Resolvent Estimates and Quantification of Nonlinear Stability

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1 Resolvent Estimates and Quantification of Nonlinear Stability Heinz Otto Kreiss Department of Mathematics, UCLA, Los Angeles, CA 995 Jens Lorenz Department of Mathematics and Statistics, UNM, Albuquerque, NM Abstract The aim of this paper is to clarify the role played by resolvent estimates for nonlinear stability. We will give examples showing that a large resolvent may lead to a small domain of nonlinear stability. In other examples the resolvent is large, but the domain of nonlinear stability is completely unrestricted. Which case prevails, depends on the details of the problem. We will also show that the size of the resolvent depends in an essential way on the norms that are used. Key words: resolvent estimates, nonlinear stability, pseudospectrum, convection dominated flow AMS subject classification: 35G2 Abbreviated Title: Quantification of Nonlinear Stability 1 Introduction In this paper we consider ordinary and partial differential equations for time dependent functions u(t) or u(x, t). We assume that there is an asymptotically stable stationary state u. Then, by definition of asymptotic stability (see [4]), if the initial perturbation u() u (or u(, ) u ( )) is sufficiently small, the solution u(t) will approach u Supported by Office of Naval Research n14 9 j Supported by NSF Grant DMS and DOE Grant DE-FG3-95ER

2 as t. The aim of this paper is to give insight into the size of the perturbation, u() u, which one is allowed to apply without losing convergence to the stationary state. More specifically, we want to illustrate the role played by the resolvent of the linearized operator (about u ) for this stability question. Our emphasis on the resolvent is motivated by the recent interest in pseudospectra [1, 3, 5, 6]. Pseudospectra give quantitative information about the size of the resolvent. For numerical purposes it is important to not only perturb the initial data, but also the differential equation. To be more specific, consider a system of ODEs du dt u t = g(u), g : C n C n, (1.1) and let g(u ) =, where u is asymptotically stable. If one solves (1.1) numerically and interpolates the numerical values, one obtains a function v(t) which solves a perturbed differential equation v t = g(v) + η(t). Here η(t) is the result of a small numerical error. If η(t) ε 1 for t T, v() u ε 2, where t T is a finite time interval, then v(t) will remain close to u provided that ε 1 and ε 2 are sufficiently small. Again, it is of interest to quantify the size of the allowed perturbations. For example, if ε 1 and ε 2 are required to be extremely small, it might become practically impossible to compute u using time stepping. The techniques that we use in this paper allow for a discussion of perturbations of the differential equation, but for simplicity we will emphasize perturbations of the initial data. In Section 2 we consider systems of ODEs u t = Au + f(u) where A is a constant n n matrix and f(u) is a smooth nonlinearity vanishing quadratically at u =. The stationary state u = is asymptotically stable if all eigenvalues of A lie in the left half plane. We will make the quantitative assumptions 1 and Re λ 2δ < for all λ σ(a) (1.2) f(u) 1 2 C f u 2 for all u C n. (1.3) 1 With σ(a) we denote the set of eigenvalues of A. 2

3 Here C f is a positive constant and u = u u denotes the Euclidean norm; the corresponding matrix norm is also denoted by. Our aim is to determine a (realistic) value for ε such that u() < ε implies u(t) as t. (1.4) If A is a normal matrix, then one can choose ε = 4δ C f, (1.5) i.e., if the nonlinearity is scaled so that 1 2 C f = 1, then the distance 2δ of the spectrum of A from the imaginary axis gives the size of the allowed perturbation. If A is not normal, but one still uses the Euclidean norm in (1.3) and in the condition u() < ε, then the restriction on ε (in terms of C f and δ) might become much stronger. To illustrate this, we consider the n n matrices A = 2δ δ 1 2δ Applying Lyapunov s technique, we will show that the condition, < δ 1. (1.6) u() < 2 δ2n 1 C f (1.7) implies lim t u(t) =. Clearly, for small δ >, the restriction (1.7) is very severe if the dimension n is large or moderate. For example, if C f = 2, δ =.1, and n = 5, then (1.7) requires u < We will make plausible that a restriction of the form (1.7) is not only sufficient, but essentially also necessary for retaining convergence, lim t u(t) =, unless one makes more specific assumptions on f(u). In Section 2.3 we treat systems u t = Au+f(u), with A given in (1.6), by the resolvent technique. The resolvent constant of A is R := sup (si A) 1 c n Re s δ, n where c n > is a constant independent of < δ 1. Thus the resolvent grows fast for δ, and one obtains a requirement for u() which is similar to (1.7), but even slightly more restrictive. For small δ >, the matrix A in (1.6) is highly nonnormal and has a large resolvent constant. One might suspect that these properties are solely responsible for the small stability region (1.7). This is not true, however, since properties of the nonlinearity 3

4 are also important; they must be taken into account if one wants to obtain realistic conditions. We will show this for PDEs. In Section 3 we illustrate strengths and weaknesses of the resolvent technique for quantifying stability for PDEs. First, a strong point of the resolvent technique is that it can provide stability for parabolic equations in finite regions by just checking an eigenvalue condition. This check can be done by numerical computations. If the lower order terms in the equation have the wrong sign, it might be impossible to obtain such a stability result by energy estimates, i.e., by Lyapunov s technique. Second, to mention a limitation of the resolvent technique, consider a nonlinear hyperbolic equation with initial and boundary conditions u t = u x + uu x, x L, t, u(x, ) = u (x), u(l, t) =. Stability can be obtained using characteristics, but the resolvent technique fails since one does not gain a derivative in the resolvent estimate, which is uniform for Re s. In contrast, the nonlinear problem u t = u x + u 2, x L, t, can be treated using the resolvent. However, with increasing domain size L, the resolvent constant grows like const L, and one obtains only a poor characterization of the admissible perturbations. The result, which is too pessimistic, can be improved by using weighted norms. In Section 4 we consider a simple model for convection dominated flow, u t = u xx + u x + uu x, x L, t, (1.8) u(x, ) = u (x), u(, t) = u(l, t) =. For large L, the corresponding pseudospectrum has been studied extensively in [5]. With increasing L, the L 2 resolvent constant grows again like const L, and a straightforward application of the resolvent technique leads to rather severe smallness requirements for u. However, using the maximum principle and energy estimates, we will show that the size of u does not have to be restricted at all. These examples show shortcomings if the resolvent technique is applied too naively. Sometimes the results can be improved by using weighted norms or by taking more specific structures of the equation into account. For example, the nonlinearity uu x = 1 2 (u2 ) x has 4

5 conservation form, and one can use this by proving a corresponding resolvent estimate. More precisely, one can estimate the solution of su = u xx + u x + F x (x), u() = u(l) =, in terms of the L 2 norm of F. In this case the resolvent constant grows only like const L, as we will show in Section 4. To summarize, the main purpose of this paper is to relate resolvent estimates to the quantification of nonlinear stability. Our results show that the relation is not simple. First, the size of the resolvent depends in an essential way on the norms that one uses. Second, the choice of norms must be related to particular properties of the nonlinearity. Even with careful choices, however, the results of the resolvent technique might still be too pessimistic. 2 Systems of ODEs Consider a system u t = Au + f(u) where f : C n C n is a C 1 function vanishing quadratically at u =. We make the eigenvalue assumption (1.2) and assume the estimate f(u) 1C 2 f u 2 for all u. This is a condition formulated in the Euclidean norm. In Section 2.1 we describe Lyapunov s technique in an appropriate H norm to quantify the size of the initial data u for which convergence u(t, u ) as t can be guaranteed. The result is applied in Section 2.2 to a class of highly non normal problems. In Section 2.3 we will show how estimates of the resolvent (si A) 1 can be used to quantify nonlinear stability. Both methods are entirely satisfactory. 2.1 Lyapunov s Technique We first recall some elementary results on Hermitian matrices; these matrices are used to construct Lyapunov functionals. For Hermitian matrices H 1, H 2 C n n we write H 1 H 2 if u H 1 u u H 2 u for all u C n. Similarly, we write H 1 < H 2 if the above inequality is strict for u. Any H > determines an inner product and a corresponding norm, u, v H = u Hv, u 2 H = u Hu. The matrix norm corresponding to the vector norm H is also denoted by H. If 5

6 γ 1 I H γ 2 I, γ j >, then γ 1 u 2 u 2 H γ 2 u 2 for all u C n. For any B C n n and any u C n one obtains thus Bu 2 H γ 2 Bu 2 γ 2 B 2 u 2 γ 2 γ 1 B 2 u 2 H, B H γ2 γ 1 B, (2.1) and similarly B γ2 γ 1 B H. (2.2) The following result of linear algebra is well known. A proof is included for completeness. Theorem 2.1 If A satisfies the eigenvalue condition Re λ 2δ < δ 1 < for all λ σ(a), (2.3) then there exists a Hermitian matrix H > with The bound (2.4) implies exponential decay of e At in H, HA + A H 2δ 1 H <. (2.4) e At H e δ 1t, t. Proof: By Schur s theorem there exists a unitary matrix U so that U AU = Λ + R = diag(λ j ) + R, Re λ j 2δ <, where R is strictly upper triangular. We set D = diag(1, ε,..., ε n 1 ), ε >, 6

7 and consider D 1 U AUD = Λ + D 1 RD. The entries in D 1 RD are O(ε). Setting S = (UD) 1 one obtains SAS 1 + (SAS 1 ) = Λ + Λ + O(ε) 2δ 1 I for sufficiently small ε. Therefore, if one defines H = S S, one obtains HA + A H = S (Λ + Λ )S + O(ε) 2δ 1 H for sufficiently small ε >. This proves (2.4). To show the bound of e At, let u(t) solve u t = Au. Then one finds that and, therefore, u(t) H e δ 1t u() H. d dt u 2 H = u, u t H + u t, u H = u, (HA + A H)u 2δ 1 u 2 H Now consider the nonlinear problem u t = Au + f(u). Assuming the eigenvalue condition (2.3) we can construct H > with (2.4). With respect to H, the nonlinearity f(u) satisfies an estimate f(u) H 1 2 C H u 2 H for all u C n, (2.5) where C H = C f,h is a positive constant. We consider the change of u(t) 2 H for a solution u(t) of u t = Au + f(u) and obtain d dt u 2 H = u, u t H + u t, u H = u, (HA + A H)u + 2 Re u, f(u) H 2δ 1 u 2 H + 2 u H f(u) H 2δ 1 u 2 H + C H u 3 H = ( 2δ 1 + C H u H ) u 2 H. 7

8 If u H < 2δ 1 C H (2.6) then the above differential inequality implies u(t) as t at an exponential rate. Thus we have proved the following result. Theorem 2.2 Assume that A and H satisfies (2.4), and let f(u) satisfy (2.5) in the corresponding norm H. Then the condition (2.6) is sufficient for convergence, u(t, u ) as t. Theorem 2.2 says that the size of the initial perturbation, which one is allowed to apply to the initial data without losing convergence to the stable state, behaves essentially like the distance of the spectrum of A from the imaginary axis if a proper norm is used. This norm depends on A. If the eigenvalue condition (1.2) is assumed and A is normal, then the bound (2.4) is satisfied with H = I and δ 1 = 2δ. Therefore, for normal A, one can work entirely with the Euclidean norm. 2.2 Application to Highly Non Normal Problems Consider a system u t = Au + f(u) with A = 2δI + J, J = , < δ 1. (2.7) We assume that the nonlinearity f(u) satisfies (1.3), a condition formulated in the Euclidean norm, and that the matrix A has size n n. What is the implication of Theorem 2.2 for such systems if we want to formulate the smallness condition for u in the Euclidean norm? First, to apply Theorem 2.1 and construct a suitable H, we have to choose δ 1 with 2δ < δ 1 <. For simplicity, we let δ 1 = δ, though this choice could be optimized as a function of n, leading to slightly improved constants. Following the proof of Theorem 2.1, we set and obtain D = diag(1, δ,..., δ n 1 ) 8

9 thus D 1 AD = 2δI + δj, u D 1 ADu δ u 2 for all u C n. The matrix H constructed in the proof of Theorem 2.1 is simply From H = D 2 = diag(1, δ 2,..., δ 2n+2 ). I H δ 2n+2 I, < δ 1, one obtains the norm estimates u u H δ n+1 u, u C n. (2.8) Therefore, assumption (1.3) implies f(u) H 1 2 δ n+1 C f u 2 H. In other words, we can choose C H = δ n+1 C f in (2.5). Consequently, Theorem 2.2 (with δ 1 = δ) implies that the condition suffices to guarantee u(t, u ) as t. satisfied if we assume To summarize, we have proved the following result. Lemma 2.1 Consider a nonlinear system u H < 2 δn C f (2.9) Because of (2.8) the condition (2.9) is u < 2 δ2n 1 C f (2.1) u t = Au + f(u), u() = u, where the n n matrix A is given in (2.7) and where f(u) 1 2 C f u 2. If (2.1) holds, then u(t, u ) as t. 9

10 For small positive δ and large or moderate values of the dimension n, the condition (2.1) on u is very restrictive. Since we made several estimates in the derivation of the condition, one might suspect (2.1) to be much too pessimistic. In general, a condition of type (2.1) is essentially also necessary, however, to guarantee convergence u(t, u ) as t unless one makes more specific assumptions on f(u). To make this plausible, we first show the following lemma about e At. Recall that denotes the matrix norm corresponding to the Euclidean vector norm. Lemma 2.2 Consider the n n matrices A = 2δI + J given in (2.7). There is a positive constant c n, independent of < δ 1, so that Proof: First, we apply Theorem 2.1 with δ 1 = δ and c n max δn 1 t eat 1. (2.11) δn 1 I H = diag(1, δ 2,..., δ 2n+2 ) δ 2n+2 I to obtain e At H e δt, t. Then (2.2) yields the upper bound for e At. To prove the lower bound, we let t = 1/δ and obtain Since J n 1 = 1, one obtains e At = e A/δ = e 2( I + 1 δ J δ n 1 (n 1)! J n 1). e A/δ = γ n δ n 1 ( 1 + O(δ) ) with γ n = e 2 /(n 1)!, and the assertion follows. Example. Consider the system u t = Au + f(u) with A given in (2.7) and f(u) = (,,...,, u 2 1) T, u IR n. The fixed point equation Au + f(u) = has the nontrivial solution u = ( (2δ) n, (2δ) n+1,..., (2δ) 2n) T, i.e., there is a nontrivial fixed point of order u = O(δ n ). If u is small and we can neglect the nonlinear term f(u) in the equation u t = Au + f(u), then we have by Lemma 2.2, max t u(t) 1 c n δ n 1 u.

11 If u(t) can become of the order of u, i.e, of the order O(δ n ), then it is plausible that convergence u(t) u can occur for t. This can only be prevented by requiring that c n δ n+1 u << O(δ n ), i.e., by requiring u C δ 2n 1 with a sufficiently small constant C. This makes it plausible that our sufficient condition (2.1) is essentially also necessary. For n = 2 these plausibility considerations can be made completely rigorous by a phase plane argument. 2.3 The Resolvent Technique We show here how bounds for the norm of the resolvent, (si A) 1, can be used to quantify nonlinear stability for the equation u t = Au + f(u). As before, we assume f(u) 1 2 C f u 2 and make the eigenvalue assumption (1.2). Then it is clear that the resolvent constant R := sup (si A) 1 (2.12) Re s is finite. We will derive a value for ε so that (1.4) holds; ε will depend on R, C f, and A. For simplicity of presentation, we will assume that f(u) is quadratic, i.e., f(u) = B(u, u) where B(u, v) is bilinear. Then, if the initial data have size u = ε, we write u(t) = εũ(t) and obtain ũ t = Aũ + εf(ũ). After dropping in our notation, the problem reads u t = Au + εf(u), u() = u, u = 1. (2.13) Thus, for convenience, we have transformed the small parameter ε to the differential equation. Let us first consider the linear problem where F L 2 (, ). We write u t = Au + F (t), u() = u, (2.14) and obtain u(t) = e t u + v(t) (2.15) v t = Av + G(t), v() =, G(t) = F (t) + (A + I)e t u, (2.16) i.e., we transform to homogeneous initial data. If 11

12 ṽ(s) = e st v(t)dt, Re s, denotes the Laplace transform, then (2.16) becomes Therefore, sṽ = Aṽ + G. ṽ(s) = (si A) 1 G(s), which shows the importance of the resolvent. Using Parseval s relation, we obtain v 2 dt R 2 v(t) 2 dt = 1 ṽ(iξ) 2 dξ, 2π ) G 2 dt R (2 2 F 2 dt + A + I 2 u 2. The last inequality is not strong enough to treat nonlinear problems since the L 2 -integral of v does not control pointvalues of v. However, since v t = Av + G, we can estimate v t in terms of v. Because of d dt v 2 2 v v t and v() =, we have sup t< v(t) 2 2 v v t dt (2 A + 1) v 2 dt + G 2 dt ( (2 A + 1)R ) G 2 dt. The values of G(t) for t > T do not change the solution v(t) for t T. Therefore, a cut off argument shows that we may replace the limit by any finite T. This leads to the following bound of u(t) = e t u + v(t) in terms of F (t) and u. Lemma 2.3 For any T >, the solution of the linear problem (2.14) satisfies with T T u 2 dt + max t T u(t) 2 4R1 2 F 2 dt + κ 2 1 u 2 (2.17) R 2 1 = 2( A + 1)R 2 + 1, κ 2 1 = 2R 2 1 A + I

13 Proof: We have Substituting the bound T T T v 2 dt + max t T v(t) 2 R1 2 G 2 dt. T G 2 dt 2 F 2 dt + A + I 2 u 2 and noting that u 2 2 v 2 + 2e 2t u 2, the estimate for u follows. Remark: In our estimate for sup v 2 we have used the bound v t A v + G, which is not useful if A is large. If A is large, it is better to bound v t and u t by using the differentiated equation u tt = Au t + F t, u t () = Au + F (). In this way one obtains better bounds than (2.17) if Au + F () is of order one. In the PDE case, where the operator corresponding to A is unbounded, such a condition on Au + F () is made implicitly through smoothness and compatibility requirements for u. In the ODE case, our implicit assumption is that A is of order one, i.e., the resolvent constant R is the only large quantity in R 1 and κ 1. Now consider the nonlinear problem (2.13) for some ε > and recall u = 1. We know that the solution exists if we can derive an a priori estimate. Clearly, there exists T > with T u(t) 2 dt + max u(t) 2 2κ 2 1. (2.18) t T Let us first assume that there exists T > so that equality holds in (2.18). We then apply (2.17) with T = T and F (t) = εf(u(t)). Since F (t) ε 2 C f u(t) 2 we obtain 2κ 2 1 = T R 2 1ε 2 C 2 f u 2 dt + max u(t) 2 t T T u 4 dt + κ 2 1 R1ε 2 2 Cf 2 max u(t) 2 t T 4R1ε 2 2 Cf 2 κ κ 2 1. T u 2 dt + κ 2 1 This estimate implies that 1 2R 1 εc f κ 1. assumption In other words, if we make the smallness ε < 1 2C f R 1 κ 1, (2.19) 13

14 then equality in (2.18) cannot occur for finite T, and we have u(t) 2 dt + sup u(t) 2 2κ 2 1. t Since we can also bound u t by using the differential equation, we obtain u(t) as t if (2.19) is assumed. Theorem 2.3 Let u(t) solve (2.13) and let R 1 and κ 1 be determined as in Lemma 2.3, i.e., R 1 = O(1 + R), κ 1 = O(1 + R) for A = O(1). Here R is the resolvent constant (2.12). If then u(t) as t. ε < 1 2C f R 1 κ 1 We shall now estimate the resolvent constant R = R(δ, n) for the n n matrices (2.7). The following general result will be used. Lemma 2.4 Suppose the matrices A and H satisfy and Then we have HA + A H 2δH < γ 1 I H = H γ 2 I, γ j >. R = sup (si A) 1 γ2 1 Re s γ 1 δ. Proof: Let Re s and set B = A si. Clearly, and Theorem 2.1 yields From HB + B H 2δH, e Bt H e δt, t. B 1 = e Bt dt one obtains B 1 H 1/δ, and then the assertion follows from (2.2). 14

15 Application to (2.7). We will show the inclusion (2δ) n R = As noted in Section 2.2, the matrix sup (si A) 1 δ n. (2.2) Re s H = diag(1, δ 2,..., δ 2n+2 ) satisfies HA + A H 2δH and I H δ 2n+2 I. Therefore, the upper bound for R follows from Lemma 2.4. To show the lower bound, we note that the equation is solved by Au = b, b = (,...,, 1) T, u = ( (2δ) n, (2δ) n+1,..., (2δ) 1), u (2δ) n. Let us apply Theorem 2.3 to a nonlinear system (2.13) with A given in (2.7). Clearly, A 3 for < δ 1, and the inclusion (2.2) yields R 1 = O(δ n ) and κ 1 = O(δ n ). Therefore, there is a (computable) constant c >, independent of δ, n, and the nonlinearity f(u), so that ε c C f implies u(t) as t. Comparing this result with Lemma 2.1, we see that the resolvent technique and Lyapunov s technique lead to similar restrictions for perturbations, but the bound derived by the resolvent technique is worse by one power of δ. 3 Examples of Partial Differential Equations In this section we discuss stability for some model PDEs. Our intention is to illustrate strengths and weaknesses of the use of the resolvent. Notations. With (u, v) = L δ 2n ū(x)v(x) dx, u = (u, u) 1/2, we denote the L 2 scalar product and norm. For positive integers k, k u 2 H = D j u 2, D = d/dx, k j= 15

16 defines the corresponding Sobolev norm based on L 2. For T >, denotes a space time maximum norm. u,t = max { u(x, t) : x L, t T } A strength of the resolvent technique is to yield stability for parabolic problems in finite domains under an eigenvalue assumption. We illustrate this in our first example. Example 1: Consider a parabolic problem with initial and boundary conditions u t = P u + u 2, x L, t, (3.1) Here u(x, ) = u (x), u(, t) = u(l, t) =. (3.2) P u = u xx + a(x)u x + b(x)u(x). (3.3) We assume the functions a(x), b(x), and u (x) to be smooth. Lemma 3.1 The following two conditions are equivalent. (EIG) The eigenvalue problem has no eigenvalue λ with 2 Re λ. (RES) There is a constant R > so that P u = λu, u() = u(l) =, su P u = r, u() = u(l) =, implies u H 2 R r for all r L 2 and all s with Re s. 2 If a(x), b(x) are real, then the eigenvalues are real. 16

17 For a proof see Sect. 6 of [2], for example. It is easy to give sufficient conditions for the eigenvalue condition (EIG). For example, if a(x) and b(x) are real and b(x) for all x or b(x) a x (x) for all x, then (EIG) is fulfilled. This follows from the maximum principle applied to P or its adjoint P. The eigenvalue condition can also be checked numerically. In general, it is not trivial to give good quantitative values for a constant R satisfying the resolvent estimate (RES). (See the example in Section 4, however.) We will assume, nevertheless, that we have a resolvent estimate 3 u H 1 R r if su P u = r, u() = u(l) =, (3.4) where r L 2 and Re s. We will quantify stability for (3.1), (3.2) in terms of R. No attempt is made to optimize constants independent of R. Write u (x) = εũ (x), u(x, t) = εũ(x, t) where ũ H 4 = 1. After dropping the notation, we obtain u t = P u + εu 2, u = u at t =, (3.5) with u H 4 = 1. The boundary conditions are always u(, t) = u(l, t) =. Consider first the linear problem u t = P u + F (x, t), u = u at t =, (3.6) and write u(x, t) = e t u (x) + v(x, t). Then v solves v t = P v + G(x, t), v = at t =, (3.7) with G = F + e t (u + P u ). Laplace transformation yields the resolvent equation and (3.4) implies sṽ(x, s) = P ṽ(x, s) + G(x, s), Re s, ṽ(, s) H 1 R G(, s), Re s. By Parseval s relation this translates into the estimate v(, t) 2 H dt 1 R2 G(, t) 2 dt. Substituting G = F + e t (u + P u ) and u = e t u + v, we obtain 3 Though one can estimate u H 2, we will only need a bound for u H 1 to treat a nonlinear term like u 2, which does not depend on u x or u xx. 17

18 with u 2 H 1 dt κ 1 + 4R 2 F 2 dt (3.8) κ 1 = κ 1 (u, R) = 2R 2 u + P u 2 + u 2 H 1. We now derive a similar estimate for u t. To this end, differentiate the equation u t = P u + F with respect to t, Therefore, with u tt = P u t + F t, u t = P u + F (x, ) =: P u + F at t =. u t 2 H 1 dt κ 2 + 4R 2 F t 2 dt (3.9) κ 2 = κ 2 (u, R) = κ 1 (P u + F, R). Our bounds for u now control sup x,t u(x, t). In fact, from d dt u 2 2 u u t one obtains u(x, t) 2 Since u = at x = we also have ( u(x, t) 2 + u t (x, t) 2) dt. thus max u(x, t) 2 u(, t) 2 x H, 1 sup u(x, t) 2 x,t ( u 2 H 1 + u t 2 H 1 ) dt. As in Sect. 2.3, we may restrict the resulting bound to any finite time interval. This yields the following estimate for the solution of the linear problem (3.6): T ( ) T ( u 2 H 1 + u t 2 H dt + u 2 1,T κ 3 + 8R 2 F 2 + F t 2) dt (3.1) with κ 3 = κ 3 (u, R) = 2κ 1 (u, R) + 2κ 2 (u, R). To treat the nonlinear problem (3.5), we consider εu 2 (x, t) = F (x, t) 18

19 as a forcing. Suppose there is a (smallest) time T > so that the left hand side of (3.1) equals 2κ 3. Then it follows that Here 2κ 3 = T ( u 2 H 1 + u t 2 H 1 ) dt + u 2,T T ( κ 3 + 8R 2 ε 2 u (u 2 ) t 2) dt. (3.11) and Therefore, from (3.11), T T u 2 2 dt u 2,T (u 2 ) t 2 dt 4 u 2,T T T u 2 dt u t 2 dt. κ 3 4R 2 ε 2 u 2,T 16R 2 ε 2 κ 2 3, T ( u 2 + u t 2) dt i.e., 1 16R 2 ε 2 κ 3. This inequality follows from the assumption that the left hand side of (3.1) equals 2κ 3. In other words, if ε 2 < 1 16R 2 κ 3, then the left hand side in (3.1) remains less than 2κ 3 for all T >, and convergence u(, t) as t follows. It remains to discuss the behavior of κ 3 in the bound given above for ε 2. We assume R 1 and ε 1 and recall u H 4 = 1 and F = εu 2. Clearly, κ 1 (u, R) C 1 u 2 H 2, κ 2 (u, R) C 2 u 2 H 4 = C 2R 2, 19

20 and, therefore, κ 3 (u, R) C 3 R 2. Here the C j are constants which depend in a simple way on bounds for the coefficients a(x), b(x) of P and on bounds for their first two derivatives. If C = max { a, a xx, b, b xx } (3.12) is of order one, then C 3 is also of order one. In terms of the original, unscaled problem (3.1), (3.2) we have shown the following stability result. Theorem 3.1 Consider the nonlinear problem (3.1), (3.2) and assume the resolvent estimate (3.4) with R 1. There exists ε > so that u H 4 < ε implies The threshold ε can be chosen as u(, t) as t. ε = C R 2 where C > is a constant which depends only on (3.12). Next we consider a simple hyperbolic problem where stability can be discussed by the method of characteristic. The resolvent technique also applies, but there are shortcomings unless one uses weighted norms. Example 2: Consider the nonlinear hyperbolic problem with initial and boundary conditions u t = u x + u 2, x L, t, (3.13) u(x, ) = u (x), u(l, t) =. (3.14) We are interested in the case where L, the length of the domain, is large. Clearly, the problem can be solved by the method of characteristics. The characteristics are the straight lines (x(t; x ), t) with dx/dt = 1, i.e., x(t; x ) = x t, and along each characteristic the solution u(x, t) satisfies d dt u(x(t; x ), t) = u 2 (x(t; x ), t). (3.15) 2

21 From the boundary condition u(l, t) = one obtains u(x, t) = for L t x L provided the solution exists. From the initial condition u(x, ) = u (x ) one obtains for u(x t, t) = u (x ) 1 tu (x ) t x, x L. Therefore, the solution exists and is zero for t > L if and only if u (x ) < 1 x for < x L. Let us now treat the problem by the resolvent technique. Proceeding as in Example 1, we consider the linear problem v t = v x + G(x, t), v(x, ) = v(l, t) =. (3.16) Laplace transformation leads to the resolvent equation for Re s. By Duhamel s principle, sṽ(x, s) = ṽ x (x, s) + G(x, s), ṽ(l, s) = Therefore, ṽ(x, s) L x G(ξ, s) dξ (L x) 1/2 G(, s). ṽ(, s) L2 G(, s) 2. (3.17) The estimate is sharp, showing that the resolvent constant (with respect to L 2 norms) grows like const L as L. The basic resolvent estimate (3.17) implies v(, t) 2 dt 1 2 L2 G(, t) 2 dt for the solution v(x, t) of (3.16). As explained in [2], one can derive similar estimates for v t, v tt, etc., and can then use these to quantify stability for the nonlinear problem (3.13), (3.14). However, since the approach is based on (3.17), the resulting conditions cannot reflect the weight 1, which for large x is suitable for the admissible perturbations. x 21

22 To obtain a better characterization of the admissible perturbations, we measure ṽ and G in different norms. Integration by parts yields and one obtains (ṽ, (x + 1)ṽ x ) = (ṽ x, (x + 1)ṽ) ṽ 2 ṽ(, s) 2, 2Re (ṽ, (x + 1)ṽ x ) + ṽ 2. Substituting ṽ x = sṽ G and noting that Re s, we find the estimate thus ṽ 2 2Re (ṽ, (x + 1) G), ṽ 2 (x + 1) G. Therefore, if one uses the L 2 norm for the solution and a weighted L 2 norm for the data, the corresponding resolvent constant remains bounded for L. The estimate then reflects the admissible perturbations realistically. Example 3: Consider the hyperbolic problem u t = u x + uu x, x L, t, (3.18) with initial and boundary conditions (3.14). We assume that u (x) is a smooth function which is compatible with the boundary condition and, for simplicity, we assume that u (x) 1 2. The problem can again be solve by the method of characteristics. Here the characteristic (x(t; x ), t) starting at (x, t) = (x, ) is the straight line with i.e., dx/dt = 1 u (x ) 1 2, (3.19) x(t; x ) = x (1 + u (x ))t. The solution u(x, t) carries the value u (x ) along the characteristic. If the characteristics do not cross inside the strip x L, t, then the solution is smooth in the strip and, because of (3.19) and u(l, t) =, the solution is zero for t 2L. Suppose two characteristics cross inside the strip. Then we have 22

23 x (1 + u (x ))t = x 1 (1 + u (x 1 ))t for some x < x 1 L and some < t 2x. Therefore, 1 1 2x t = u (x 1 ) u (x ) =: r.h.s. (3.2) x 1 x It is easy to give conditions on the derivative u x (x) which prevents such a crossing of characteristics. For example, assume that u x (x) α (x + 1) 2, x L, (3.21) where α >. Then the right hand side of (3.2) is bounded by r.h.s. α (x + 1)(x 1 + 1) < α x. This makes (3.2) impossible for α = 1. To summarize, if the initial function u 2 satisfies the bound (3.21) with α = 1, then the solution remains smooth and is zero for t 2L. 2 The resolvent technique cannot be used for Example 3, because, when applying the technique, we regard the nonlinear term as forcing. However, for hyperbolic equations, one does not gain a derivative in the resolvent estimate which is uniform for Re s and, therefore, one cannot treat nonlinearities depending on u x. In Section 4 we replace (3.18) by the parabolic equation u t = u x + uu x + u xx. For the parabolic problem the resolvent technique applies. However, as we will see, the stability results are too pessimistic. 4 A Model for Convection Dominated Flow For large L, the initial boundary value problem u t = u xx + u x + uu x, x L, t, (4.1) u(x, ) = u (x), u(, t) = u(l, t) =, (4.2) may be considered as a simple model for convection dominated flow. We assume the initial function u (x) to be real, smooth, and compatible with the boundary conditions. (Because of smoothing in time, it is easy to relax smoothness 23

24 and compatibility assumptions.) Before discussing resolvent estimates, let us show that u(, t) as t, without any restriction on the size of u (x). The elementary proof combines the maximum principle with energy estimates. Lemma 4.1 The solution of (4.1), (4.2) satisfies u(x, t) u. Proof: We may assume that u >. For w(x, t) = e αt u(x, t), α >, one obtains w t = w xx + w x + w( α + e αt w x ). Therefore, a maximum of w(x, t) over x L, t T, can only be attained at t =. This yields w(x, t) < w for t >, and the assertion follows for α. The following result shows that the maximum norm of u decays exponentially in time. (For large L, the decay is slow.) Theorem 4.1 The solution of (4.1), (4.2) satisfies the estimate u(, t) 2 2e αt u { 1 4 K2 u 2 + u x 2} 1/2 where K = 1 + u, α = π 2 /L 2. Proof: The usual energy estimate reads i.e., 1 2 d dt u(, t) 2 = (u, u t ) = (u, u xx ) + (u, u x ) + (u, uu x ) = u x (, t) 2, (4.3) t u(, t) u x (, τ) 2 dτ = u 2. (4.4) For any function g H 1 (, L) with g() = g(l) = we have g x 2 α g 2, α = π 2 /L 2, as follows by sin expansion. Therefore, (4.3) yields d dt u(, t) 2 2α u(, t) 2, 24

25 i.e., u(, t) e αt u. Furthermore, using (4.1), (4.2) and the previous lemma, 1 2 Integration in time yields d dt u x(, t) 2 = (u x, u xt ) = (u xx, u t ) = u xx 2 (u xx, (1 + u)u x ) u xx 2 + K u x u xx 1 4 K2 u x (, t) 2. u x (, t) 2 u x 2 1 t 2 K2 u x (, τ) 2 dτ 1 4 K2 u 2, where the last estimate follows from (4.4). Finally, since u(, t) = and (u 2 ) x = 2uu x, we obtain This proves the theorem. u(, t) 2 2 u(, t) u x (, t) 2e αt u { 1 4 K2 u 2 + u x 2} 1/2. Resolvent Estimates. If one wants to apply the resolvent technique to (4.1), (4.2), one must estimate the solution u(x) of the BVP su = u xx + u x + g(x), u() = u(l) =, Re s, (4.5) in terms of g(x). (Note that the functions u and g are now complex valued.) An elementary approach is to bound the Green s function. One obtains the following estimate. Lemma 4.2 The Green s function of (4.5) satisfies G(x, y) 4 1 e L =: C L. 25

26 Therefore, u C L L g 4L g. (4.6) Linear growth of the resolvent constant (w.r.t. L 2 norms) with increasing L cannot be improved (though the constant 4 might not be optimal). Proof: a) The roots λ 1,2 of the characteristic equation λ 2 + λ s = satisfy Setting we have Re λ 1 1, Re λ 2 for Re s. (4.7) a(x) = e λ 1x e λ 2x, b(x) = e λ 1(x L) e λ 2(x L), G(x, y) = where W (y) is the Wronskian of a(y), b(y), a(y)b(x) W (y), y x L, b(y)a(x) W (y), x y L, W (y) = e y (λ 1 λ 2 )(e λ 2L e λ 1L ). Consider the case y x, for example. (The case x y is treated similarly.) Noting that λ 1 + λ 2 = 1, we can write G(x, y) = (eλ 1y e λ 2y ) (e λ 1(x L) e λ 2(x L) ) e y (λ 1 λ 2 ) (e λ 2L e λ 1L ) = (e λ 2y e λ 1y ) (e λ 1x e λ 2(x L)+λ 1 L ) (λ 1 λ 2 ) (e (λ 1 λ 2 )L 1) = eλ 1x λ 2 y e λ1(x y) e λ 2(x L y)+λ 1 L + e λ 2(x L)+λ 1 (L y) (λ 1 λ 2 ) (e (λ 1 λ 2 )L 1) N =: D. Each of the four terms in the numerator N has the form e q with Re q, which implies that N 4. Also, because of (4.7), D 1 e L. This shows the pointwise bound G(x, y) for the Green s functions of (4.5). Therefore, 4 1 e L =: C L 26

27 u C L L g. (4.8) b) Consider (4.5) for large L with s = and g(x) 1. The solution is u(x) x L, except for an initial layer. Since u L3 and g 2 = L, it follows that the resolvent constant grows at least like L/ 3 as L. Using integration by parts, one can extend the previous lemma and obtain a bound of u H 2 in terms of g. The constant C L 4 in the proof below is defined in Lemma 4.2. Theorem 4.2 There is a constant K (of order one) so that if (4.5) is assumed and L 1. u H 2 KL g (4.9) Proof: Multiply (4.5) by ū(x) and integrate by parts to obtain s u 2 + u x 2 = (u, u x ) + (u, g). (4.1) Here (u, u x ) = (u x, u) is purely imaginary. Taking real parts in (4.1) yields Re s u 2 + u x 2 u g. Since Re s and u C L L g we obtain u x 2 C L L g 2 (4.11) and Re s u g. (4.12) Taking imaginary parts in (4.1) yields These bounds imply that Im s u u x + g. s 2 u 2 g 2 + ( u x + g ) 2 g 2 (3 + 2C L L). Finally, using the differential equation for u, u xx g + u x + s u 27

28 and the theorem is proved. g ( 1 + C L L C L L ), Remark: The proof shows that one can choose K = 4 + ε, ε >, in (4.9) if L is sufficiently large. Using the resolvent estimate provided by Theorem 4.2, one can prove a nonlinear stability result for (4.1), (4.2) and also for more general equations u t = u xx + u x + f(u, u x, u xx ) with initial and boundary conditions (4.2). Here it is assumed that f(u, u x, u xx ) is a smooth nonlinearity which vanishes quadratically at (u, u x, u xx ) =. One obtains nonlinear stability, i.e., u(, t) as t, if the initial data are restricted in size as u H k c L 2. (4.13) Here c > is a sufficiently small constant and the index k is sufficiently large. To prove such a result one can basically argue as in Section 3, Example 1. For details we refer to [2]. For the particular nonlinearity f = uu x in (4.1), a restriction of type (4.13) is not necessary, however, as shown in Theorem 4.1. We will now prove that the size of the resolvent constant, which is KL in Theorem 4.2, can be reduced to K 1 L if one uses different norms. Since the nonlinearity in (4.1) has conservation form, f = 1 2 (u2 ) x, we consider a resolvent equation su = u xx + u x + F x (x), u() = u(l) =, Re s. (4.14) Theorem 4.3 There is a constant K 1 (of order one) so that if (4.14) is assumed and L 1. u x + u K 1 L F (4.15) Proof: a) Multiply (4.14) by ū(x) and integrate by parts to obtain s u 2 + u x 2 = (u, u x ) (u x, F ). (4.16) Here (u, u x ) is purely imaginary. Taking real parts in (4.16) yields Re s u 2 + u x 2 u x F, 28

29 thus u x F and Re s u 2 u x F F 2. Taking imaginary parts in (4.16) yields Im s u 2 u x F + u u x F 2 + u F. Adding the estimates for Re s u 2 and Im s u 2, we obtain This implies that s u 2 2 F 2 + u F 2 F 2 + s 2 u s F 2. If s 1/ L, then we obtain s u 2 ( ) F 2. s 1 L u 2 (4 + L) F 2, s 1 L. (4.17) Since b) To estimate the solution u of (4.14) for small s, let U(x) denote the solution of = U x + U + F (x), U() =. (4.18) it is easy to show that x U(x) = e (x y) F (y)dy U F and U(L) F. (4.19) We write u = U + v and obtain for the new variable v(x), sv = v xx + v x su(x), v() =, v(l) = U(L). (4.2) 29

30 Let v = v 1 + v 2 where By Lemma 4.2 and (4.19), sv 1 = v 1xx + v 1x su(x), v 1 () = v 1 (L) =, sv 2 = v 2xx + v 2x, v 2 () =, v 2 (L) = U(L). v 1 C L L s U C L L s F. The function v 2 (x) solves a homogeneous differential equation. Therefore, v 2 (x) = c 1 e λ 1x + c 2 e λ 2(x L), where λ 1,2 are the roots of λ 2 + λ s =, thus Re λ 1 1, Re λ 2. The coefficients c 1,2 are determined by the boundary conditions for v 2. It is not difficult to show that for L 1. Therefore, c 1 2 U(L) and c 2 2 U(L) v 2 2 U(L) ( e λ 1x + e λ 2(x L) ) 4 L U(L) 4 L F. Since u = U + v 1 + v 2, our estimates show that u F ( 1 + C L L s + 4 L ). We use this bound for s 1/ L to obtain u F ( 1 + (4 + C L ) L ), s 1 L. Since we have already shown (4.17) and since u x F, the assertion follows. Using the resolvent estimate provided by Theorem 4.3, one obtains nonlinear stability for (4.1), (4.2) under a restriction of the form u H k c L. Here c > is a sufficiently small constant and the index k is sufficiently large. 3

31 Generalizations. The estimates in Theorems 4.2 and 4.3 are based on Lemma 4.2, which uses an explicit bound for a Green s function. For more general equations and systems of equations, such bounds will be difficult to obtain. One can prove resolvent estimates also without using the Green s function, however. A systematic approach is to write the equation as first order system and to diagonalize. Then, essentially, one needs to discuss the roots of the characteristic equation and one has to estimate the solutions of scalar first order equations. With this approach one can also derive resolvent estimates for half space problems. We refer to [2] for details. 5 Conclusions and Remarks We have demonstrated how resolvent estimates can be used to quantify the domain of nonlinear stability of stationary solutions for initial value problems. For ODEs and for parabolic differential equations on bounded domains with O(1) coefficients, the technique can give satisfactory results. In other situations, e.g. for u t = u xx + u x + uu x in a large x interval, the result is too pessimistic. Hyperbolic problems like u t = u x + uu x cannot be treated at all by the resolvent technique as presented in the paper since one does not gain a derivative in the resolvent estimate uniformly for Re s. A major difficulty is that resolvent estimates are very sensitive to the choice of norms. To get good results, the norms should be related to the form of the nonlinearity and to properties of the initial disturbance. For example, one should take the conservation form of the nonlinearity uu x = 1 2 (u2 ) x into account by showing a resolvent estimate with right hand side F x. In another example, u t = u x + u 2, we have shown how the use of weighted norms may improve the result. However, even with carefully chosen norms, the final result may still be too pessimistic. Concerning pseudospectra, we believe that they may be useful in the matrix case and for parabolic equations in bounded domains with O(1) coefficients. In the latter case, the dynamics is essentially restricted to a subspace of finite dimension, and L 2 L 2 resolvent estimates are appropriate. (If the nonlinearity depends also on u x or u xx, one needs resolvent estimates from L 2 to H 1 or H 2.) In these cases pseudospectra may give valuable insight into the size of the resolvent. For problems on large or infinite domains the relevant resolvent estimates are rather delicate. It is not clear, at present, how numerical techniques (discretization and numerical computation of pseudospectra of matrices) can be used in these cases. 31

32 References [1] T. Braconnier and N.J. Higham, Computing the field of values and pseudospectra using the Lanczos method with continuation, BIT, 36 (1996), pp [2] H.-O. Kreiss and J. Lorenz, Stability for time dependent differential equations, Acta Numerica 1998, Cambridge U. Press, 1998, pp [3] S.-H. Lui, Computation of pseudospectra by continuation, SIAM J. Sci. Comput., 18 (1997), pp [4] A. M. Lyapunov, General problem of the stability of motion, Collected works, Vol.2, Izdat. Akad. Nauk SSSR, Moscow, (Russian). [5] S. Reddy and L. Trefethen, Pseudospectra of the convection diffusion operator, SIAM J. Appl. Math. 54 (1994), [6] L. Trefethen, Pseudospectra of linear operators, SIAM Review 39 (1997),