Global Dissipativity and Inertial Manifolds for Diffusive Burgers Equations with Low-Wavenumber Instability

Global well-posedness, existence of globally absorbing sets and existence of inertial manifolds is investigated for a class of diffusive Burgers equations. The class includes diffusive Burgers equation with nontrivial forcing, the Burgers-Sivashinsky equation and the Quasi-Stedy equation of cellular flames. The global dissipativity is proven in 2D for periodic boundary conditions. For the proof of the existence of inertial manifolds, the spectral-gap condition, which Burgers-type equations do not satisfy in its original form is circumvented by the Cole-Hopf transform. The procedure is valid in both one and two space dimensions.


Introduction
We study a class of diffusive Burgers equations with low-wavenumber instability. These are modified Burgers equations, which in addition to the usual Burgers nonlinearity and the diffusive term contain an additional linear term responsible for linearized instability of low wavenumbers. The most prominent example of an equation of this type is the Kuramoto-Sivashinsky equation (KSE), ∂ t U + (U · ∇)U + (∆ 2 + α∆)U = ∆U, subject to appropriate initial and boundary conditions. KSE has been introduced in [25] as a model for the flame front propagation in combustion theory. Ever since, the equation has been the subject of intense scientific interest due to its complicated and interesting dynamical behavior, which for a range of parameters, is characterized by high-dimensional cellular chaos. Many of the basic questions for the KSE in one space dimension subject to periodic boundary conditions, such as the global dissipativity and the existence of inertial manifolds have since been resolved (see e.g. [4,14,10]). In two space dimensions, the equation is not known to be globally well-posed. Most of the recent research activity has shifted towards obtaining the best bounds on the absorbing sets and the dimensions of the global attractor and the inertial manifolds, since the numerical simulations suggest better estimates than the ones which have been obtained analytically.
KSE is not the only equation of this type which is of interest. The socalled Burgers-Sivashinsky equation (BSE), ∂ t U + (U · ∇)U = ∆U + (α − 1)U, was introduced as the simplest prototype of diffusive Burgers equations with low-wavenumber instability. As it turns out, the similarity between KSE and BSE due to the instability of low wavenumbers, high wavenumber damping, and nonlinear stabilization via energy transfer from the low to the high modes is rather superficial. The BSE is a gradient system, and as such it displays fairly simple dynamics characterized only by steady-states. This fact suggests that whatever the cause of cellular-chaotic dynamics may be, it is not simply due to the interplay between diffusivity, Burgers energy transfer between Fourier modes, and low-wavenumber instability, but there is something more subtile at play. It also suggests that estimates on the attractor obtained using methods which simultaneously apply to both KSE and BSE are not sharp, and certain mechanisms unique to KSE need to be identified and explored.
In recent years, yet another system of the type described above has become subject of interest. The so-called Quasi-Steady equation (QSE), ∂ t U + (U · ∇)U = ∆U + α(I − (I − ∆) −1 )U was introduced by Brauner et al. in [2] as the weakly nonlinear version of a certain truncation of the κ − θ model of cellular flames (see [13]). However, despite its ad-hoc introduction, it turns out that its dispersive relation most closely mimics the dispersion relation of the original combustion problem. Just like KSE, QSE also models the onset of the cellular-chaotic dynamics; additionally however, it may better capture some important features of the global dynamics of the original combustion problem. Unlike KSE and BSE, QSE is an integro-differential equation and it is nonlocal. In some respect, it is more benign than the KSE; as we shall prove later in this paper, it is globally well posed and globally dissipative in two space dimensions, when subjected to periodic boundary conditions. Dissipative Burgers equations with low-wavenumber instability are yet another example of parabolic PDEs, which exhibit long-term dynamics with properties typical of finite-dimensional dynamical systems. The global attractor, often considered the central object in the study of long-term behavior of dynamical systems, appears to be inadequate in capturing this finite-dimensionality, even when its Hausdorff dimension is finite. This is mainly due to two facts. Firstly, the global attractor can be a very complicated set, not necessarily a manifold; the question whether the dynamics on it can be described by a system of ODEs is yet to be resolved in the literature. Secondly, although all solutions approach this set, they do so at arbitrary rates, algebraic or exponential, and, consequently, the dynamics outside the attractor is not tracked very well on the attractor itself. When they exist, inertial manifolds emerge as most adequate objects to capture the finite-dimensionality of a dissipative parabolic PDE. Introduced by Foias et al. in [11], they are defined to remedy the shortcomings of the global attractor just described: they should be finite dimensional positive-invariant Lipschitz manifolds which attract all solutions exponentially, and on which the solutions of the underlying PDE are recoverable from solutions of a system of ODEs, termed 'inertial form'. The existence of an inertial manifold does not merely have a theoretical value, but a practical one as well: using a system of ODEs instead of a system of PDEs facilitates computations and numerical analysis.
Unfortunately, however useful, inertial manifolds seem unattainable for the vast majority of physically relevant parabolic PDE-s due to a rather re-strictive spectral-gap condition that the system at hand is required to satisfy. Despite the fact that this condition emerges naturally in any approach to the theory of inertial manifolds, the question of its actual importance remains unclear. This can be demonstrated by comparing KSE and BSE. The prior equation is known to satisfy the spectral-gap condition and therefore possess inertial manifolds despite its complicated attractor, while the second with its very simple dynamical behavior is not known to possess them. One of the goals of this paper is to demonstrate that the spectral gap condition which the BSE does not satisfy is actually an artifact in this case. The equation can be transformed into a form for which the condition will be satisfied. Using the Cole-Hopf transform, a large class of diffusive Burgers equations including BSE and QSE will be shown to possess inertial manifolds in both one and two space dimensions. The idea to transform the equation was also developed and used by the author to prove the existence of inertial manifolds for a class of nonlinear Fokker-Planck equations, which appear in the modeling of nematic polymers (see [28,29,30]). Rather than the Cole-Hopf transform, the author developed a nonlinear nonlocal transform which also eliminates the first-order derivatives from the equation, thus allowing the equation to satisfy the spectral-gap condition.
Let us remark here that the inertial manifold result of this paper also applies to the diffusive Burgers equation with forcing. In the literature on inertial manifolds, this equation is often cited as an example of an equation with trivial dynamics, for which the existence of inertial manifolds is not known due to the spectral-gap condition. Let also be noted here that the transformation approach is not an entirely new approach. In the paper [18], the author uses a different kind of transformation to prove the existence of inertial manifolds for the Burgers equation with forcing; the result, however, has since been shown to be flowed.
The paper is structured as follows. In Section 2, we introduce a more general framework for the study of diffusive Burgers equations with lowwavenumber instability subject to periodic boundary conditions; we shift our attention to a subclass of such equations, for which the low-wavenumber instability is due to a term T U, where T : L p → L p is a bounded Fourier multiplier for some p > 2. Note that KSE is no longer in this class, and the approach presented in this paper does not provide an alternative proof of the existence of inertial manifolds for KSE. In Section 3, expanding on the ideas of [21], we prove the global well-poedness and the global dissipativity for equations in this class in two space dimensions. The procedure in one space dimension is simpler, and we omit it. Finally, in Section 4, we transform the equation into a form which satisfies the spectral-gap condition, and prove the existence of inertial manifolds in both one and two space dimensions.

Class of Diffusive Burgers Equations with Low-Wavenumber Instability
We examine global well-posedness, global dissipativity and the existence of inertial manifolds for a class of diffusive Burgers equations in one or two space dimensions, which assume the general form and we write (2.1) as a system in the following form: We consider the equation on Q = [−L/2,L/2] d for some L > 0 in space dimension d = 1 or d = 2, subject to periodic boundary conditions, is the canonical basis of IR d , and subject to the zero mean condition, The term T U is responsible for the instability of low wavenumbers. The operator T is assumed to be a Fourier multiplier associated with a given bounded symbol m : ZZ d → IR with zero mean (m(0) = 0), and G is a given Lperiodic function with zero mean. Due to the fact that the Fourier multipliers commute, the equation is often written in an integrated form This form will be useful to us in proving the existence of inertial manifolds. However, this equation does not preserve zero mean, and the the dissipativity of (2.1) does not imply the dissipativity of this integrated form. There are two possibilities to circumvent this difficulty. One is to use the following integrated form instead, which does preserve the mean: Another approach, which we shall adopt in this paper, is to use the following integrated form, for which the dissipativity is implied by the dissipativity of (2.1): Observe that the mean then satisfies As discussed in the introduction, there are several different equations of type (2.1), which are of interest. The simplest example is the diffusive Burgers equation, i.e. the case when T = 0. This equation is dynamically very simple; the attractor consists of a single steady-state. However, even in this trivial case, the existence of an inertial manifold has yet not been established for a nontrivial forcing. The attractor is not necessarily an inertial manifold, since it does not attract all solutions at an exponential rate. As already discussed in the introduction, dynamically more interesting is the Burgers-Sivashinski equation, m(k) = (α − 1), α > 1, and much more interesting is the Quasi-Steady equation of cellular flames, Both systems resemble the Kuramoto-Shivasinski equation,

Mathematical Setting and Existence of Absorbing Sets
For the study of the system ( endowed with the usual · H s := · Ḣs norm. In case d = 2, let us also define and, symmetrically, H s x 2 . For simplicity, the L p , p ≥ 1 norm will be denoted | · | L p , and the L 2 norm simply | · |. We also denote H = H 0 and H = H 0 .
The standard Galerkin prcedure can be used to establish local existence of solutions, and without proof we state the following Lemma 1 Suppose that T is the Fourier multiplier associated with a bounded symbol m, and G ∈ H 1 . For a given initial datum The global well-posedness of systems of type (2.1) and the existence of a globally absorbing ball is a subtile matter due to instability of low wavenumbers. These questions have been resolved for the 1D Kuramoto-Sivashinsky equation, 1D Burgers-Sivashinsky equation, 1D Quasi-Steady equation and, in space dimension two, for the 2D Burgers-Sivashinsky equation (see [1,4,14,21]). Here, we state and prove a theorem on global wellposedness and dissipativity for the equations of the type (2.1) in two space dimensions. The proof expands on the proof of [21] for the 2D Burgers-Sivashinsky equation. For the sake of simplicity, we shall not concern ourselves with the size of constants with respect to the periodicity L, and we shall assume that G = 0.
Similarly to other proofs of the dissipativity of the systems with instability of low wavenumbers, our proof is also based on the following Poincaré-type inequality, which we adopt from [21]. It follows from the inequality for the 1D case obtained in [4] after integrating with respect to the second space variable.
Proposition 1 For any β > 0 there exists a 'gauge' function Φ ∈ H ∞ x 1 depending on x 1 only and a constant γ > 0, so that for every v ∈ H 1 x 1 , the following inequality holds: We are now ready to state and prove the global well-posedness and dissipativity result.
Theorem 1 Suppose that the symbol m is such that T is an L p multiplier for some p > 2. Then the Cauchy problem for the equation (2.1) is globally well-posed for any initial datum U 0 = (u 0 1 ,u 0 2 ) ∈ H 1 and there exists ρ > 0, and a positive time (3.6) Proof : As already indicated, the proof expands on the proof of [21], and we shall omit some details contained in that paper, and we shall emphasize the differences arising from replacing the term U by the more general term, T U, where T is an L p multiplier for some p > 2. Let K q := |T | L q →L q , whenever it is finite. The proof consists of two parts. Firstly, one obtains a priori estimates on the quantities |(∂ x 1 u 1 ) + | and |(∂ x 2 u 2 ) + |, which suffice to prove the global well-posedness. Subsequently, one imployes the Poincaré-type inequality from Proposition 1 to prove the existence of a globally absorbing ball in H. The existence of a globally absorbing ball in H 1 follows easily.
Let U = (u 1 ,u 2 ) be a solution of (2.2) to the Cauchy problem U(0) = U 0 for an initial datum U 0 = (u 1 0 ,u 2 0 ) ∈ H 1 , and let us assume that for some T * < ∞, the interval [0,T * ) is the maximal interval of existence of that solution. In particular, this means that limsup t→T − * |∇U(t)| = ∞. Multiplying (2.1) by U, integrating by parts, and employing Hölder inequality imply The Ladyzhenskaya inequality, and Young's inequality imply further that for some constant C > 0 large enough, we have On the other hand, multiplication by −∆U, integrating by parts, and using the fact that ∂ x 1 u 2 = ∂ x 2 u 1 imply Similarly as before, we have It appears crucial to obtain some control over the term |(div U) + |. To this end, let us denote w 1 = (∂ x 1 u 1 ) + and w 2 = (∂ x 2 u 2 ) + . Multiplying the first equation of (2.2) by −∂ x 1 (w 1 ) p−1 , integrating by parts, and using the fact that ∂ x 1 u 2 = ∂ x 2 u 1 , one obtains Proceeding symmetrically with the second equation, then adding the two inequalities, and using Young's's inequality, one has Denoting α p := |w 1 | p L p + |w 2 | p L p , one obtains the inequalitẏ which further implies sup t∈[t 0 ,T * ) In particular, this implies that |(div U) + | is uniformly bounded on [0,T * ). This fact together with (3.7) implies that |U| is uniformly bounded on [0,T * ). This, in turn, together with (3.8), integration by parts and Young's inequality implies that |∇U| is uniformly bounded on [0,T * ). We have arrived at a contradiction to the assumption that limsup t→T − * |∇U(t)| = ∞, and the global well-posdness follows.
Let us now assume that T * = +∞. In order to prove the global dissipativity using (3.7) and (3.8), one would like to obtain an a priori bound over |(div U) + | independent of |(div U 0 ) + |. One easily checks that is an upper solution to inequality (3.9), thus yielding the following inequality As a consequence, there also exists a constant c p > 0 depending on L and p so that max(|w 1 (t)|,|w 2 (t)|) ≤ c p , t ≥ 1. (3.10) The next step is to use the Poincaré inequality from Proposition 1 to circumvent the difficulties arising from the instability of low wavenumbers. To this end, let Φ ∈ H ∞ x 1 initially be any function depending on the variable x 1 only, and let v 1 = u 1 − Φ. Then, v 1 satisfies Multiplying by v 1 , integrating over Q and noticing Using the Ladyzhenskaya inequality, one obtains and Young's inequality implies where the constant C 1 depends on p and L, and the constant C 2 on Φ. According to Proposition 1, we can chose Φ ∈ H ∞ x 1 depending on x 1 and γ > 0 so that for any v ∈ H 1 Using the fact that Proceeding symmetrically with the second equation of (2.2) and adding the two equations yields the existence of the time T 0 = T 0 (|U 0 |) > 0 so that for all t ≥ T 0 , |U(t)| < ρ 1 . From this fact, inequality (3.10), and Young's inequality, the conclusion of the theorem follows easily from inequality (3.8).
2 Let us make the following remarks: Corollary 1 Suppose that the symbol m is such that T is an L p multiplier for some p > 2. Then the Cauchy problem for the equation (2.4) is globally well-posed for any initial datum φ 0 ∈ H 2 and there exist constants r > 0 and r ∞ > 0, and a positive time In particular, the set B := B H 2 (r) is an absorbing set for the equation (2.4).
Proof : The first inequality of the corollary follows immediately from Theorem 1 and the equation for the mean (2.5), using the Poincaré inequality. The second one follows then from Agomon inequality in two space dimensions,

Inertial Manifolds
Let us first recall the definition of inertial manifolds and a theorem on their existence. Let us denote A = −∆ : H 2 → H. Following the standard procedure, from the Fourier modes, e 2πik·x/L , k ∈ ZZ d , one constructs the complete set of eigenfunctions w k , k = 0,1,2,... of the operator A corresponding to the eigenvalues λ k which belong to the set where (Λ k ) ∞ k=0 is an increasing sequence. The application of existing theory of inertial manifolds is contingent on the existence of sufficiently large gaps in this set. If d = 1, then Λ k = (2π/L) 2 k 2 , and the spectral gaps Λ k+1 − Λ k = (2π/L) 2 (2k + 1). If d = 2, it is known that Λ k ∼ k and, after passing to a subsequence, logk ∼ < Λ k+1 − Λ k (see [22]). Therefore, there are arbitrarily large gaps in space dimensions one and two.
Consider an evolution equation of the form 12) where N : H α → H β , 0 ≤ α − β ≤ 1, is locally Lipschitz-continuous. We define the projection operators and Q n = I − P n .

Definition 1 An inertial manifold M is a finite-dimensional Lipschitz manifold which is positively invariant,
and exponentially attracts all solutions uniformly on any bounded set U ⊂ H α of initial data, The inertial manifold is said to be asymptotically complete, if for any solution u(t) there exists v 0 ∈ M such that There are various methods for proving the existence of inertial manifolds. Almost of them, except for some very special cases, require some kind of Lipschitz continuity of the nonlinearity N and make use of a very restrictive spectral gap property of the linear operator A. In the approach we adopt here, these two conditions yield the strong squeezing property, which, in turn, yields the existence of an inertial manifold which is obtained as a graph of a Lipschitz mapping. We state the following Theorem 2 Suppose that the nonlinearity N in (4.12) satisfies the following three conditions Suppose that the eigenvalues of A satisfy the spectral gap condition, for some n ∈ IN. Then there exists an asymptotically complete inertial manifold which is obtained as the graph of a Lipschitz function Φ : Restricting (4.12) to M yields the ordinary differential equation for p = P n u dp dt + Ap = P n N(p + Φ(p)) termed the inertial form.
Proof : As already mentioned, there are several different proofs of some version of this theorem. For an elegant improvement of the geometric approach developed in [7] see also [23,24].
2 The above theorem does not apply to Burgers equation in its original form. To illustrate this fact, let us proceed in the usual fashion and denote N(U) = −(U · ∇)U + T U + ∇G. The first three conditions of the theorem could be circumvented in the usual fashion by 'preparing' the equation; instead of N, one checks the conditions of the theorem for N P which is obtained from N in a manner which leaves it unchanged on a bounded absorbing set; it is declared equal to zero outside a larger ball which contains the absorbing set, and in between, it is defined in a fashion which would insure the global Lipschitz continuity. It has been proven that this procedure does not change the dynamics inside the absorbing ball which all solutions enter exponentially fast. In order to be able to prepare the equation, on can easily see that we would have to choose α = 1 and β = 0. However, it can be easily checked that the spectral gap condition (4.13) is not satisfied for a large Lipschitz constant C in either space dimension d = 1 or d = 2.
The main idea of this section is to circumvent the difficulty presented by the spectral-gap condition by performing the change of variables, in (2.4) to obtain the equation This is the Cole-Hopf transform which eliminates the Burgers nonlinearity from the equation. In case of the non-forced diffusive Burgers equation, it transforms it into a linear heat equation. In this case, however, the result of the transform is a new nonlinear parabolic equation. Let us denote Φ = Ψ −1 .
, where r and r ∞ were introduced in Corollary 1. Then B ⊂ H 2 is a bounded set and Ψ : (B, · H 1 ) → (B, · H 1 ) is a Lipschitz homeomorphism.
Proof : Firstly, observe that r 0 := e −r∞/2 ≤ ψ ≤ e r∞/2 =: r 1 holds pointwise for all ψ ∈ B. Secondly, for ψ = e −φ/2 ∈ B, in view of the Ladyzhenskaya inequality, and we easily conclude that there exists r 2 > 0 so that B ⊂ B H 2 (r 2 ). To see that the mapping Ψ is a Lipschitz homeomorphism, let ψ 1 = e −φ 1 /2 ∈ B and ψ 2 = e −φ 2 /2 ∈ B. We then have In the last step we again used the Ladyzhenshaya and Young's inequalities, and the Lipschitz continuity of Ψ| B follows. On the other hand, and the Lipschitz-continuity of Φ| B follows. 2 From Corollary 1 follows the following Corollary 2 Suppose that the symbol m is bounded and that T is an L p multiplier for some p > 2. Then the Cauchy problem for the equation (4.14) is globally well-posed for any positive initial datum ψ 0 ∈ H 2 , and there exists a positive time T = T ( log(ψ 0 ) H 2 ) such that for all t ≥ T we have ψ(t) ∈ B. In view od the fact that that Φ : B → H 1 is a Lipschitz continuous, it is easy to check that N| B : B → H 1 is Lipschitz continuous with respect to norm · H 1 .
We are now in the position to prepare the equation as it is customary in the literature. First, we modify the nonlinear term: N P : B ∪ (H 1 \B H 1 (2r 2 )) → H 1 is clearly a Lipschitz continuous. Denote by C > 0 its Lipschitz constant. Following [31], a Lipschitz-continuous function defined on a subset of a Hilbert space can be extended to a Lipschitz continuous function defined on the entire Hilbert space, even preserving the Lipschitz constant C > 0. Without changing the notation, let us by N P : H 1 → H 1 denote such an extension. The prepared equation reads now d dt ψ + Aψ = N P (ψ). (4.15) This equation clearly satisfies the first three conditions of the Theorem 2, where we chose α = β = 1. The spectral gap condition (4.13) reads in this case λ n+1 − λ n ≥ 4C, which is clearly satisfied for some large enough n ∈ IN in both space dimensions one and two, as there exist arbitrary large spectral gaps of A in both of these dimensions. Thus, the prepared equation (4.15) possesses an asymptotically complete inertial manifold M P . Following the general procedure of [10], from the existence of an inertial manifold M p for the equation (4.15) one infers the existence of an asymptotically complete inertial manifold M for the transformed equation (4.14). Finally, since Ψ is a Lipschitz-homeomorphism in H 1 , an asymptotically complete inertial manifold in H 1 for the integrated equation ( is an asymptotically complete inertial manifold in H for the equation (2.1). Thus, we have proved the following Theorem 3 Suppose that the symbol m is such that T is an L p multiplier for some p > 2, and G ∈ H 2 . Then the system (2.1) possesses an asymptotically complete inertial manifold.