parts and using the boundary conditions (iii’) on the boundary y = 0, we get t ∞ I1 = G(x, t − s; y)∂s U (y, s) dy ds t ∞ = ∂s G(x, t − s; y)U (y, s) dy ds ∞ + ∞ G(x, 0; y)U (y, t) dy − where note that G(x, t; y)U (y, 0) dy ∞ G(x, 0; y)U (y, t) dy U (x, t) = and also t ∞ I2 = G(x, t − s; y)(−Ly )U (y, s) dy ds t ∞ = G(x, t − s; y)((AU )y − (BUy )y )(y, s) dy ds t ∞ (−Gy A − (Gy B)y )U (y, s) dy ds = − t t Gy (x, t − s; 0)BU (0, s)ds − G(x, t − s; 0)AU (0, s)ds ˜ y B since HB ≡ 0, Combining these estimates, and noting that Gy B = G we obtain (236) by rearranging and integrating by parts the last term of t (241) ∞ G(x, t − s; y)Q(U, Uy )y (y, s) dy ds t ∞ = 0 ˜ (H + G)(x, t − s; y)Q(U, Uy )y (y, s) dy ds STABILITY OF BOUNDARY LAYERS 53 As an expression for Ux , we obtain the following Lemma 5.2 (Integral formulation for Ux ) We have (242) t ∞ Gx (x, t; y)U0 (y) dy − Ux (x, t) = t t ∞ (Hx − Hy )(x, t − s; y)Π1 Q(U, Uy )y (y, s) dy ds + 0 t − H(x, t − s; 0)Π1 Q(U, Uy )y (0, s) ds ˜ xy (x, t − s; 0)BU (0, s) + Gx (x, t − s; 0)AU (0, s) ds G + − ∞ H(x, t − s; y)Π1 Q(U, Uy )yy (y, s) dy ds t−1 ∞ 0 ∞ t + t−1 ˜ xy (x, t − s; y)Π2 Q(U, Uy )(y, s) dy ds G ˜ x (x, t − s; y)Π2 Q(U, Uy )y (y, s) dy ds G where U (y, 0) = U0 (y) Proof Differentiating the formulation (236) for U (x, t) with respect to x and noting that t t ∞ ∞ Hx φ dy ds = 0 t − t ∞ (Hx − Hy )φ dy ds t ∞ H(x, t − s; y)φy (y, s) dy ds − 0 H(x, t − s; 0)φ(0, s)ds and t−1 ˜ xy ψ dy ds = G 0 t − t−1 ∞ ∞ ˜ xy ψ dy ds G ˜ x ψy dy ds − G t t−1 ˜ x (x, t − s; 0)ψ(0, s)ds G are valid for any smooth functions φ, ψ, we obtain the lemma 5.2 Convolution estimates To establish stability, we use the following lemmas proved in [HZ, HR, RZ] Lemma 5.3 (Linear estimates I) Under the assumptions of Theorem 1.7, +∞ +∞ (243) ˜ t; y)|(1 + |y|)−3/2 dy ≤ C(θ + ψ1 + ψ2 )(x, t), |G(x, ˜ x (x, t; y)|(1 + |y|)−3/2 dy ≤ C(θ + ψ1 + ψ2 )(x, t), |G and so the latter is dominated by ψ1 + ψ2 , for ≤ t ≤ +∞, some C > 54 T NGUYEN AND K ZUMBRUN Lemma 5.4 (Linear estimates II) Under the assumptions of Theorem 1.7, if |U0 (x)| + |∂x U0 (x)| ≤ E0 (1 + |x|)−3/2 , E0 > 0, then, for some θ > 0, +∞ +∞ (244) H(x, t; y)U0 (y) dy ≤ CE0 e−θt (1 + |x|)−3/2 , Hx (x, t; y)U0 (y) dy ≤ CE0 e−θt (1 + |x|)−3/2 , and so both are dominated by CE0 (ψ1 + ψ2 ), for ≤ t ≤ +∞, some C > Lemma 5.5 (Nonlinear estimates I) Under the assumptions of Theorem 1.7, t +∞ t−1 (245) +∞ 0 ˜ y (x, t − s; y)|Ψ(y, s) dyds ≤ C(θ + ψ1 + ψ2 )(x, t), |G ˜ xy (x, t − s; y)|Ψ(y, s) dyds ≤ C(θ + ψ1 + ψ2 )(x, t), |G for ≤ t ≤ +∞, some C > 0, where Ψ(y, s) := (θ + ψ1 + ψ2 )2 (y, s) (246) Lemma 5.6 (Nonlinear estimates II) Under the assumptions of Theorem 1.7, t t H(x, t − s; y)Υ(y, s) dyds ≤ C(ψ1 + ψ2 )(x, t) +∞ (247) +∞ (Hx − Hy )(x, t − s; y)Υ(y, s) dyds ≤ C(ψ1 + ψ2 )(x, t) t +∞ t−1 ˜ x (x, t − s; y)|Υ(y, s) dyds ≤ C(ψ1 + ψ2 )(x, t) |G for all < t < +∞, some C > 0, where Υ(y, s) := s−1/4 (θ + ψ1 + ψ2 )(y, s) (248) We require also the following estimate accounting boundary effects Lemma 5.7 (Boundary estimates I) Under the assumptions of Theorem 1.7, if |h(t)| + |h′ (t)| ≤ E0 (1 + t)−1 , t t (249) H(x, t − s; 0)h(s) ds ≤ CE0 (ψ1 + ψ2 )(x, t) Hx (x, t − s; 0)h(s) ds ≤ CE0 (ψ1 + ψ2 )(x, t), for ≤ t ≤ +∞, some C > STABILITY OF BOUNDARY LAYERS 55 Proof Note that H(x, t; 0) ≡ for the outflow case A∗ < Consider the inflow case A∗ > (and thus a ¯∗ > 0) We have t H(x, t − s; 0)h(s) ds = e−η0 x/¯a∗ |h(− ≤ e−η0 |x| (1 + |x − a ¯∗ t|)−1 ≤ CE0 (ψ1 + ψ2 )(x, t), t (x − a ¯∗ t))| a ¯∗ Hx (x, t − s; 0)h(s) ds (x − a ¯∗ t))| a ¯∗ ≤ e−η0 |x| (1 + |x − a ¯∗ t|)−1 ≤ CE0 (ψ1 + ψ2 )(x, t), ≤ e−η0 x/¯a∗ |h| + |h′ | (− which completes the proof of the lemma Lemma 5.8 (Boundary estimates II) Under the assumptions of Theorem 1.7, if |h(t)| + |ht (t)| ≤ E0 (1 + t)−1 , t (250) t ˜ y (x, t − s; 0)Bh(s)+G(x, t − s; 0)Ah(s) ds G ≤ CE0 (θ + ψ1 + ψ2 )(x, t) ˜ xy (x, t − s; 0)Bh(s)+Gx (x, t − s; 0)Ah(s) ds G ≤ CE0 (θ + ψ1 + ψ2 )(x, t) for ≤ t ≤ +∞, some C > t−1 ˜ xy (x, t − s; 0) is nonProof The estimate on , where Gy (x, t − s; 0), G singular, follows readily by estimates similar to but somewhat simpler than those of Lemma (5.5), which we therefore omit t To bound the singular part t−1 , we integrate (238) in y from to +∞ to obtain ˜ y B + GA = − (251) G +∞ +∞ G(x, t − s; y)Ay dy + Gs (x, t − s; y) dy Substituting in the lefthand side of (250), and integrating by parts in s, we obtain (252) t t−1 +∞ ˜ y B + GA)h(s) ds = (G 0 +∞ − +∞ + Ay (y)G(x, τ ; y) dy h(t − τ ) dτ G(x, τ ; y) dy h′ (t − τ ) dτ G(x, 1; y) dy h(t − 1), 56 T NGUYEN AND K ZUMBRUN which by |G|dy ≤ C is bounded by max0≤τ ≤1 (|h| + |h′ |)(t − τ ) Combining this with the following more straightforward estimate (for large x, |x| > a+ n t) (253) t t−1 ˜ y (x, t − s; 0)Bh(s) ds ≤ G ˜ y (x, τ ; 0)|Bh(t − τ ) dτ |G ≤ C max |h(t − τ )| 0≤τ ≤1 τ −1 e−|x| /Cτ dτ = C|x|−2 max |h(t − τ )| 0≤τ ≤1 × (|x|2 /τ )e−|x| /Cτ dτ ≤ C max |h(t − τ )||x|−2 , 0≤τ ≤1 (254) t t−1 ˜ t − s; 0)Ah(s) ds ≤ G(x, ˜ τ ; 0)|Ah(t − τ ) dτ |G(x, ≤ C max |h(t − τ )| 0≤τ ≤1 0≤τ ≤1 τ −1 e−|x| −2 ≤ C max |h(t − τ )||x| 0≤τ ≤1 /Cτ dτ ≤ C max |h(t − τ )| τ −1/2 e−|x| /Cτ dτ , ˜ and the estimate (249) for H term (thus together with (254) for G = G+H), t we find that the contribution from t−1 has norm bounded by max (|h| + |h′ |)(t − τ )(1 + |x|)−2 ≤ CE0 (ψ1 + ψ2 )(x, t) 0≤τ ≤1 t−1 Combining this estimate with the one for , we obtain the first inequality in (250) For second inequality, we first differentiate (252) with respect to x to get (255) t +∞ ˜ xy B + Gx A)h(s) ds = (G 0 t−1 +∞ − +∞ + 1 Ay (y)Gx (x, τ ; y) dy h(t − τ ) dτ Gx (x, τ ; y) dy h′ (t − τ ) dτ Gx (x, 1; y) dy h(t − 1), which, by |Gx |dydτ ≤ C τ −1/2 dτ ≤ C, is bounded by max0≤τ ≤1 (|h|+ |h′ |)(t − τ ), similarly as above STABILITY OF BOUNDARY LAYERS 57 For the large x, clearly we still have similar estimates as (253) and (254) ˜ xy and G ˜ x These, estimate (249) for Hx , and (255) yield the contribufor G t t−1 tion from t−1 as above, which together with the estimate for completes the proof of (250) 5.3 Linearized stability In this subsection, we shall give the proof of Theorem 1.6 We first need the following estimates: Lemma 5.9 ([MaZ4]) Under the assumptions of Theorem 1.6, +∞ ˜ t; y)f (y) dy G(·, +∞ (256) H(·, t; y)f (y) dy Lp ≤ C(1 + t)− (1−1/r) |f |Lq , Lp ≤ Ce−ηt |f |Lp , for all t ≥ 0, some C, η > 0, for any ≤ q ≤ p and f ∈ Lq ∩ Lp , where 1/r + 1/q = + 1/p Lemma 5.10 Under the assumptions of Theorem 1.6, if |h(t)| ≤ E0 (1 + t)−1 , t (257) ˜ y (x, t − s; 0)Bh(s)+G(x, t − s; 0)Ah(s) ds G Lp ≤ CE0 (1 + t)− (1−1/p) for ≤ t ≤ +∞, some C > Proof This follows at once by the boundary estimate (250) and the fact that |(θ + ψ1 + ψ2 )(·, t)|Lp ≤ C(1 + t)− (1−1/p) Proof of Theorem 1.6 Sufficiency of (D) for linearized stability (the main point here) follows easily by applying the above lemmas to the following representation for solution U (x, t) of the linearized equations (10) ∞ G(x, t; y)U0 (y) dy U (x, t) = t + ˜ y (x, t − s; 0)BU (0, s) + G(x, t − s; 0)AU (0, s) ds G where U (y, 0) = U0 (y) and |U (0, s)| ≤ C|h(s)| ≤ C(1 + s)−1 by (11) in the inflow case, and |BU (0, s)| ≤ C|h(s)| ≤ C(1 + s)−1 by (12) in the outflow case, noting that G(x, t; 0) ≡ in this case Necessity follows by a much simpler argument, restricting x, y to a bounded set and letting t → ∞, noting that G is given by the ODE evolution of the spectral projection onto the finite set of zeros of D in ℜλ ≥ 0, necessarily nondecaying, plus an O(e−ηt ) error, η > 0, from which we find that asymptotic decay implies nonexistence of any such zeros; see Proposition 7.7 and Corollary 7.8, [MaZ3] for details 58 T NGUYEN AND K ZUMBRUN 5.4 Nonlinear argument In this subsection, we shall give the proof of Theorem 1.7 In fact, with the above preparations, the proof of nonlinear stability is also straightforward Lemma 5.11 (H local theory) Under the hypotheses of Theorem 1.7, then, for T sufficiently small depending on the H −norm of U0 , there exists a unique solution U (x, t) ∈ L∞ (0, T ; H (x)) of (231) satisfying |U (t)|H ≤ C|U0 |H (258) for all ≤ t ≤ T Proof Short-time existence, uniqueness, and stability are described in [Z2, Z4], using a standard (bounded high norm, contractive low norm) contraction mapping argument We omit the details Lemma 5.12 Under the hypotheses of Theorem 1.7, let U ∈ L∞ (0, T ; H (x)) satisfy (231) on [0, T ], and define (259) ζ(t) := sup x,0≤s≤t (|U | + |Ux |)(θ + ψ1 + ψ2 )−1 (x, t) If ζ(T ) and |U0 |H are bounded by ζ0 sufficiently small, then, for some ǫ > 0, (i) the solution U , and thus ζ extends to [0, T + ǫ], and (ii) ζ is bounded and continuous on [0, T + ǫ] Proof Boundedness and smallness of |U (t)|H on [0, T ] follow by Proposition 4.1, provided smallness of ζ(T ) and |U0 |H By Lemma 5.11, this implies the existence, boundedness of |U (t)|H on [0, T + ǫ], for some ǫ > 0, and thus, by Lemma 4.3, boundedness and continuity of ζ on [0, T + ǫ] Proof of Theorem 1.7 We shall establish: Claim For all t ≥ for which a solution exists with ζ uniformly bounded by some fixed, sufficiently small constant, there holds (260) ζ(t) ≤ C2 (E0 + ζ(t)2 ) From this result, provided E0 < 1/4C22 , we have that by continuous induction (261) ζ(t) < 2C2 E0 for all t ≥ From (261) and the definition of ζ in (259) we then obtain the bounds of (21) Thus, it remains only to establish the claim above Proof of Claim We must show that (|U |+|Ux |)(θ +ψ1 +ψ2 )−1 is bounded by C(E0 + ζ(t)2 ), for some C > 0, all ≤ s ≤ t, so long as ζ remains sufficiently small First we need an estimate for U (0, s) and Us (0, s) For the inflow case, by boundary condition estimate (234) and by the hypotheses on h(s), we have (262) |U (0, s)| ≤ C(h(s) + |U (0, s)|2 ) ≤ C(E0 (1 + s)−1 + |U (0, s)|2 ) STABILITY OF BOUNDARY LAYERS 59 from which by continuity of |U (0, t)| (Remark 4.4) and smallness of E0 , we obtain a similar estimate to (261): |U (0, s)| ≤ CE0 (1 + s)−1 (263) Similarly for an estimate of Ut (0, t), by taking the derivative of (234), we get |Us (0, s)| ≤ C(h′ (s) + |U ||Us |(0, s)) (264) ≤ C(E0 (1 + s)−1 + |U (0, s)||Us (0, s)|) which by the same argument as above yields |Us (0, s)| ≤ CE0 (1 + s)−1 (265) Next, for the outflow case with boundary condition (235), we have |BU (0, s)| ≤ CE0 (1 + s)−1 + O(|U (0, s)|2 ) (266) |(BU )s (0, s)| ≤ CE0 (1 + s)−1 + O(|U ||Us |(0, s)) Now by (259), we have for all t ≥ and some C > that |U (x, t)| + |Ux (x, t)| ≤ ζ(t)(θ + ψ1 + ψ2 )(x, t), (267) and therefore |Q(U, Uy )(y, s)| ≤ Cζ(t)2 Ψ(y, s) (268) |Π1 Q(U, Uy )y (y, s)| ≤ Cζ(t)2 Ψ(y, s) with Ψ = (θ + ψ1 + ψ2 )2 as defined in (246), for ≤ s ≤ t As an estimate for U (x, t), we use the representation (236) of U (x, t): ∞ G(x, t; y)U0 (y) dy |U (x, t)| = t + 0 ˜ y (x, t − s; 0)BU (0, s) + G(x, t − s; 0)AU (0, s)) ds (G t ∞ H(x, t − s; y)Π1 Q(U, Uy )y (y, s) dy ds + 0 t ∞ + 0 ˜ y (x, t − s; y)Π2 Q(U, Uy )(y, s) dy ds , G where by applying Lemmas 5.3-5.6 together with (268), we have ∞ G(x, t; y)g(y) dy (269) ∞ ≤ E0 ˜ t; y)| + |H(x, t; y)|)(1 + |y|)−3/2 dy (|G(x, ≤ CE0 (θ + ψ1 + ψ2 )(x, t) 60 T NGUYEN AND K ZUMBRUN t ∞ (270) ˜ y (x, t − s; y)Q(U, Uy )(y, s) dy ds G t ≤ Cζ(t)2 ∞ ˜ y (x, t − s; y)|Ψ(y, s) dy ds |G ≤ Cζ(t)2 (θ + ψ1 + ψ2 )(x, t) t ∞ 0 H(x, t − s; y)Π1 Q(U, Uy )y (y, s) dy ds ≤ Cζ(t)2 (271) ≤ Cζ(t)2 t ∞ t H(x, t − s; y)(θ + ψ1 + ψ2 )2 dy ds ∞ H(x, t − s; y)Υ(y, s) dy ds ≤ Cζ(t)2 (θ + ψ1 + ψ2 )(x, t) and, for the boundary term, we apply the estimate (263) and Lemma 5.8, yielding t (272) ˜ y (x, t − s; 0)BU (0, s) + G(x, t − s; 0)AU (0, s)) ds (G ≤ C(E0 + ζ(t)2 )(θ + ψ1 + ψ2 )(x, t) for the inflow Whereas, for the outflow case, noting that G(x, t − s; 0) ≡ in the outflow case, we apply the estimate (266), (267) and Lemma 5.8 to give the same estimate as above, yielding t ˜ y (x, t − s; 0)BU (0, s) ds ≤ C(E0 + ζ(t)2 )(θ + ψ1 + ψ2 )(x, t) G where we used (267) for |U (0, s)| ≤ ζ(t)(1 + t)−1/2 Therefore, combining the above estimates, we obtain (273) |U (x, t)|(θ + ψ1 + ψ2 )−1 (x, t) ≤ C(E0 + ζ(t)2 ) To derive the same estimate for |Ux (x, t)|, we first obtain by using Proposition 4.1, t |U (t)|2H ≤ Ce−θt |U0 |2H + C ≤ C(E0 + ζ(t)2 )t −1/2 e−θ(t−τ ) |U (τ )|2L2 + Bh (τ ) dτ , where Bh is the boundary function defined in Proposition 4.1, and thus by the one dimensional Sobolev embedding: |U (t)|W 3,∞ ≤ C|U (t)|H , (274) |Q(U, Ux )x | ≤ C(ζ (t) + 4C E02 )Υ |Q(U, Ux )xx | ≤ C(ζ (t) + 4C E02 )Υ where Υ = t−1/4 (θ + ψ1 + ψ2 ) STABILITY OF BOUNDARY LAYERS 61 Now again applying Lemmas 5.3-5.8 together with 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ed McGraw-Hill Series in Mechanical Engineering McGraw-Hill Book Co., Inc., New York, 1960 STABILITY OF BOUNDARY LAYERS [SZ] [YZ] [Z2] [Z3] [Z4] [ZH] 63 Serre, D and Zumbrun, K., Boundary layer stability in real vanishing-viscosity limit, Comm Math Phys 221 (2001), no 2, 267–292 Yarahmadian, S and Zumbrun, K., Pointwise Green function bounds and longtime stability of large-amplitude noncharacteristic boundary layers, Preprint (2008) K Zumbrun Multidimensional stability of planar viscous shock waves In Advances in the theory of shock waves, volume 47 of Progr Nonlinear Differential Equations Appl., pages 307516 Birkhă auser Boston, Boston, MA, 2001 K Zumbrun Stability of large-amplitude shock waves of compressible NavierStokes 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 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 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: nguyentt@indiana.edu Department of Mathematics, Indiana University, Bloomington, IN 47402 E-mail address: kzumbrun@indiana.edu ... nonlinear stability of sufficiently large- amplitude boundary- layers of the compressible Navier–Stokes equations of isentropic ideal gas dynamics, with adiabatic index γ ≥ 1, the first such result for. .. http://www.dfrc.nasa.gov/Gallery/photo/F-16XL2/index.html STABILITY OF BOUNDARY LAYERS particular, stability properties appear to be quite important for the understanding of this phenomenon For further discussion, including the related issues of. .. the stability of boundary layers assuming that the boundary layer solution is noncharacteristic, which means, roughly, that signals are transmitted into or out of but not along the boundary In