Nonlinear Stability of Rotating Patterns

Transcription

1 Dynamics of PDE, Vol.5, No.4, 349-4, 28 Nonlinear Stability of Rotating Patterns Wolf Jürgen Beyn and Jens Lorenz Communicated by Yannis Kevrekidis, received August 21, 28. Abstract. We consider 2D localized rotating patterns which solve a parabolic system of PDEs on the spatial domain Ê 2. Under suitable assumptions, we prove nonlinear stability with asymptotic phase with respect to the norm in the Sobolev space H 2. The stability result is obtained by a combination of energy and resolvent estimates, after the dynamics is decomposed into an evolution within a three dimensional group orbit and a transversal evolution towards the group orbit. The stability theorem is applied to the quintic cubic Ginzburg Landau equation and illustrated by numerical computations. Contents 1. Introduction and Main Result Decomposition of the Dynamics Resolvent Estimates for L Fredholm Theory Estimates for e tl and e tl Proof of Nonlinear Stability Existence and Estimate of D φ u The Quintic Cubic Ginzburg Landau Equation 387 Appendix A. Perturbation Theorem for C -Semigroups 395 References Introduction and Main Result 1.1. Rotating Patterns. Consider a system of reaction diffusion equations for a vector function U(x, t), 1991 Mathematics Subject Classification. 35B35, 35B4, 35K57. Key words and phrases. Rotating patterns, asymptotic stability, nonlinear stability, relative equilibria, group action, Ginzburg Landau equation. Supported by SFB 71 Spectral Structures and Topological Methods in Mathematics, Bielefeld University. 349 c 28 International Press

2 35 WOLF JÜRGEN BEYN AND JENS LORENZ (1.1) U t = A U + f(u), x Ê 2, U(x, t) Ê m, where A Ê m m is a positive definite matrix 1 and f : Ê m Ê m, f C 4, is a smooth nonlinearity. We set ( ) ( ) cosθ sinθ 1 R θ =, J = R sin θ cosθ π/2 =, 1 and assume a solution U (x, t) of (1.1) of the form (1.2) U (x, t) = u (R ct x) where c Ê, c, and u : Ê 2 Ê m is a smooth function. Writing (1.2) in polar coordinates, (1.3) U pol (r, φ, t) = u pol (r, φ ct), we see that the solution U (x, t) = U pol (r, φ, t) given in (1.2) describes a pattern that rotates with angular velocity c about the origin, x =. The aim of this paper is to give conditions for the pattern u (x) and the linearization of equation (1.1) about u (x) which guarantee nonlinear stability with asymptotic phase of the rotating pattern under small initial perturbations. A precise stability statement is formulated in Theorem 1.1 below. In Section 8 we apply the stability theorem to the cubic quintic Ginzburg Landau equation. An essential assumption is that the pattern u (x) is localized in the following sense: Assumption 1: For some constant vector u Ê m we have 2 (1.4) (1.5) (1.6) sup u (x) u as R, x R sup D α u (x) as R for 1 α 2, x R u u H 2 (Ê, Ê m ). Remark 1: A class of patterns that are not localized in the above sense are Archimedian spirals. These satisfy u pol (r, φ) u (κr + φ) as r where u (ξ) is a non constant 2π periodic function. Spectral stability and instability of Archimedean spirals is discussed in [17]. An extension of the nonlinear stability results of the present paper to Archimedian spirals is non trivial, however, and will be the subject of future work. The following assumption greatly facilitates our stability proof: Assumption 2: Let u Ê m denote the constant asymptotic state of the pattern introduced in Assumption 1. Then the matrix B := f (u ) is negative definite: 1 The Euclidean inner product on m and Ê m is denoted by u, v = P ū j v j with corresponding norm u = u, u 1/2. The matrix A Ê m m is assumed to satisfy u, 1 2 (A+AT )u 2β A u 2 for all u Ê m where β A >. We then write A 2β A I >. 2 With D α we denote a partial derivative of order α. For the Sobolev spaces used see Section 2.1.

3 NONLINEAR STABILITY OF ROTATING PATTERNS 351 B 2βI <. Henceforth we will assume u = for convenience and without loss of generality. Let us transform equation (1.1) to a co rotating frame: The function U(x, t) = u(r ct x, t) solves (1.1) if and only if u(x, t) solves (1.7) u t = A u + cd φ u + f(u) with D φ = x 2 D 1 + x 1 D 2, D j = / x j. The function u (x) determining the rotating pattern (1.2) is a time independent solution of (1.7), i.e., if we define (1.8) L u = A u + cd φ u then L u + f(u ) =. In other words, u is an equilibrium for equation (1.7). A fundamental and well recognized difficulty of any stability result of an equilibrium u is that u is not an isolated equilibrium. Besides u, any function u(x) = u (R θ x) also satisfies (1.9) A u + cd φ u + f(u) =. It is more important, however, that any solution u of (1.9) gives rise to a three parameter family of solutions U(x, t) of (1.1): If η Ê 2 and θ Ê are arbitray, then (1.1) U(x, t) = u (R ct θ (x η)) solves (1.1), representing a pattern obtained from U (x, t) = u (R ct x) by shifting and rotating the x coordinates. For fixed t, the spatial patterns x U(x, t) in (1.1) form the three dimensional group orbit of u ; see Section 1.3. For the perturbed dynamics near U (x, t) one must expect different decay behaviour within and towards the group orbit of u, and any stability analysis must take this into account. We remark that the three parameter family of patterns x U(x, t) in (1.1) typically do not solve the equilibrium equation (1.9). Nevertheless, under suitable assumptions, the linearization of (1.9) about u has a three dimensional invariant subspace corresponding to three eigenvalues on the imaginary axis; these eigenvalues are and ±ic Outline and Discussion. The main result of the paper is the Stability Theorem, Theorem 1.1. To motivate and formulate it, we present some background material on group action in Section 1.3. The presentation is specialized to our application, namely to take the 3 dimensional group orbit of u into account. A major step of the stability analysis is, then, to decompose the perturbed solution u(x, t) into two parts: One part evolves within the 3 dimensional group orbit of u, the other part in a linear complementary subspace W of codimension 3. Roughly speaking, we show that the evolution in W decays exponentially; this allows us,

4 352 WOLF JÜRGEN BEYN AND JENS LORENZ a posteriori, to control the dynamics within the group orbit and to prove that it settles to a precise asymptotic form. To establish exponential decay of the (linearized) dynamics within W turns out to be quite delicate. Our approach to obtain this result may be of some independent interest and is formulated as a theorem on C semigroups in Section A. The issue is the following: Suppose one has established exponential decay for some C semigroup, e ta Ce βt, t, β >, and wants to show exponential decay for a semigroup e t(a+b) where B is a bounded linear operator. Of course, in general, exponential decay for e t(a+b) does not hold. Suppose, however, that Re λ β < for all eigenvalues λ of A+B. In general, exponential decay of the semigroup e t(a+b) still cannot be concluded since the operator e t(a+b), t >, may have continuous spectrum reaching into the right half plane (see [13] or [12] for an example 3 and [8, Ch.IV] for a recent overview of the problem). However, if we add the assumption that the operators Be ta, t >, are compact, then exponential decay of e t(a+b) does follow. We will prove this in Appendix A. Using this abstract result, exponential decay of the linearized dynamics in the complementary space W follows and the Stability Theorem can be proved. We refer to Section 1.7 for a discussion of related techniques in the literature, in particular for proving stability of traveling waves Relative Equilibria and Group Action. We will explain the notions in the context of equation (1.1). The Euclidean Group SE(2). Let SE(2) = Ê 2 S 1 denote the Euclidean group consisting of all pairs with group operation γ = (η, θ), η Ê 2, θ S 1, γ γ = (η, θ) ( η, ( θ) = η + R θ η, θ + θ ). Here S 1 = Ê/(2π) denotes the circle group. The unit element in SE(2) is denoted by ½ = (, ). 3 Theorem of [12] gives an example of an operator A with a value µ in the continuous spectrum of e A for which all solutions λ of the equation e λ = µ lie in the resolvent of A.

5 NONLINEAR STABILITY OF ROTATING PATTERNS 353 Group Action. Let u : Ê 2 Ê m denote any function and let γ = (η, θ) SE(2). One defines the group action by (1.11) ( ) a(γ)u (x) = u(r θ (x η)), x Ê 2, γ = (η, θ). Our analysis below will be carried out in the function space H 2 Eucl = {u H2 : D φ u L 2 }. (For definitions of spaces and norms, see Section 2.1.) It is easy to see that u H 2 Eucl implies a(γ)u H2 Eucl and ( ) a(γ γ)u = a( γ) a(γ)u. Let us define the operator F : H 2 Eucl L2 by F(u) = A u + f(u). It is well known that F is equivariant with respect to the action of the group, i.e., (1.12) F(a(γ)u) = a(γ)f(u), γ SE(2), u H 2 Eucl. Definition 1.1. A relative equilibrium of the system (1.1) is a solution U (x, t) of the form (1.13) U (x, t) = [a(γ(t))u ](x), x Ê 2, t, where u H 2 Eucl and γ C1 ([, ), SE(2)). By definition, the group orbit of a function u HEucl 2 consists of all functions a(γ)u ( ), γ SE(2). Using this terminology, a relative equilibrium of the system (1.1) is a solution U (x, t) of (1.1) that moves within the group orbit of some fixed pattern u HEucl 2. To ensure that the function (1.13) solves (1.1), the motion γ(t) on the group is by no means arbitrary. In fact, it always has the form γ(t) = exp(gt) for some element g in the Lie algebra, but we will not make explicit use of this fact, see [4, Theorem 7.2.4]. In our case, assume that the functions (1.14) ϕ 1 = D 1 u, ϕ 2 = D 2 u, ϕ 3 = D φ u are linearly independent. Then one can show that a relative equilibrium (1.13) is either a rotating wave, (1.15) U (x, t) = u (R ct (x x ) + x ) for some c, x Ê 2, or a drifting wave, (1.16) U (x, t) = u (x tv ) for some v Ê 2. For the solution (1.15) of (1.1) the point x is the center of rotation. For the original solution u (R ct x) in (1.2), the origin x = is the center of rotation. Perturbing the initial data u (x) will generally excite rotational as well as translational modes of the solution and, in particular, one must expect the center of rotation to move out of the origin. For further details of the perturbed solution, see Theorem 1.1.

6 354 WOLF JÜRGEN BEYN AND JENS LORENZ 1.4. Eigenvalues of the Linearized Operator L. Introduce the linear operator (1.17) Lv = A v + cd φ v + B(x)v with B(x) = f (u (x)) which is obtained by linearizing the equation (1.18) L u + f(u ) = (with L v = A v + cd φ v) about u. We consider L and L as operators from H 2 Eucl to L2. Applying D φ to the equation (1.18), one finds that D φ u is an eigenfunction of L to the eigenvalue zero. Also, applying D 1 and D 2 to (1.18), one finds that L has the invariant subspace (1.19) span{d 1 u, D 2 u } with corresponding eigenvalues ±ic (see Lemma 2.3). Remark 2: One can obtain these results also by differentiating the equation (1.2) (A + cd φ )(a(γ)u ) + f(a(γ)u ) = for all γ = (η, θ) SE(2) with respect to the group variables θ, η 1, and η 2. Here one should note that the group SE(2) is not Abelian. Under the action of an Abelian Lie group with 3 dimensional Lie algebra, one can expect a zero eigenvalue of multiplicity at least three. We also note that by differentiating (1.11) w.r.t. η 1, η 2, and θ and evaluating at (η, θ) = (, ) one obtains that the space is tangent at u to the group orbit of u. Φ := span{d 1 u, D 2 u, D φ u } Assumption 3: The functions D 1 u, D 2 u, D φ u lie in HEucl 2, are nontrivial, and the corresponding eigenvalues ±ic and of L : HEucl 2 L2 are algebraically simple. As we will show, Assumptions 1-2 guarantee that the operator L from (1.17) has its essential spectrum in the region Re s 2β. In fact, we will prove that the operator L v = A v + cd φ v + B v has resolvent for Res 2β; here the assumption B 2βI < is crucial. A compact perturbation argument then shows that the essential spectrum of L lies in Re s 2β. The existence of eigenvalues s of L with Re s 2β and s / {ic, ic, } will be excluded explicitly: Assumption 4: The operator L : H 2 Eucl L2 (see (1.17)) has no eigenvalue s with Re s 2β, except for the eigenvalues s 1,2 = ±ic and s 3 = mentioned in Assumption 3.

7 NONLINEAR STABILITY OF ROTATING PATTERNS Stability Theorem. Consider the initial value problem (1.21) u t = A u + cd φ u + f(u) = L u + f(u), u(x, ) = u (x) + v (x) where v HEucl 2 is small w.r.t. H2. Our main result is the following. Theorem 1.1. Under the Assumptions 1-4 there exist constants ε > and C > so that the following holds for v H 2 < ε : (1) the solution u(x, t) of (1.21) exists for all t ; (2) the solution u(x, t) can be written in the form ) (1.22) u(x, t) = u (R θ(t) (x η(t)) + w(x, t) (3) where η C 1 ([, ), Ê 2 ), θ C 1 ([, ), S 1 ), (1.23) w(, t) H 2 Ce βt v H 2 ; (4) η() + θ() C v H 2 ; (5) there exist η Ê 2, θ S 1, depending on v, such that (1.24) η(t) R ct η + θ(t) θ Ce βt v H 2. Remark 3: The constants η and θ specify the so called asymptotic phase of the perturbed solution. For the original variable U(x, t) = u(r ct x, t), solving (1.1), one obtains U(x, t) = u (R θ(t) (R ct x η(t))) + w(r ct x, t) = a(γ(t))u + w(x, t). Here, using the group action (1.11) we define (1.25) γ(t) = ( η(t), θ(t) + ct), η(t) = R ct η(t), w(x, t) = w(r ct x, t). Then the estimates (1.23) and (1.24) can be written as follows: η(t) η + θ(t) θ + U(, t) a(γ(t))u H 2 Ce βt U(, ) u H 2. That is, U(, t) approaches, in H 2 -norm, a pattern a(γ(t))u similar to the unperturbed pattern u (R ct x), but the perturbed pattern rotates about the center η and has a phase shift θ. The angular velocity c is the same for the unperturbed solution u (R ct x) and the asymptotic solution a(γ(t))u. The decay of the error term, w(, t), to zero and the approach of the group variables θ(t) and η(t) to their limit values θ and η is exponential as t. Remark 4: In Theorem 1.1 we have left the notion of a solution imprecise. In fact, to prove the theorem, we will use an integral formulation and a contraction argument w.r.t. H 2, leading to a mild solution of (1.21). This is a function u C 1( [, ), L 2) C ([, ), H 2) satisfying the integral version of (1.21), (1.26) u(, t) = e tl (u + v ) + t e (t τ)l f(u(, τ))dτ.

8 356 WOLF JÜRGEN BEYN AND JENS LORENZ Because of the simple structure of the nonlinearity, the contraction argument does neither need nor provide any control of D φ u(, t) L 2. However, using the assumption v H 2 Eucl, one can show, a posteriori, that D φu(, t) L 2 exists and grows at most exponentially in time. In fact, the constructed solution u(x, t) is a classical solution of (1.7) and, for all t, we have u t (, t), u(, t), D φ u(, t) L 2 (Ê 2, Ê m ). See Section 7 for details. Remark 5: We believe that, using Assumption 2, one can show that the equilibrium pattern u (x) and the solution u(x, t) of (1.7) decay exponentially as x. Moreover, we expect that (1.5) can be deduced from (1.4) under Assumption 2. However, we have not carried out detailed arguments Outline of Proof and Functional Analytical Setting. We will use the linear operators (1.27) (1.28) (1.29) with Lv = A v + cd φ v + B(x)v, B(x) = f (u (x)) L v = A v + cd φ v L v = A v + cd φ v + B v, B = f (u ) = f () D φ v = x 2 D 1 v + x 1 D 2 v. We consider L, L, and L as operators from H 2 Eucl into L2. Since the operator D φ has unbounded coefficients, some of the results derived below do not seem to follow directly from standard theorems. In Section 3 we consider the resolvent equation L v sv = h, h L 2, Re s β, and construct a solution v HEucl 2. For Re s β we also show the resolvent estimates (L s) 1 h 2 H 2 Eucl (L s) 1 h L 2 ( C 1 + Im s 2) h 2 L 2, 1 2β + Re s h L 2. These imply that L : H 2 Eucl L2 L 2 is a closed operator. Since H 2 Eucl is dense in L 2 one obtains existence of the C semigroup e tl : L 2 L 2 and for the generator the domain of definition The operator D(L ) = H 2 Eucl. Lv = L v + (B(x) B )v differs form L by a term which is a bounded operator from L 2 into L 2. One obtains the semigroup e tl : L 2 L 2. More generally, we show that the same results hold for Sobolev spaces H n, n 1, with H n, H n+2 Eucl replacing L2, H 2 Eucl.

9 NONLINEAR STABILITY OF ROTATING PATTERNS 357 Using the energy technique, we will show exponential decay estimates for the semigroup e tl in H 2 in Section 5. A main difficulty is to extend these decay estimates for e tl partially to the semigroup e tl : The purely imaginary eigenvalues ±ic and of L must be taken into account. To do this we decompose the space H 2 as (1.3) H 2 = Φ (H 2 Ψ ) where Φ is the 3 dimensional space (1.19), corresponding to the eigenvalues ±ic and of L, and Ψ is the corresponding space for L. Here the adjoint L and the orthogonal complement Ψ are taken w.r.t. the L 2 inner product. In Section 4 we justify the application of Fredholm theory. In (1.3) the space H 2 is decomposed into invariant subspaces for L and we show the decay estimate (1.31) e tl w H 2 Ce βt w H 2, w H 2 Ψ, in Section 5. A corresponding estimate holds for H 2 Eucl but we will not use this. Using the projector P from L 2 onto Ψ along Φ and an implicit function argument, we decompose the solution u(x, t) in the form (1.22) and derive (coupled) evolution equations for the group variable γ(t) = (η(t), θ(t)) and the error term w(x, t). The evolution equations take the form (1.32) (1.33) γ E c γ w t Lw = r [γ] (γ(t), w(, t)) = r [w] (γ(t), w(, t)) with coupling terms r [γ] and r [w] ; see Section 2 for details. Here the error term w(x, t) evolves in the space H 2 Ψ, and the decay estimate (1.31) can be used for the inhomogeneous equation (1.33). Using an integral formulation of (1.33) (see (6.4)) and careful estimates of the coupling terms r [γ] and r [w], a contraction argument is then used to prove Theorem 1.1. The details are provided in Section Remarks on the literature. Structurally, our approach for splitting the dynamics as in (1.32), (1.33) follows Henry s method [11, Ch.5] for proving stability of traveling waves with asymptotic phase. A few generalizations and variations of this technique have been developed since, and we refer to [18] for a recent survey. Most results use analyticity of the semigroup and prove exponential decay by using the integral representation of the semigroup and resolvent estimates. As noted above, this approach does not apply when the linearized operator generates only a C -semigroup. For this case Bates and Jones [2] set up an invariant manifold theory allowing to decompose the dynamics near a traveling wave into a center manifold (formed by the translates of the wave) and a stable manifold. Exponential decay of the semigroup is obtained in a similar but somewhat more involved way by comparing with a constant coefficient semigroup, see the application to the FitzHugh Nagumo system in [2, Sect.4]. A general abstract principle that allows to reduce the dynamics near a relative equilibrium to a center manifold is derived in [16]. Moreover, using the arguments from [2] the authors prove that the center

10 358 WOLF JÜRGEN BEYN AND JENS LORENZ manifold is exponentially attracting for rotating waves satisfying our assumptions. However, stability with asymptotic phase is not discussed in [16]. Finally, during revision of the manuscript we learnt of the recent work [9] which uses compact perturbation techniques for C -semigroups (see Appendix) for proving nonlinear stability of traveling waves for some combustion problems. Acknowledgement: The authors are particularly grateful to Vera Thümmler for her excellent numerical work on the Ginzburg-Landau equations in Section 8. Not only did her results illustrate the theory at an intermediate stage, they also stimulated completion of the spectral investigations. 2. Decomposition of the Dynamics In this section, the solution u(x, t) of (1.7) with initial condition u(x, ) = u (x) + v (x) will be decomposed as u(x, t) = u (R θ(t) (x η(t))) + w(x, t), w(, t) H 2 Ψ, where w(x, t) will be shown to decay exponentially Spaces and Norms. On L 2 = L 2 (Ê 2, m ) we define the inner product (u, v) L 2 = u(x), v(x) dx Ê 2 with u, v = ū j v j. j For brevity we often write (u, v) = (u, v) L 2. On the Sobolev space H n (Ê 2, m ) we have a semidefinite and a definite inner product: (u, v) H n = n! α! (Dα u, D α v) L 2 (2.1) (2.2) α =n = ((u, v)) H n = 2 i 1,...,i n=1 α n = with corresponding (semi) norms (D i1 D in u, D i1 D in v) L 2 α! α! (Dα u, D α v) L 2 n (u, v) H k k= u 2 H n = (u, u) H n and u 2 H n = ((u, u)) H n. A simple calculation shows that the inner products are invariant under orthogonal transformations of the independent variable, i.e., (u, v) H n = (u Q, v Q) H n if Q T Q = I. Thus we have for all γ SE(2) and all u, v H n : (2.3) (2.4) (a(γ)u, a(γ)v) H n = (u, v) H n, a(γ)u H n = u H n, ((a(γ)u, a(γ)v)) H n = ((u, v)) H n, a(γ)u H n = u H n.

11 NONLINEAR STABILITY OF ROTATING PATTERNS 359 Finally, for n 2 we introduce the space HEucl n = {u H n : D φ u H n 2 } which is a Hilbert subspace of H n with inner product (u, v) H n Eucl = ((u, v)) H n + ((D φ u, D φ v)) H n 2. The following lemma generalizes the rule (2.5) (u, D φ u) =. (2.6) Lemma 2.1. For n Æ we have α =n if u, D φ u H n (Ê, Ê m ). Proof. From n! α! (Dα u, D α D φ u) = (u, D φ u) H n = we have D φ u = x 2 D 1 u + x 1 D 2 u and, by induction, D 1 D φ = D φ D 1 + D 2 (2.7) D k 1D φ = D φ D k 1 + kd k 1 1 D 2. Similarly, (2.8) D l 2D φ = D φ D l 2 ld 1 D l 1 2. Combining these equations we have (2.9) D k 1D l 2D φ = D φ D k 1D l 2 + kd k 1 1 D l+1 2 ld k+1 1 D l 1 2. From (2.5) and (2.9) we obtain n! n ( n α! (Dα u, D α D φ u) = (D j) j 1 Dn j 2 u, D j 1 Dn j 2 D φ u) = α =n j=1 j= n j=1 j= ( n j (D j) j 1 Dn j 2 u, D j 1 1 D n j+1 2 u) n 1 ( n (n j) (D j) j 1 Dn j 2 u, D j+1 1 D n j 1 2 u). Shifting the index in the second sum we end up with n ( ( ( )) n n j (n j + 1) (D j) j j 1 1 Dn j 2 u, D j 1 1 D n j+1 2 u) =.

12 36 WOLF JÜRGEN BEYN AND JENS LORENZ Lemma 2.2. Let u, D φ u H n (Ê 2, m ). Then we have (2.1) α =n n! α! Re (Dα u, D α D φ u) = Re (u, D φ u) H n =. Proof. This follows from the previous lemma since we can write u = u 1 +iu 2 with u j H n (Ê 2, Ê m ) and obtain (u, D φ u) H n = (u 1, D φ u 1 ) H n + (u 2, D φ u 2 ) H n i(u 2, D φ u 1 ) H n + i(u 1, D φ u 2 ) H n The Eigenvalues and ±ic of L. Recall that U (x, t) = u (R ct x) denotes a solution of (1.1), thus u (x) is a stationary solution of (1.7). The operator obtained by linearizing the stationary equation (1.7) about u is (2.11) Lv = A v + cd φ v + B(x)v, B(x) = Df(u (x)). Lemma 2.3. Let U (x, t) = u (R ct x) denote a rotating pattern solving (1.1) with u H 2 Eucl, c, and nontrivial functions ϕ 1 = D 1 u, ϕ 2 = D 2 u, ϕ 3 = D φ u H 2 Eucl. Then ±ic are eigenvalues of L with eigenfunctions ϕ 1 ± iϕ 2, and is an eigenvalue of L with eigenfunction ϕ 3. Proof. Applying D 1 and D 2 to the stationary equation (1.18) leads to = L(D 1 u ) + cd 2 u, = L(D 2 u ) cd 1 u, from which the first assertion follows. Similarly, we apply D φ to (1.18) and, using D φ = D φ, obtain the second assertion. By Assumption 1 we have u (x) u = as x ; therefore, the linear operator (2.12) L v = A v + cd φ v + B v, B = f (u ) = f (), also plays a role in our analysis. We consider L, L, and L as linear operators defined on H 2 Eucl taking values in L2. The formal adjoint of L is defined by (2.13) L u = u cd φ u + B( ) T u, L : H 2 Eucl L 2, and satisfies (2.14) (Lu, v) L 2 = (u, L v) L 2 for all u, v H 2 Eucl. (We will see below that the formal adjoint L does, in fact, agree with the abstract adjoint of L as defined, for example, in [19].) The following lemma summarizes important properties of the operator L. Lemma 2.4. For all complex s with Re s > 2β the operator L s : (H 2 Eucl, H 2 Eucl ) (L2, L 2) is a linear homeomorphism and the operator L s : (H 2 Eucl, H 2 Eucl ) (L 2, L 2)

13 NONLINEAR STABILITY OF ROTATING PATTERNS 361 is Fredholm of index. Moreover, there exist adjoint eigenfunctions ψ 1, ψ 2, ψ 3 H 2 Eucl (Ê2, Ê m ) with (2.15) L (ψ 1 ± iψ 2 ) = ic(ψ 1 ± iψ 2 ), L ψ 3 =. For s 1 = ic, s 2 = ic, s 3 = the inhomogeneous equation (2.16) (L s j )u = h, h L 2, j = 1, 2, 3, has a solution u HEucl 2 if and only if (ψ 1 + iψ 2, h) L 2 =, j = 1, (ψ 1 iψ 2, h) L 2 =, j = 2, (ψ 3, h) L 2 =, j = 3. If the orthogonality condition is satisfied, then one can select a solution u of (2.16) with where C does not depend on h L 2. u H 2 C h L 2 Proof. Resolvent estimates for L are shown in Section 3. Based on these, the Fredholm property of L s follows essentially from the Riesz Fréchet Kolmogorov compactness criterion; see Section 4 for details. Lemma 2.4 is written in spaces of complex valued functions. Since we have real differential operators we can express the same results in terms of real-valued functions. If necessary we write the corresponding real function spaces as LÊ 2, H2 Ê, H2 Ê,Eucl. Lemma 2.4 can be used to decompose the spaces LÊ 2, H2 Ê and H2 Ê,Eucl as follows. First, define (2.17) Φ = span{ϕ 1, ϕ 2, ϕ 3 }, Ψ = span{ψ 1, ψ 2, ψ 3 }, W = Ψ, where the orthogonal complement of Ψ is taken in LÊ 2. Then, for n 2, introduce the subspaces (2.18) W n = H n W, W n Eucl = Hn Eucl W. With these settings we have (2.19) L 2 Ê = Φ W and H2 Ê = Φ W 2, H 2 Ê,Eucl = Φ W 2 Eucl. The decompositions (2.19) are compatible with the operator L : H 2 Eucl L2 in the sense that (2.2) L(Φ) Φ and L(WEucl 2 ) W. We can carry this argument further to show that (2.21) HÊ,Eucl 4 = Φ WEucl 4 and L(WEucl) 4 W 2. Note that Φ HÊ,Eucl 4 follows by Theorem 3.2 from L ϕ j = s j ϕ j + (B B( ))ϕ j j = 1, 2, 3 and Assumptions 1 and 3.

14 362 WOLF JÜRGEN BEYN AND JENS LORENZ Using the simplicity of the eigenvalues ±ic we have (ψ 1 + iψ 2, ϕ 1 + iϕ 2 ) L 2 and, therefore, may assume (ψ 1 + iψ 2, ϕ 1 + iϕ 2 ) L 2 = 2. Combined with the orthogonality of left and right eigenfunctions, this implies the normalization (2.22) (ψ j, ϕ k ) L 2 = δ j,k for j, k = 1, 2, 3. The projector P from LÊ 2 onto W along Φ is then given by 3 (2.23) Pu = u ϕ j (ψ j, u) L 2. This projector P also maps H n Ê,Eucl onto W n Eucl and Hn onto W n (n = 2, 4). j= Decomposition of the Solution u(x, t). Let u(x, t) denote the solution of (1.21), i.e., u solves the differential system (1.7) (with stationary solution u (x)), but the initial data are perturbed, u(x, ) = u (x) + v (x). A major step is to decompose u(x, t) into two parts: One part moves within the group orbit of u and the other part, which we call w(x, t), moves within the space W = Ψ : (2.24) where u(x, t) = w(, t) W, (a(γ(t))u ) (x) + w(x, t) = u ( R θ(t) (x η(t)) ) + w(x, t), γ(t) = (η(t), θ(t)) SE(2). This decomposition follows the approach for traveling waves in [11, Ch.5] and corresponds to a transformation to a local coordinate system. The local coordinate system uses the group variables γ = (η, θ) and the subspace W = Ψ defined above. Note that W is transversal to the space Φ where Φ is the tangent space, at u, to the group orbit of u (cf. the slice theorem in [4]). A rigorous formulation of the change of coordinates uses the derivative of the group action { SE(2) L 2 a( )u : γ a(γ)u defined by (1.11). As follows from Assumption 3, this derivative exists and can be evaluated at γ = ½ = (, ) as follows: 3 (2.25) D[a(½)u ]µ = µ j ϕ j for µ Ê 3. j=1 Here we have identified the Lie algebra T ½ SE(2) = se(2) with Ê 3. The following result is then a consequence of the inverse function theorem. Lemma 2.5. Let P denote the projector (2.23). The map { SE(2) Φ (2.26) Γ : γ (I P)(a(γ)u u ) satisfies Γ(½) = and is a local diffeomorphism near γ = ½ with derivative 3 (2.27) DΓ(½)µ = µ j ϕ j, µ Ê 3. j=1

15 NONLINEAR STABILITY OF ROTATING PATTERNS 363 Moreover, the transformation { SE(2) W L 2 (2.28) T : (γ, w) T(γ, w) = a(γ)u u + w is a local diffeomorphism near (½, ). The solution of T(γ, w) = v is given by (2.29) γ = Γ 1 ((I P)v), w = v + u a(γ)u. Proof. It suffices to note that the map (µ, w) DT(½, )(µ, w) = D[a(½)u ]µ + w is a linear homeomorphism from Ê 3 W onto L 2 and that T(γ, w) = v and w W imply (I P)(a(γ)u u ) = (I P)v; hence we have γ = Γ 1 ((I P)v) by the first part of the lemma The Decomposed System. Consider a solution u(, t) of (1.21) for a time interval t < t where t is chosen so that u(, t) lies completely in a neighborhood of u where the transformation T (see (2.28)) can be inverted, i.e., (2.24) can be re-written as (2.3) u(, t) u ( ) = T(γ(t), w(, t)), t < t. (The time interval t < t may equal t < ; in fact, our arguments will show that the choice t = can be made if the initial perturbation is small enough.) We derive differential equations and initial conditions for γ(t) and w(, t) W. At t = we have hence by Lemma 2.5 (2.31) u(, ) = u + v = a(γ())u + w(, ), γ() = γ := Γ 1 ((I P)v ) w(, ) = w := v + u a(γ )u. In the following, we use the abbreviation γ(t)x = R θ(t) (x η(t)) if γ(t) = (η(t), θ(t)). Insert u(, t) from (2.24) into (1.21) and obtain for t < t : Now we use that = d dt (a(γ(t)u )) + w t (, t) A (a(γ(t))u ) A w(, t) ( ) cd φ (a(γ(t))u ) cd φ w(, t) f a(γ(t))u + w(, t) [ = Du (γ(t)x) R π 2 θ(t) (x η(t)) θ(t) ] R θ(t) η(t) A u (γ(t)x) + w t (, t) A w(, t) cd φ w(, t) f(a(γ(t))u + w(, t)) cdu (γ(t)x)r θ(t)+ π 2 x. Du (γ(t)x)r θ(t)+ π x = Du π (γ(t)x)r 2 2 and evaluate (1.2) at γ = γ(t) to obtain ( γ(t)x + R θ(t) η(t) ) A u (γ(t)x) = cdu (γ(t)x)r π 2 γ(t)x + f(a(γ(t))u ).

16 364 WOLF JÜRGEN BEYN AND JENS LORENZ Therefore, = Du (γ(t)x) [ R θ(t) ( η(t) + cr π 2 η(t))] D φ u (γ(t)x) θ(t) +w t (, t) A w(, t) cd φ w(, t) B(x)w(, t) Introducing the remainder [f(u (γ(t)x) + w(, t)) f(u (γ(t)x)) Df(u (x))w(, t)]. (2.32) r [f] (γ, w) = f(a(γ)u + w) f(a(γ)u ) Df(u ( ))w we finally arrive at the equation w t (, t) = Lw(, t) + r [f] (γ(t), w(, t)) (2.33) [ + Du (γ(t) )R θ(t) η(t) + cr π η(t)] + D 2 φ u (γ(t) ) θ(t). For γ SE(2) we define the linear mapping Ê 3 Φ [ ( ) ] (2.34) S(γ) : µ1. µ (I P) Du (γ ) + D µ φ u (γ )µ 3 2 Since S(½) agrees with DΓ(½) from (2.27) we find that the inverse S(γ) 1 : Φ Ê 3 exists and depends smoothly on γ in a neighborhood of ½. Applying I P to (2.33) and using the inverse of S(γ(t)), t < t, we obtain the following differential equation for γ(t) = (η(t), θ(t)): (2.35) ( η + cr π 2 γ E c γ η ) = r θ [γ] (γ, w), ( ) ( ) r [γ] Rθ cr (γ, w) = S(γ) 1 1 (I P)r [f] π (γ, w), E c = 2. In the next step we apply the projector P to (2.33) to obtain the w-equation: (2.36) w t Lw = ( P P ( Du (γ ) D φ u (γ ) ) S(γ) 1 (I P) ) r [f] (γ, w) =: r [w] (γ, w). Together with initial conditions (2.31), equations (2.35) and (2.36) constitute the transformed system. Working the way backwards, we see that any solution (γ(t), w(, t)) of the transformed system that stays in a neighborhood of (½, ) leads to a solution of (1.22) via the transformation (2.24). In Section 6 we will show that both remainders r [w] and r [γ] are Lipschitz bounded with respect to w H 2 with Lipschitz constants that become small as γ ½ and w H 2. This will lead to the proof of nonlinear stability. Recall the definition 3. Resolvent Estimates for L (3.1) L v = A v + cd φ v + B v, v HEucl 2, where A, B Ê m m are constant matrices satisfying (3.2) u, Au 2β A u 2, u, B u 2β u 2 for all u Ê m and

17 NONLINEAR STABILITY OF ROTATING PATTERNS 365 β A >, β >, c Ê, c. Theorem 3.1. Let s = s 1 +is 2 with s j Ê and s 1 β. Then, for every h L 2, the equation L v sv = h has a unique solution v H 2 Eucl and (3.3) (3.4) v L 2 1 2β + s 1 h L 2 v 2 H 2 + c2 D φ v 2 L 2 C(1 + s2 2) h 2 L 2. The constant C depends only on β A, β, and B, A. The theorem is proved in three steps. First, in the next section, we give a formal derivation of the resolvent estimate (3.4), assuming existence of v. In Section 3.2 we prove existence of a solution for a scalar equation like (3.1). The proof uses Fourier expansion in φ and the solution of inhomogeneous Bessel equations. In Section 3.3 we generalize the existence argument from the scalar case to the case of a system (3.1). If h H n (instead of h L 2 ) one can extend the estimates of the previous theorem to higher derivatives of v where v is the solution of the resolvent equation (3.5) A v + cd φ v + B v sv = h. Note that this extension is not completely trivial since the term D φ v has variable, unbounded coefficients. Theorem 3.2. Let s = s 1 +is 2 with s j Ê and s 1 β. Let h H n and let v HEucl 2 denote the solution of (3.5) of Theorem 3.1. Then v Hn+2 Eucl and 1 (3.6) v H n h H n 2β + s 1 v 2 H n+2 + c2 D φ v 2 H n C(1 + s2 2 ) h 2 H n. (3.7) Remark: We found it instructive to derive detailed resolvent estimates via the energy method. In Section 5 we use the resolvent estimates to construct the corresponding semigroup e tl. An alternative approach is to start with the known semigroups generated by + B and D φ and to use their commutativity to construct and estimate e tl. Then resolvent estimates for L can be obtained from the integral representation (L s) 1 = exp(t(l s))dt Formal Derivation of the Resolvent Estimates. We first note a generalization of the estimates (3.2) to complex vectors. If u = u 1 + iu 2, u j Ê m, then we have u 1 + iu 2, A(u 1 + iu 2 ) = u 1, Au 1 + u 2, Au 2 i u 2, Au 1 + i u 1, Au 2. Therefore, taking real parts, Re u, Au 2β A u 2 for all u m.

18 366 WOLF JÜRGEN BEYN AND JENS LORENZ Formal Derivation of the Estimates of Theorem 3.1: Taking the L 2 inner product of (3.8) A v + cd φ v + B v sv = h with v one obtains (v, A v) L 2 + c(v, D φ v) L 2 + (v, B v) L 2 s v 2 L 2 = (v, h) L 2. Integrating by parts and taking real parts yields (3.9) 2β A v 2 H 1 2β v 2 L 2 s 1 v 2 L 2 Re (v, h) L 2. Taking absolute values one obtains 2β A v 2 H 1 + (2β + s 1) v 2 L 2 v L 2 h L 2 which yields (3.3). Since β + s 1 one also obtains This implies that 2β A v 2 H 1 + β v 2 L 2 v L 2 h L 2. v 2 H 1 + v 2 L 2 C h 2 L 2, C = C(β A, β). Next, take the L 2 inner product of (3.8) with v, ( v, A v) L 2 + c( v, D φ v) L 2 + ( v, B v) L 2 s( v, v) L 2 = ( v, h) L 2. Taking real parts and using Lemma 2.1 yields 1 2 ( v, (A + AT ) v) L 2 j Re (D j v, B D j v) L 2 + s 1 v 2 H 1 = Re( v, h) L 2. Here and one finds that Re(D j v, B D j v) L 2 2β D j v 2, 2β A v 2 L 2 + (2β + s 1) v 2 H 1 v L 2 h L 2. Since β + s 1 one obtains (3.1) 2β A v L 2 h L 2. Finally, to estimate D φ v L 2 we take the L 2 inner product of (3.8) with D φ v: (3.11) (D φ v, A v) L 2 + c D φ v 2 L 2 + (D φv, B v) L 2 s(d φ v, v) L 2 = (D φ v, h) L 2. Here (D φ v, v) L 2 is purely imaginary. Therefore, taking real parts in (3.11) and then taking absolute values yields ) c D φ v 2 L ( B 2 + s 2 D φ v L 2 v L 2 + A D φ v L 2 v L 2 + D φ v L 2 h L 2. Divide by D φ v L 2 and use the estimates (3.1) and v L 2 C h L 2 to obtain

19 NONLINEAR STABILITY OF ROTATING PATTERNS 367 ( ) c D φ v L 2 C 1 + s 2 h L 2. This completes the formal derivation of the resolvent estimate (3.4). Formal Derivation of the Estimates of Theorem 3.2: Take the H n inner product of the resolvent equation (3.8) with v to obtain (v, A v) H n + c(v, D φ v) H n + (v, B v) H n s v 2 H n = (v, h) H n. Then take the real part of the resulting equation and note that by Lemma 2.2. One finds that Re (v, D φ v) H n = (3.12) 2β A v 2 H n+1 2β v 2 H n s 1 v 2 H n Re (v, h) H n. Taking absolute values the estimate (3.6) follows. Similarly, taking the H n inner product of (3.8) with v the estimate (3.1) generalizes to (3.13) 2β A v H n h H n. Finally, taking the H n inner product of (3.8) with D φ v leads to (3.7) Existence in the Scalar Case. Consider a scalar equation (3.14) v + cd φ v (2β + s)v = h where h is a given complex valued function in the plane and c Ê, c, β >, s = s 1 + is 2, s 1 β. To show solubility of the equation, we may put strong assumptions on h because we can derive estimates of the solution v in terms of h and then can make an approximation argument. If written in Cartesian coordinates, we will assume that h C (Ê 2 \ {}), i.e., h is a C function compactly supported in Ê 2 \ {}. The set of these functions h is dense in L 2. To show solubility of (3.14) we will treat the equation in polar coordinates. For simplicity of notation, we simply write h = h(r, φ) for the right hand side and v = v(r, φ) for the solution to be constructed. Using Fourier expansion in φ we write and obtain, formally, h(r, φ) = h n (r)e inφ, v(r, φ) = v n (r)e inφ (3.15) v n r v n n2 r 2 v n (2β + s icn)v n = h n (r), r >, n. Note that

20 368 WOLF JÜRGEN BEYN AND JENS LORENZ If we abbreviate then h 2 L = 2π r h 2 n (r) 2 dr. q 2 = 2β + s icn = 2β + s 1 + is 2 icn We will prove: Lemma 3.1. Consider the ODE Re q 2 = 2β + s 1 β >, Re q >. (3.16) and assume 1 r (rv n) n2 r 2 v n q 2 v n = h n (r) Re q 2 β >, Re q >, h n C (, ). The ODE has a unique solution v n C [, ) which decays exponentially as r. This solution satisfies (3.17) v n (r) = O(r n ) as r and v n (r) e 1 2 Re qr as r and satisfies the estimates (3.18) r v n 2 dr + n 2 1 r v n 2 dr + r v n 2 dr C β r h n 2 dr. Proof: Estimate and Uniqueness. First assume existence of a solution v n of (3.16), (3.17). Multiply (3.16) by r v n and integrate by parts to obtain r v n 2 dr n 2 1 r v n 2 dr q 2 r v n 2 dr = r v n h n dr. Taking real parts and using that Re q 2 β > the estimate (3.18) follows. Existence. First consider the homogeneous equation (3.19) r 2 v n + rv n q 2 r 2 v n n 2 v n =. Setting v n (r) = y(iqr) one obtains Bessel s equations of order n: The function z 2 y (z) + zy (z) + (z 2 n 2 )y(z) =, z = iqr. solves (3.19) and satisfies v (1) n (r) = J n (iqr), r, Write (3.19) in the form v (1) n (r) = O(r n ) as r.

21 NONLINEAR STABILITY OF ROTATING PATTERNS 369 v n + 1 r v n q 2 v n n2 r 2 v n =. For large r one obtains the constant coefficient approximation v n q 2 v n = which has the decaying solution e qr, Re q >. A perturbation argument shows that (3.19) has a nontrivial, exponentially decaying solution v (2) n (r) with v (2) n (r) Ce 1 2 Re qr, r 1. The two solutions v n (1) and v n (2) of (3.19) are linearly independent since otherwise v n (1) by (3.18). A solution v n of (3.16), (3.17) can be constructed as follows. Assuming that the right hand side h n (r) is supported in < a < r < b, let v n,par (r) denote the particular solution of (3.16) with thus v n,par (a) = v n,par(a) =, We construct v n in the form v n,par (r) = for r a. v n (r) = v n,par (r) + αv (1) n (r) for suitable α. In fact, consider the boundary operator R b v n = γ 1 v n (b) + γ 2 v n(b) where (γ 1, γ 2 ) (, ) is chosen so that R b v (2) n = R b v n = R b v n,par + αr b v (1) n. =. Then choose α so that (Note that R b v n (1) since v n (1), v n (2) are linearly independent.) With this choice of α, the function v n = v n,par + αv n (1) is a multiple of v n (2) for r b. Consequently, (3.17) holds.. If h (N) (r, φ) = N n= N h n(r)e inφ is a finite Fourier approximation of h(r, φ), we obtain the corresponding solution v (N) (r, φ) = N n= N v n (r)e inφ of (3.14) with right hand side h (N). The estimate (3.18) yields v (N) 2 H 1 + v(n) 2 L 2 C h 2 L 2. Clearly, the full estimate corresponding to (3.4), which we have derived formally in Section 3.1, is valid for the finite Fourier sum v (N). For N we obtain the solution v HEucl 2 of (3.14) and the estimate (3.4).

22 37 WOLF JÜRGEN BEYN AND JENS LORENZ 3.3. Existence in the Systems Case. Consider the equation L v sv = h under the assumptions specified in Theorem 3.1 but assume h C (Ê 2 \ {}). As in the scalar case, we write h(r, φ) = h n (r)e inφ, v(r, φ) = v n (r)e inφ, and obtain, for each integer n, the second order system (3.2) A(v n + 1 r v n n2 r 2 v n) Pv n = h n (r), r >, with P = B icni + si. Lemma 3.2. All eigenvalues µ of A 1 P lie in Proof: If A 1 Pu = µu, u, then \ (, ]. thus µau = Pu = B u + s 1 u + i(s 2 cn)u, Re (µ u, Au ) = Re u, B u + s 1 u 2 β u 2 >. Since Re u, Au >, it is not possible that µ (, ]. To solve (3.2) let us first assume that A 1 P is diagonalizable, T 1 A 1 PT = D = diag(µ 1,...,µ m ). Then, if v n = Tw n, the system (3.2) transforms to (3.21) w n + 1 r w n n2 r 2 w n Dw n = T 1 h n (r), r >, The above system consists of n scalar equations of the form (3.22) α + 1 r α n2 r 2 α µα = g(r), g C Here µ \ (, ] and we can write (, ). µ = q 2, Req >. We cannot directly apply Lemma 3.1 since we do not know if Re µ = Re (q 2 ) >. However, we know from the scalar case that (3.21) has two nontrivial solutions α (1) (r), α (2) (r), where α (1) (r) = O(r n ) for r and where α (2) (r) decays exponentially as r. If the function α (2) (r) would be a multiple of α (1) (r) then we would obtain a nontrivial, exponentially decaying solution v n (r) of the homogeneous system (3.2)

23 NONLINEAR STABILITY OF ROTATING PATTERNS 371 with v n (r) = O(r n ). This would contradict an energy estimate that one can derive for (3.2) as in the scalar case. Thus, we conclude that the two functions α (1) (r) and α (2) (r) are linearly independent. We can then construct an exponentially decaying solution of the inhomogeneous equation (3.22) as in the proof of Lemma 3.1. These arguments show the existence of an exponentially decaying solution v n C [, ) of the system (3.2) under the assumption that A 1 P is diagonalizable. By summing the Fourier series, we obtain a solution v HEucl 2 of the equation L v sv = h. If A 1 P is not diagonalizable, we approximate A and P by A ε and P ε so that Aε 1 P ε is diagonalizable. Since the estimates do not depend on ε, we can make a small perturbation argument and obtain a solution also in the case where A 1 P is not diagonalizable. This completes the proof of Theorem 3.1. Recall the definition 4. Fredholm Theory Lv = A v + cd φ v + B(x)v, v H 2 Eucl. We show the Fredholm property for L using the Riesz Fréchet Kolmogorov Theorem (see, for example, [1, Ch.2], [2, Ch. X]). Theorem 4.1. (Riesz Fréchet Kolmogorov) A set V L 2 (Ê d ) is precompact with respect to the L 2 -norm if and only if the following three conditions hold: (1) (2) (3) lim sup v(x) 2 dx = ; R v V x R sup v L 2 < ; v V lim sup v(x + ξ) v(x) 2 dx =. ξ v V Ê d Lemma 4.1. Let k and let M : Ê d Ê m m be a matrix valued C k -function satisfying (4.1) sup D γ M(x) as R, γ k. x R Then the operator of multiplication (4.2) is compact. M : { H k+1 (Ê d, Ê m ) H k (Ê d, Ê m ) u( ) M( )u( ) Proof. Consider first the case k = and apply Theorem 4.1 to the set V = {Mu : u H 1, u H 1 1}. From (4.1) we have M(x)u(x) 2 dx sup{ M(x) : x R} u 2 L 2 x R sup{ M(x) : x R} as R.

24 372 WOLF JÜRGEN BEYN AND JENS LORENZ Clearly, V is bounded in L 2. For all u H 1 1 and ξ Ê d we have 1 2 u(x + ξ) u(x) 2 dx = Du(x + sξ)ds ξ dx Ê d Ê d 1 ξ 2 Du(x + sξ) 2 dsdx Ê d 1 = ξ 2 Du(x + sξ) 2 dxds Ê d = ξ 2 Ê d Du(x) 2 dx ξ 2 u 2 H 1. Therefore we can estimate for ξ 1, u H 1 1 M(x + ξ)u(x + ξ) M(x)u(x) 2 dx Ê d ( 2 M(x + ξ) 2 u(x + ξ) 2 + M(x) 2 u(x) 2) dx + 2 x R x R ( M(x + ξ) M(x) 2 u(x + ξ) 2 + M(x) 2 u(x + ξ) u(x) 2) dx 4 sup M(x) sup M(x + ξ) M(x) 2 x R 1 x R + 2 M( ) L u(x + ξ) u(x) 2 dx. Ê d Given ε >, choose R large so that the first term is ε and then choose ξ small such that the last two terms are ε. The proof for general k uses the first step and the Leibniz rule for D γ (M(x)u(x)). We proceed with the Proof. of Lemma 2.4 The homeomorphism L s : H 2 Eucl L2 is established in Section 3. Then we can write (4.3) L s = L s + B( ) B = (I + (B B )(L s) 1 )(L s). By classical Fredholm theory (see, for example, [19, Ch.IV13]) it is sufficient to show that the operator (B B )(L s) 1 is compact in L 2. But this follows from Lemma 4.1 using Assumption 1 and the fact that (L s) 1 maps L 2 into H 2 Eucl H1. Since the Fredholm index of L s j is we infer that codimr(l s j ) = 1 for all three eigenvalues s j, see (2.16). Moreover, the relations dim(n(l s j )) = 1, j = 1, 2, 3 and the Fredholm alternative for (2.16) follow from [19, Ch.IV11] provided we show that the abstract adjoint L : D(L ) L 2 L 2, defined there, coincides with the formal adjoint L : H 2 Eucl L2 defined in (2.13). Recall that (4.4) D(L ) = {v L 2 : w L 2 s.t. (Lu, v) L 2 = (u, w) L 2 u D(L) = H 2 Eucl }, where L v is defined as w given by (4.4). Then (2.14) implies that H 2 Eucl D(L ) and that L is an extension of L. By Lemma 2.4 and Assumption 4 we have that L s : H 2 Eucl L2 is a homeomorphism for all s >. Applying the same theory to the differential operator L (note that B T satisfies Assumption 2 as well) we

25 NONLINEAR STABILITY OF ROTATING PATTERNS 373 find for s > B T that L s : HEucl 2 L2 is also a homeomorphism. Now take v D(L ) with w = L v L 2 as in (4.4) and let ṽ HEucl 2 be the unique solution of (L s)ṽ = w sv. Then we obtain for all u HEucl 2 ((L s)u, v) L 2 = s(u, v) L 2 + (u, w) L 2 (4.5) = (u, (L s)ṽ) L 2 = ((L s)u, ṽ) L 2. Since L s is onto, we conclude v = ṽ HEucl 2 and hence D(L ) = D(L ). Recall the definition 5. Estimates for e tl and e tl Here L u = A u + cd φ u + B u, u H 2 Eucl. B 2βI, β >, and A 2β A I, β A >. The operator L has constant coefficients, except for the term D φ u which reads D φ u = x 2 D 1 u + x 1 D 2 u. In Sections we consider the Cauchy problem u t = L u, u(x, ) = u (x), and denote the solution by u(, t) = e tl u. The proofs given below are formal in the sense that we assume u to be sufficiently regular and to have sufficient decay as x. Note that in this section we work with real valued functions only. Recall the norms and spaces from Section 2.1, and for brevity let (u, v) = (u, v) L Decay Estimates in H n Norm. Using Lemmas 2.1 and 2.2 we obtain the decay from energy estimates. Theorem 5.1. For n Æ we have 1 d (5.1) 2 dt u(, t) 2 H 2β A u(, t) 2 n H 2β u(, n+1 t) 2 H. n Proof. For n = this is the usual energy estimate: 1 d 2 dt u 2 = (u, u t ) = (u, A u) + c(u, D φ u) + (u, B u) 2β A u 2 H 1 2β u 2 where we have used that (u, D φ u) =. For n 1 the proof proceeds in the same way using Lemma 2.1.

26 374 WOLF JÜRGEN BEYN AND JENS LORENZ Clearly, the previous theorem implies 1 d (5.2) 2 dt u(, t) 2 H 2β u(, 2 t) 2 H. 2 This yields the decay estimate (5.3) e tl H 2 H 2 e 2βt. The same estimate holds if we replace the H 2 norm by the H 1 norm or by the L 2 norm Boundedness of e tl from H 2 to H 3. Theorem 5.2. There is a constant C > so that for any t > : (5.4) u(, t) 2 H 3 C t u 2 H 2. Proof. First integrate (5.1) for n = 2 to obtain thus (5.5) Consider u(, t) 2 H 2 u 2 H 2 + 4β A t t u(, s) 2 H 3 ds, u(, s) 2 H 3 ds 1 4β A u 2 H 2. (5.6) d ) (t u(, t) 2 H = u(, t) 2 dt 3 H + t d 3 dt u(, t) 2 H. 3 From Theorem 5.1 with n = 3 we obtain Using this in (5.6) we find and integration yields d dt u(, t) 2 H 3. d ) (t u(, t) 2 H u(, t) 2 dt 3 H 3 This proves the theorem. t t u(, t) 2 H u(, s) 2 3 H ds 3 1 4β A u 2 H 2

27 NONLINEAR STABILITY OF ROTATING PATTERNS 375 Combining the result of the previous theorem with the decay estimate of u(, t) H 2 we have In particular, u(, t) 2 H 3 C t u 2 H 2 for < t. (5.7) e tl H 2 H 3 C t for < t Estimates of D φ u. The function D φ u satisfies D φ u t = A D φ u + cd φ D φ u + B D φ u, D φ u(, ) = D φ u. The basic energy estimate applied to v = D φ u yields We have shown: Theorem 5.3. If u H 2 Eucl then D φ u(, t) L 2 e 2βt D φ u L 2. ) u(, t) 2 H + D φu(, t) 2 2 L e 4βt( u 2 2 H + D φu 2 2 L. 2 Moreover, the same arguments as in the previous section yield u(, t) 2 H 1 C t u 2 L 2 for < t. We may apply this estimate to D φ u instead of u: D φ u(, t) 2 H 1 C t D φu 2 L 2 for < t Application of Theorem A.1. We apply Theorem A.1 to show that Lu = L u + (B(x) B )u generates an exponentially decaying semigroup. Theorem 5.4. Under Assumptions 1-4 the operator L generates a C semigroup e tl on H 2. For any β < 2β there exists a constant C = C( β) such that (5.8) e tl u H 2 Ce t β u H 2 for all u W 2 = Ψ H 2. Proof. We use Theorem A.1 with X = H 2, A = L, D(A) = HEucl 4 and the operator of multiplication Bu = (B( ) B )u. Since C HEucl 4 we obtain that HEucl 4 is dense in H2. Closedness of L follows from the resolvent estimate (3.7) for n = 2. The estimate (3.6) then guarantees via the Hille Yosida Theorem that L generates a C semigroup on H 2 which is of type ω A 2β; see also (5.3). Next, we note that Assumption 1 implies sup D j (B(x) B ) = sup D 2 f(u (x))d j u (x) as R, x R x R and, in a similar way, sup D γ (B(x) B ) as R for γ 2. x R

28 376 WOLF JÜRGEN BEYN AND JENS LORENZ From (5.7) we have that e tl, t > maps H 2 into H 3 and by Lemma 4.1 (k = 2) we obtain that the operator u (B(x) B )e tl u, t >, from H 2 into itself is compact. Therefore, L also generates a C semigroup on H 2 with 4 (5.9) σ ess (e L ) e 2β < 1. There is the slight complication that the eigenvalues λ of L satisfy Re λ 2β < only if we restrict L to WEucl 2 = H2 Eucl W. The space W Eucl 2 is not invariant under L. Therefore, we repeat the arguments from the proof of the second assertion in Theorem A.1. For a fixed β < 2β we have from (5.9) σ ess (e L+ β) e (2β β) =: q < 1 where the essential spectrum is taken in H 2. Now, H 2 = Φ W 2 is a decomposition of H 2 into invariant subspaces of e L (cf. (2.2),(2.21)) where e L has three simple eigenvalues 1, e ±ic in Φ by Assumption 3. In particular, if we restrict e L to W 2 the essential spectrum does not grow, i.e. (5.1) σ ess ((e L+ β) W 2) q < 1. Moreover, by Assumption 4 the operator L W 2 has no eigenvalues λ with Re λ 2β. Then the spectral mapping theorem for point spectra [13, Theorem 2.4] shows and thus σ point ((e L ) W 2) exp(σ point (L W 2)) {} σ point ((e L+ β) W 2) q. Combining this with (5.1) we obtain σ((e L+ β) W 2) q < 1. The remaining arguments are as in Section A: There is a norm + on W 2, equivalent to H 2, so that (e L+ β) W Therefore, (e t(l+ β) ) W 2 + C, and (e tl ) W 2 + Ce βt for t. Because of the norm equivalence: (e tl ) W 2 H 2 C 1 e βt. 6. Proof of Nonlinear Stability In this section we complete the proof of Theorem 1.1 by a nonlinear stability argument applied to the integral equations related to (2.35), (2.36). 4 The essential spectrum σess(a) of an operator A is defined in Definition 8.1 below. We write σ ess(a) q if s q for all s σ ess(a).

29 NONLINEAR STABILITY OF ROTATING PATTERNS Integral Equation Formulation. Recall the subspace from (2.18) (6.1) W 2 = {u H 2 : (ψ j, w) L 2 =, j = 1, 2, 3} and the linear operator from (2.11) (6.2) Lv = A v + cd φ v + B(x)v, B(x) = Df(u (x)), v H 2 Eucl. In Section 5 we showed that L generates a C -semigroup e tl on H 2 that has W 2 as an invariant subspace and satisfies (6.3) e tl w H 2 C W e βt w H 2 for w W 2, t. Here we can take β β < 2β arbitrary with C W depending on β. By Duhamel s principle the integral formulation of (2.36) is (6.4) (6.5) w(t) = e tl w + t w() = w = v + u a(γ )u, where γ is given by (2.31) as follows e (t τ)l r [w] (γ(τ), w(τ))dτ, (6.6) γ() = γ = Γ 1 (I P)v. From (2.35) the integral formulation of the γ-equation is (6.7) γ(t) = e tec γ + t e (t τ)ec r [γ] (γ(τ), w(τ))dτ. We will estimate the remainders occuring in these equations Estimate of Group Action. The topology on the manifold SE(2) = Ê 2 S 1 is generated by the metric (( ) ( )) η1 η2 d(γ 1, γ 2 ) = d, = η θ 1 θ 1 η 2 + d(θ 1, θ 2 ), 2 (6.8) d(θ 1, θ 2 ) = min n θ 1 θ 2 2πn. For convenience we use the same symbol d for the metric on S 1 and on SE(2). We show that the action of the group SE(2) on u is Lipschitz with respect to the metric on SE(2). Lemma 6.1. There exists a constant C a 1 such that for all γ 1, γ 2 SE(2) (6.9) a(γ 1 )u a(γ 2 )u H 2 C a d(γ 1, γ 2 ). Remark. The proof will show that the constant C a depends only on the semi norms D φ u H k, D j u H k for j, k = 1, 2 the existence of which is guaranteed by Assumption 3. Proof. From the triangle inequality a(γ 1 )u a(γ 2 )u H 2 a(η 1, θ 1 )u a(η 2, θ 1 )u H 2+ a(η 2, θ 1 )u a(η 2, θ 2 )u H 2 it is clear that it suffices to prove the inequalities (6.1) (6.11) a(η 1, θ)u a(η 2, θ)u H 2 C η 1 η 2 for all θ S 1 a(η, θ 1 )u a(η, θ 2 )u H 2 Cd(θ 1, θ 2 ) for all η Ê 2.

30 378 WOLF JÜRGEN BEYN AND JENS LORENZ Moreover, by invariance of the H 2 norm with respect to the group action (see (2.3), (2.4)) it is sufficient to consider (6.1) for θ =, η 2 = and (6.11) for η =, θ 2 =, i.e. we must show (6.12) (6.13) a(η, )u u H 2 C η for η Ê 2 a(, θ)u u H 2 Cd(θ, ) for θ S 1. For θ S 1 take θ ( π, π] such that d(θ, ) = θ. We have u (R θ x) u (x) = u (R θx) u (x) = 1 d [ u (R dt θt x) ] 1 dt = D φ u (R θt x)dt θ. Integrating with respect to x and using Cauchy Schwarz and Fubini yields 1 u (R θ x) u (x) 2 dx D φ u (R t θx) 2 dt θ 2 dx 1 = d(θ, ) 2 D φ u (R t θx) 2 dxdt = d(θ, ) 2 D φ u 2. In a similar way 1 u (x η) u (x) 2 dx 1 Du (x tη) 2 dx η 2 dt = Du 2 η 2. Du (x tη)ηdt 2 dx Summarizing, we have proved L 2 estimates (6.12) with C = D φ u and (6.13) with C = Du. Estimates of derivatives proceed in a similar fashion. Note that implies D(a(, θ)u u )(x) = (Du (R θx) Du (x))r θ + Du (x)(r θ I) D(a(, θ)u u ) ( D φ Du + Du )d(θ, ). Using the commutator relations (2.7),(2.8) leads to the estimate D(a(, θ)u u ) ( D φ u H Du H 1)d(θ, ). Combining this with the estimate D(a(η, )u u ) D 2 u η we obtain the estimates in (6.12),(6.13) for the H 1 norm. Finally, the second derivative satisfies D 2 (a(, θ)u u ) This leads to the estimate = (D 2 u (R θ ) D 2 u )[R θ, R θ] + D 2 u [R θ I, R θ] + D 2 u [I, R θ I]. D 2 (a(, θ)u u ) (2 D 2 u + D φ D 2 u )d(θ, ). Again the commutator relations (2.7),(2.8) allow to estimate the constants in terms of Du H 1 and D φ u H 2 which are finite by Assumption 3. The proof of (6.12) is completed by the estimate D 2 (a(η, )u u ) D 3 u η.

31 NONLINEAR STABILITY OF ROTATING PATTERNS Estimates of Nonlinearities. Recall the Sobolev embedding estimate (6.14) u C e u H 2 for u H 2 (Ê 2 ). In the following we consider functions (6.15) u = a(γ)u + w, γ SE(2), w H 2, w H 2 ε, where ε will be made small during the proofs. Our first condition is C e ε 1 so that (6.14) and (6.15) imply (6.16) u u + 1. Next introduce the constants (6.17) M k = max{ D k f(ξ) : ξ u + 1}, k =,...,4. Lemma 6.2. There exist constants C, C 1, C 2 > and neighborhoods U = {γ SE(2) : d(γ, ½) ρ } W = {w H 2 Eucl : w H 2 ε } such that the following estimates hold for all w, w 1, w 2 W, γ, γ 1, γ 2 U : (6.18) (6.19) (6.2) (6.21) r [f] (γ, w 1 ) r [f] (γ, w 2 ) H 2 ) C (d(γ, ½) + max( w 1 H 2, w 2 H 2) w 1 w 2 H 2, ) r [f] (γ, w) H 2 C (d(γ, ½) + w H 2 w H 2, r [w] (γ, w 1 ) r [w] (γ, w 2 ) H 2 ) C 1 (d(γ, ½) + max( w 1 H 2, w 2 H 2) w 1 w 2 H 2, ) r [w] (γ, w) H 2 C 1 (d(γ, ½) + w H 2 w H 2, (6.22) r [γ] (γ 1, w 1 ) r [γ] (γ 2, w 2 ) + r [w] (γ 1, w 1 ) r [w] (γ 2, w 2 ) H 2 C 2 (d(γ 1, γ 2 ) + w 1 w 2 H 2). Proof. Since r [f] (γ, ) = it is obvious that (6.19) and (6.21) follow from (6.18) and (6.2) by setting w 1 = w, w 2 =. It is useful to introduce the mapping ρ [f] : SE(2) Ê m Ê 2 Ê m : (6.23) ρ [f] (γ, w, x) = f(a(γ)u (x) + w) f(a(γ)u (x)) Df(u (x))w. The corresponding Nemitzky operator from (2.32) is then given by r [f] (γ, w)(x) = ρ [f] (γ, w(x), x), x Ê 2 for γ SE(2), w H 2. Consider γ SE(2) and w j Ê m, w j 1, j = 1, 2 and define w = w 1 w 2. Then we obtain from (6.17) (6.24) ρ [f] (γ, w 1, x) ρ [f] (γ, w 2, x) = 1 Df(a(γ)u (x) + w 2 + tw) Df(u (x))dtw M 2 ( a(γ)u (x) u (x) + max( w 1, w 2 )) w.

32 38 WOLF JÜRGEN BEYN AND JENS LORENZ Similarly, (6.25) D w ρ [f] (γ, w 1, x) D w ρ [f] (γ, w 2, x) M 2 w and (6.26) D 2 w ρ[f] (γ, w 1, x) M 2. Consider now w 1, w 2 W, let w = w 1 w 2 and obtain from (6.24), (6.14) and Lemma 6.1: = r [f] (γ, w 1 ) r [f] (γ, w 2 ) 2 r [f] (γ, w 1 ) r [f] (γ, w 2 ) 2 dx M 2 2 ( a(γ)u u + max( w 1, w 2 )) 2 w(x) 2 dx M 2 2C 2 e ( a(γ)u u H 2 + max( w 1 H 2, w 2 H 2)) 2 w 2 M 2 2C 2 ec 2 a (d(γ, ½) + max( w 1 H 2, w 2 H 2)) 2 w 2 H 2. We estimate the derivative D(r [f] (γ, w 1 ) r [f] (γ, w 2 )) = (D w ρ [f] (γ, w 1, x) D w ρ [f] (γ, w 2, x))dw 1 (x) + D w ρ [f] (γ, w 1, x)(dw 1 Dw 2 )(x) + (Df(a(γ)u + w 1 ) Df(a(γ)u + w 2 ))D(a(γ)u u )(x) + 1 D 2 f(a(γ)u + w 2 + tw) D 2 f(u )dt [Du, w 1 w 2 ] = T 1 + T 2 + T 3 + T 4. Using (6.25) and Lemma 6.1 the estimates are T 1 (x) 2 dx M2 2 w(x) 2 Dw 1 (x) 2 dx T 2 (x) 2 dx M2C 2 e 2 w 2 H 2 w 1 2 H 2, D w ρ [f] (γ, w 2, x) 2 Dw(x) 2 dx M2 2 C2 e w 2 2 H 2 w 2 H 2, T 3 (x) 2 dx M2 2 w(x) 2 D(a(γ)u u ) 2 dx M2 2 w 2 a(γ)u u 2 H 2 M2C 2 ec 2 a w 2 2 H 2d(γ, ½)2, T 4 (x) 2 dx M3 2 ( a(γ)u u + max( w 1, w 2 )) 2 Du 2 w 2 dx M 2 3 C4 e Du 2 H 2 ( a(γ)u u H 2 + max( w 1 H 2, w 2 H 2)) 2 w 2 ( M 3 C 2 ec a (d(γ, ½) + max( w 1 H 2, w 2 H 2)) w H 2) 2.

33 NONLINEAR STABILITY OF ROTATING PATTERNS 381 We evaluate the second derivative D 2 (r [f] (γ, w 1 )) = D 2 wρ [f] (γ, w 1, x)[dw 1, Dw 1 ] + D w ρ [f] (γ, w 1, x)d 2 w 1 + 2D x D w ρ [f] (γ, w 1, x)[dw 1, I] + D 2 xρ [f] (γ, w 1, x) = T 1 (w 1 ) + T 2 (w 1 ) + T 3 (w 1 ) + T 4 (w 1 ). For the first term we use the Gagliardo Nirenberg estimate (6.27) Du 4 dx C G u D 2 u 3 C G u 4 H for u Ê 2 H2. 2 With the abbreviation j = (a(γ)u + w j )(x), j = 1, 2 we have T 1 (w 1 ) T 1 (w 2 ) = D 2 f( 1)[Dw, Dw 1 ] + D 2 f( 1)[Dw 2, Dw] + (D 2 f( 1) D 2 f( 2))[Dw 2, Dw 2 ] = T 11 + T 12 + T 13, T 11 (x) 2 dx M2 2 Dw 2 Dw 1 2 dx M 2 2 ( Dw 4 dx M 2 2 C G w 2 H 2 w 1 2 H 2. ) 1 Dw dx The estimate of T 12 is analogous and for T 13 we have T 13 (x) 2 dx M3 2 w 2 Dw 2 2 dx M 2 3C 2 ec G w 2 H 2 w 2 2 H 2. Setting j = (γ, w j (x), x), j = 1, 2 we find with (6.25) ( ) T 2 (w 1 ) T 2 (w 2 ) = D w ρ [f] ( 1)D 2 w + D w ρ [f] ( 1) D w ρ [f] ( 2) D 2 w 2 = T 21 + T 22, T 21 (x) 2 dx M 2 2 ( a(γ)u u + w 1 H 2) 2 D 2 w 2 dx M2C 2 ec 2 a 2 (d(γ, ½) + w 1 H 2) 2 w 2 H 2, T 22 (x) 2 dx M2 2 w(x) 2 Dw 2 (x) 2 dx M2 2 C2 e w 2 H 2 w 2 2 H 2. With t j = (a(γ)u + tw j )(x), j = 1, 2 we write the T 3 terms as follows T 3 (w 1 ) T 3 (w 2 ) = D x (Df( 1) Df(u ))[Dw 1, I] D x (Df( 2) Df(u ))[Dw 2, I] = 1 1 D 3 f(t 1)dt[Da(γ)u, w 1, Dw 1 ] D 3 f(t 2)dt[Da(γ)u, w 2, Dw 2 ].

34 382 WOLF JÜRGEN BEYN AND JENS LORENZ Telescoping in the usual way and using f C 4 and Da(γ)u = Du C e Du H 2 C (cf. Lemma 6.1) we can estimate T 3 (w 1 ) T 3 (w 2 ) 2 dx C ( w 1 2 w 2 H 2 + w 2 H 2 w 2 2 H 2 + w 2 w 2 2 w 2 2 H 2 ). Finally, we have T 4 (w 1 ) = D 3 (t 1)dt [Da(γ)u, w 1, D(a(γ)u u )] D 2 f(t 1)dt [w 1, D 2 (a(γ)u u )] D 3 f(t 1)dt [D(a(γ)u u ), Du, w 1 ] ( D 3 f(t 1) D 3 f(u ) ) dt [Du, Du, w 1 ] ( D 2 f(t 1) D 2 f(u ) ) dt [D 2 u, w 1 ] = T 41 + T 42 + T 43 + T 44 + T 45. All terms T 4j (w 1 ) T 4j (w 2 ), j = 1,...,5 can be handled by telescoping as before. Note that w-differences are estimated by w 1 w 2 C e w 1 w 2 H 2 whereas D j (a(γ)u u ), j = 1, 2 stays under the integral so that Lemma 6.1 applies. We do not give the details. Note that the small Lipschitz estimate (6.18) transfers directly from r [f] to r [w]. By (2.36) we have for w H 2 Eucl (6.28) r [w] (γ, w) = ( P + P ( Du (γ ) D φ u (γ ) ) S(γ) 1 (I P) ) r [f] (γ, w), and the assertion follows from the uniform bound of Da(γ)u H 2 and of S(γ) 1 for γ U, see (2.34) and Lemma 6.1. Since r [γ] from (2.35) is of a similar structure as r [f] we get a Lipschitz estimate for r [γ] with respect to w as well. Finally, we obtain a Lipschitz estimate for ρ [f] with respect to γ from Lemma 6.1 = ρ [f] (γ 1, w, x) ρ [f] (γ 2, w, x) 1 Df(a(γ 1 )u (x) + tw(x)) Df(a(γ 2 )u (x) + tw(x)) dt M 2 a(γ 1 )u (x) a(γ 2 )u (x) w M 2 C e a(γ 1 )u a(γ 2 )u H 2 w M 2 C e C a d(γ 1, γ 2 ) w. Since S(γ) 1 is Lipschitz bounded this leads to (6.22) via equations (6.28) and (2.35) Gronwall Estimate. The Gronwall type estimate taylored to our needs is the following. Lemma 6.3. Let ε, C, C, β be positive constants such that C 1 and (6.29) ε β 16 CC,

35 NONLINEAR STABILITY OF ROTATING PATTERNS 383 and let ϕ : [, t ) [, ), < t, be a continuous function satisfying for t < t (6.3) ϕ(t) Cεe βt + C t Then ϕ satisfies the exponential bound e β(t τ) ( ϕ 2 (τ) + εϕ(τ) ) dτ. (6.31) ϕ(t) < 2Cεe 3 βt 4, t < t. Proof. First note that (6.31) holds at t =. If (6.31) does not hold, let T (, t ) denote the smallest time at which equality occurs. Then we have 2Cε exp( 3 βt 4 ) = ϕ(t) Cεe βt + C T e β(t τ) (ϕ 2 (τ) + εϕ(τ))dτ T Cεe βt + 2C Cε 2 e βt 2C exp( βτ βτ ) + exp( 2 4 )dτ ( ) Cεe βt 2 + 2C Cε C exp( βt) + exp( 4 β 3 βt 4 ) ( ) < 2Cε exp( 3 βt 4 ) 1 8C Cε +. 2 β This estimate contradicts (6.29), and the lemma is proved Proof of Theorem 1.1. Let ρ > and ε > denote constants specifying the neighborhoods in Lemma 6.2. For the initial value v from (1.21) we assume (6.32) v H 2 ε 1, and we impose several restrictions on ε 1 in the sequel. By Lemma 2.5 and Lemma 6.1 the initial values (6.6),(6.5) satisfy for sufficiently small ε 1 (6.33) d(γ, ½) = d(γ 1 (I P)v, Γ 1 ()) C Γ I P H 2 H 2ε 1 ρ, w H 2 v H 2 + u a(γ )u H 2 ε 1 + C a d(γ, ½) (1 + ĈC a)ε 1 ε. where Ĉ = C Γ I P H 2 H2. Then Lemma 6.2 applies. We also introduce the constants (cf. (2.34)) (6.34) C g = C S I P H 2 H 2, where C S = sup γ U S(γ) 1. As usual, the Lipschitz bound (6.22) ensures the existence of a local solution in C([, t ), H 2 ) for the system (6.4),(6.5),(6.6),(6.7). We use (6.3) with β = 4 3 β and C W = C W ( β). Let C = C W C 1 max(ĉ, 1), where C 1 is from Lemma 6.2, and impose the following conditions on ε 1 in (6.32) (6.35) ε 1 min ( β 12C 1 C W C, ε 4C, βρ 2(Ĉβ + 2C gcε (1 + 2C)) ).

36 384 WOLF JÜRGEN BEYN AND JENS LORENZ Let [, t ) be the maximal domain of existence for the solution (w(t), γ(t)) in W U, cf. Lemma 6.2. From (6.4) and (6.3) we have the estimate w(t) H 2 C W e t β w t H 2 + C W C 1 e β(t τ) (Ĉε 1 + w(τ) H 2) w(τ)) H 2dτ. Lemma 6.3 applies with ε = ε 1, ϕ(t) = w(t) H 2, since condition (6.29) follows from (6.35). From (6.31) we obtain the estimate (6.36) w(t) H 2 2Cε 1 e βt, t < t. The condition (6.35) guarantees that w(t) stays in an ε 2 ball in H2 and thus stays away from the boundary of W. With (6.36) and (6.19) we obtain from (6.7) the estimate d(γ(t), ½) d(e tec γ, ½) + t r [γ] (γ(τ), w(τ) dτ t d(γ, ½) + C g r [f] (γ(τ), w(τ)) H 2dτ Ĉε 1 + C g (ε + 2Cε 1 ) t 2Cε 1 e βτ dτ ε 1 (Ĉ + 2 β C gcε (1 + 2C)) ρ 2, where the last inequality is a consequence of (6.35). Thus we conclude that t =, and the exponential estimate (6.36) holds for all t. In fact, for any v satisfying (6.32) we can repeat the a-priori estimate above with ε 2 = v instead of ε 1. With the same constants we then find (6.37) w(t) H 2 2C v H 2e βt, t <, which proves the w-estimate in Theorem 1.1. The smallness estimate for γ = γ() = (η, θ ) follows directly from (6.33). We define the asymptotic phase γ = (η, θ ) by (6.38) γ = γ + e τec r [γ] (γ(τ), w(τ))dτ. Note that the integral exists because of e τec = 1 and the exponential decay shown above, (6.39) r [γ] (γ(τ), w(τ)) C γ e βτ, C γ = C g ε (1 + 2C)ε 1. This leads to d(γ, e tec γ(t)) = d((η, θ ), (R ct η(t), θ(t))) d(γ, γ + t C γ e βt. t e τec r [γ] (γ(τ), w(τ))dτ) e τec r [γ] (γ(τ), w(τ))dτ) As before, the value ε 1 that appears in C γ can be replaced by v H 2, and this proves the final estimate in Theorem 1.1.

37 NONLINEAR STABILITY OF ROTATING PATTERNS Existence and Estimate of D φ u In this section we denote the L 2 norm on L 2 (Ê 2, Ê m ) by. We assume f C to avoid counting of derivatives. The arguments in this section are only needed to show existence of and bounds for D φ u. The results of Section 6 yield existence of a solution u(x, t) in of the integral equation C 1( [, ), L 2) C ([, ), H 2) if u(t) = e tl u + t e (t τ)l f(u(τ))dτ and v H 2 is small. u = u + v H 2 Theorem 7.1. If u H 2 Eucl then u(t) H2 Eucl for all t and (7.1) D φ u(t) Ce γt D φ u where C and γ are independent of t. Proof. 1. We know that L : H 2 Eucl L2 L 2 is closed and, therefore, using properties of the resolvent, It follows that e tl : H 2 Eucl H2 Eucl and D(L ) = H 2 Eucl. e tl v H 2 Eucl Ce γt v H 2 Eucl for v HEucl 2 ( ). 2. Let r C [, ), HEucl 2 and let v HEucl 2. Set v(t) = e tl v + ( ) Then v C [, ), HEucl 2 and t (7.2) v(t) H 2 Eucl Ce γt v H 2 Eucl + C e (t τ)l r(τ)dτ. t e γ(t τ) r(τ) H 2 Eucl dτ. 3. Let f (w) Q for all w Ê m. Let r be as above. We have Define Obtain the estimate D φ f(r(t)) Q D φ r. v(t) = e tl + t e (t τ)l f(r(τ))dτ.

38 386 WOLF JÜRGEN BEYN AND JENS LORENZ t (7.3) D φ v(t) Ce γt D φ v + CQ e γ(t τ) r(τ) dτ. 4. For u H 2 Eucl define the sequence un by u (t) = e tl u u n+1 (t) = e tl u + t e (t τ)l f(u n (τ))dτ Fix a time interval < t 1 t t 2. Parabolic smoothing estimates hold and one obtains u n C ( ) Ê 2 [t 1, t 2 ] and the sequence u n is uniformly smooth on Ê 2 [t 1, t 2 ]. This means that all derivatives are bounded independently of n. (This follows from bounds for u n (t) H k for all k. Time derivatives and mixed derivatives can be expressed by space derivatives using the differential equation u n+1 t = L u n+1 + f(u n ).) Iterating the estimate (7.3) one obtains (7.4) D φ u n (t) C e γt D φ u. 5. Fix a compact region Ω = B R [t 1, t 2 ] in which the sequence u n is uniformly smooth. By Arzela Ascoli we obtain a smooth function u on Ω and a subsequence of u n so that (for any fixed K) max (x,t) Ω Dα u n (x, t) D α u(x, t) as n, n Æ1, α K. Here D α is a space-time derivative. (We do not know yet that this limit u is the solution of the integral equation constructed above, but will show this below.) A contraction argument in a weak norm (see below) will show that any possible limit of any subsequence of u n converges to the same limit on Ω. Therefore, all derivatives of the whole sequence u n converge in maximum norm on Ω to the corresponding derivatives of u. One obtains that u C (Ê 2 (, )). One also finds that max (x,t) Ω D φu n (x, t) D φ u(x, t) as n. 6. From (7.4) we have B R D φ u n (x, t) 2 dx C 2 e 2γt D φ u 2 and find for n : B R D φ u(x, t) 2 dx C 2 e2γt D φ u 2. Now let R to obtain

39 NONLINEAR STABILITY OF ROTATING PATTERNS 387 (7.5) D φ u(t) C e γt D φ u. 7. The sequence u n satisfies u n+1 t = L u n+1 + f(u n ), u n (x, ) = u (x). All derivatives converge pointwise on Ê 2 (, ) to the corresponding derivatives of u. Therefore, u t = L u + f(u), u(x, ) = u (x). This implies that the function u which is constructed here as the limit of u n agrees with the solution constructed in Section The contraction argument is as follows. For obtain with Therefore, δ n+1 = t δ n = u n+1 u n e (t τ)l M n (x, t)δ n (τ)dτ M n (x, t) Q. max τ t δn+1 (τ) C 1 t max τ t δn (τ). If C 1 t a geometric sum argument shows that any limit of any subsequence of the u n is unique on [t 1, t 2 ] for < t 1 < t 2. One can then restart the argument at t 2. The constant C 1 does not change. 8. The Quintic Cubic Ginzburg Landau Equation Consider the quintic cubic Ginzburg Landau equation (QCGL) (8.1) u t = α u + u(µ + β u 2 + γ u 4 ) where x Ê 2, u(x, t) and µ Ê, α, β, γ. Since u is complex valued the equation corresponds to a real system with two variables. In Section 8.2 we identify the essential spectrum of the differential operator obtained by linearizing about a spinning soliton. The result is consistent with the numerical findings in Section Spinning Solitons for the QCGL Equation. Spinning solitons are solutions that rotate at constant speed and converge to u = as x. Such solutions are described in [6],[7], for example. We use the parameters (8.2) α = 1 2 (1 + i), β = i, γ = 1.1i, µ = 1 2. For the numerical computation we used Comsol Multiphysics TM [5]. The problem is discretized on balls B R () = {x Ê 2 : x R} with Neumann boundary conditions and piecewise linear finite elements of maximum diameter h max. The

40 388 WOLF JÜRGEN BEYN AND JENS LORENZ initial solution is obtained from the freezing approach in [3]. Then the following boundary value problem is solved (cf. the time dependent equation (1.7)): = A u + f(u) + cd φ u + η 1 D 1 u + η 2 D 2 u in B R, (8.3) = u n on B R. Here the parameter values c, η 1, η 2 are included as additional unknowns and three phase conditions are added in order obtain a well-posed boundary value problem accessible to the package [5]. Real and imaginary parts of the solution are shown in Figure 1. With the numerical solution at hand, the same code is used to compute a prescribed number neig of eigenvalues of the linearized finite element operator. The code uses the package ARPACK with a prescribed real shift σ Ê and is expected to give the neig eigenvalues closest to σ. We have B = µi, which satisfies Assumption 2 since µ <. Assumptions 3 and 4 will be checked numerically. Our reference discretization uses the values (8.4) R = 3, h max =.25, neig = 4, σ = 3 which leads to a matrix eigenvalue problem with about 1 5 eigenvalues. The spectral picture corresponding to this choice is Figure 2 showing neig = 4 eigenvalues close to σ = 3. We observe a zig-zag type cluster of eigenvalues which one expects to correspond to essential spectrum. In fact, the structure will be explained by Theorem 8.1 below. In addition, the three eigenvalues ±ic and on the imaginary axis are clearly visible in Figure 2. Moreover, Figure 2 suggests that there are eight additional complex conjugate pairs of eigenvalues lying between the imaginary axis and the zig-zag structure. For these eight pairs we have boxed the 8 eigenvalues with positive imaginary part. Note that one of the boxed eigenvalues near the top is rather close to the zig-zag structure, but does not belong to it. The eigenvalues ±ic and are also boxed. In addition, we have boxed two (numerical) eigenvalues appearing near the center at two tips of the zig-zag structure. Figure 3 plots the real parts of the eigenfunctions corresponding to the 12 boxed eigenvalues. Note that the eigenfunctions numbered 1 to 1 are localized; these eigenfunctions correspond to the eigenvalues and ic as well as to the eight isolated eigenvalues not belonging to the zig-zag structure. The eigenfunctions numbered 11 and 12 correspond to the two boxed eigenvalues at the tips of the zig-zag stucture; as expected, these eigenfunctions are not localized since they correspond to essential spectrum. To validate our findings we have varied two parameters in the reference configuration (8.4): (1) Decrease radius to R = 2. The spectral picture is Figure 4 (left); the eigenfunctions are similar to those in Figure 3 (not shown). (2) Coarsen the grid to h max =.5. The spectral picture is Figure 4 (right); the eigenfunctions are similar to those in Figure 3 (not shown). As already mentioned, the zig-zag structure in Figure 2 corresponds to essential spectrum of L. Therefore, our tests confirm that there are eight pairs of isolated eigenvalues between the essential spectrum and the imaginary axis. Since the corresponding eigenfunctions are localized and since there are no unstable eigenvalues,

41 NONLINEAR STABILITY OF ROTATING PATTERNS 389 Figure 1. Real and imaginary part of spinning solitons in the QCGL-system Assumptions 3 and 4 are confirmed. As one expects, variation of the size of the domain has a strong impact on the clusters that approximate the essential spectrum while refining the mesh does no change the clusters very much.

42 39 WOLF JÜRGEN BEYN AND JENS LORENZ 6 R=3, hmax=.25, neig= Figure 2. Spectrum of linearization about spinning soliton On the other hand, looking at numerical values (not shown) one finds that convergence towards the isolated eigenvalues is best observed when the mesh-size is varied. We refer to the work by Sandstede and Scheel [17],[14] on absolute spectra, which is relevant when studying perturbations such as truncation to a bounded domain. For the three eigenvalues ±ic and on the imaginary axis we have also compared the numerical eigenfunctions ϕ j,h with the eigenfunctions D 1 u ± id 2 u and φ 3 = D φ u given by Lemma 2.3. Here we have used the computed numerical approximation for u (as a solution of (8.3)) and have evaluated its derivatives numerically. The resulting errors are ϕ 1,h ϕ 1 L 2 = inf θ [,2π] ϕ 2 e iθ (ϕ 2,h + iϕ 3,h ) L 2 = In view of the tolerances used, these results give satisfactory tests. Remark : We note that equation (8.1) happens to have an extra S 1 symmetry given by (8.5) F(e iθ u) = e iθ F(u), θ S 1, where F denotes the right hand side of (8.1). Numerical computations suggest that the rotating wave u satisfies u (R θ x) = e iθ u (x). Then u is also a relative equilibrium with respect to the group action of G = S 1 Ê 2 given by (cf. (1.11)) (a(η, θ)u)(x) = e iθ u(x η).

43 NONLINEAR STABILITY OF ROTATING PATTERNS 391 Figure 3. Eigenvalues and eigenvectors corresponding to boxes in Figure 2, the first two correspond to and ic, the next eight to isolated and stable eigenvalues, the last two to spectral values at the tip of the zig-zag structure This leads to a simpler linearization where D φ is not present and, therefore, a simpler stability analysis is possible than provided by our Theorem. However, this special situation can be easily avoided by destroying the symmetry (8.5) in (8.1). For example, we can perturb the complex factor γ in the two-dimensional real version of (8.1) so that it no longer corresponds to multiplication by a complex number. Numerical experiments show that the spinning solitons and the structure of the spectrum persist for the modified system On the Essential Spectrum of L. We use the following terminology ([11, Ch.5]):