Final Dynamics of Systems of Nonlinear Parabolic Equations on the Circle

We consider the class of dissipative reaction-diffusion-convection systems on the circle and obtain conditions under which the final (at large times) phase dynamics of a system can be described by an ODE with Lipschitz vector field in $\mathbb{R}^{N}$. Precisely in this class, the first example of a parabolic problem of mathematical physics without the indicated property was recently constructed.


Introduction
The problem of describing the final (at large times) dynamics of dissipative semilinear parabolic equations (SPE) ∂ t u = G(u) ( * ) (see [5]) with a Hilbert phase space X by an ordinary differential equation (ODE) in R N has been attracting researcher's attraction for a long time. In fact, it is required to separate finitely many "determining" degrees of freedom of an infinite-dimensional dynamical system. In this case, the key geometric object is the so-called (global) attractor [1,13,16], i.e., the connected compact invariant set A ⊂ X that uniformly attracts bounded subsets X as t → +∞.
The required ODE can sometimes be implemented as an inertial form [13,16,19] obtained by restricting the initial equation to an inertial manifold, i.e, a finite-dimensional invariant C 1 -surface M ⊂ X containing the attractor and exponentially attracting (with asymptotic phase) all trajectories of ( * ) as t → +∞. The theory of inertial manifolds originally encountered systematic difficulties, and several alternative concepts of finitedimensional reduction of SPE have therefore been developed starting from [3,12,14,15].
Following [14], we shall say that the dynamics of ( * ) on the attractor (final dynamics) is finite-dimensional if there exists an ODE in R N with Lipschitz vector field, resolving flow { Θ t } t∈R , and invariant compact set K ⊂ R N such that the phase half-flows { Φ t } t≥0 of equation ( * ) on A and { Θ t } t≥0 are Lipschitz conjugate on K. The existence of the inertial manifold implies that the dynamics is finite-dimensional on the attractor and, in general, looks like a more attractive property. Indeed, in the first case, the inertial form provides an exponential asymptotics of any solution of the equations at large times, and in the second case, we have an ODE reproducing the original dynamics only on the itself. Nevertheless, the fact that the dynamics is finite-dimensional on A means that the structure of limit regimes of SPE with infinitely many degrees of freedom is no more complicated than the structure of similar regimes of an ODE with Lipschitz vector field in R N .
In this paper, we consider the problem of whether the final dynamics is finitedimensional for one-dimensional systems of reaction-diffusion-convection equations where u = (u 1 , . . . , u m ) and f and g are sufficiently smooth matrix and vector functions.We assume that x ∈ J, where J is a circle of length 1. The matrix of diffusion coefficients D is assumed to be diagonal, D = diag{d j }, d j > 0. As the phase space we choose an appropriate space X ⊂ C 1 (J, R m ) in the Hilbert semiscale {X α } α≥0 generated by a linear positive definite operator u → u − Du xx in X = L 2 (J, R m ). We postulate that evolution equation (1.1) is dissipative in X and there exists the attractor A ⊂ X consisting of functions u = u(x), u ∈ C 1 (J, R m ). The algebraic structure of the "convection matrix" f = f (x, u), f = {f ij }, i, j ∈ 1, m, on the convex hull co A ⊂ X will play an important role.
For scalar equations of the form (1.1), the fact that the dynamics is finite-dimensional on the attractor was established in [15]. In the vector case, the final dynamics of systems (1.1) with scalar diffusion matrix D and spatially homogeneous nonlinearity f (u)∂ x u + g(u) was studied in [8], and the second restriction seems to be technical. The existence of an inertial manifold was proved in [8] for the scalar equation (m = 1), and for m > 1, it was proved under the assumption that the function matrix f (u) is diagonal with a unique nonzero element in a convex neighborhood of the attractor. The results obtained in [8] are based on a non-local change of the phase variable u which "decreases" the dependence of the nonlinear part (1.1) on ∂ x u and allows using the well-known "spectral gap condition".
Generalizing and developing the approach in [15], we study whether the dynamics is finite-dimensional on the attractor, but we do not consider the problem of existence of an inertial manifold for systems of equations (1.1). At the same time, we here consider the case of nonscalar diffusion matrix D and spatially nonhomogeneous nonlinearity with f = f (x, u), g = g(x, u). We prove that the limit dynamics is finite-dimensional for wide classes of systems (1.1). Now, omitting the details related to the choice of phase space and dissipativity conditions, we formulate the main results of the paper as follows.
The phase dynamics on the attractor of system (1.1) is finite-dimensional if any of the following three conditions is satisfied. We see that the scalar diffusion is favorable for the finite-dimensional reduction of (1.1).
The assumptions that the matrices are commutative can conditionally be formulated as the consistency of convection with diffusion and the self-consistency of convection on the convex hull of the attractor. Usually, the attractor A of system (1.1) can be localized in a ball B ⊂ X centered at zero. Since the embedding X → C(J, R m ) is continuous, it is actually sufficient to verify the conditions on f = f (x, u) in assertions (A), (B), and (C) for x ∈ J, u ∈ R m : |u| < r with an appropriate r > 0.
In the class of one-dimensional systems (1. The results of the paper can be generalized to systems on the circle of the form with a smooth vector function f = (f 1 , . . . , f m ). Such systems with various boundary conditions can be reduced (see [7,8]) to the form (1.1) by the termwise differentiation and an appropriate change of the variable.
We here do not consider the Dirichlet and Neumann boundary conditions for (1.1), this can be studied in a subsequent paper. The existence of an inertial manifold is proved in a similar situation in [7] for systems of general form (1. 2) with f = f (u, u x ) and a scalar diffusion matrix.
The paper is organized as follows. Section 2 contains necessary information about abstract SPE and the conditions for their final dynamics to be finite-dimensional. In Section 3, it is shown how these conditions can be applied to parabolic systems (1.1).
The main results are obtained in Section 4. In the short Section 5, we present several examples of system (1.1) which admit a finite-dimensional final dynamics. Finally, in Section 6, we discuss alternative approaches to the problem of finite-dimensional reduction of systems (1.1).

General information
First, we consider the abstract dissipative SPE in a real separable Hilbert space X with scalar product (·, ·) and the norm · . We assume that the the unbounded positive definite linear operator A with domain of definition D(A) ⊂ X has a compact resolvent. We assume that X α = D(A α ) with α ≥ 0. Then u α = A α u , X 0 = X, and X 1 = D(A). For arbitrary Banach spaces Y 1 and Y 2 , we let BC ν (Y 1 , Y 2 ), ν ∈ N 0 , denote the class of C ν -smooth mappings Y 1 → Y 2 that are bounded on balls. We assume that a nonlinear function F belongs to BC 2 (X α , X) for some α ∈ [0, 1) and equation (2.1) is dissiparive, i.e., generates a resolving semiflow {Φ t } t≥0 in the phase space X α and there exists a retracting ball B a = {u ∈ X α : u α < a} such that Φ t B r ⊂ B a for any ball B r : u α < r for t > t * (r). In this case, the semiflow {Φ t } inherits [5] the C 2 -smoothness, and there exists the compact attractor A ⊂ B a consisting of all bounded complete trajectories {u(t)} t∈R ⊂ X α and uniformly attracting balls X α as t → +∞. In fact, A ⊂ X 1 due to the smoothing action of the parabolic equation [5, The embeddings X σ ⊂ X α with α < σ < 1 are dense and compact, and u α ≤ c u σ , c = c(α, σ), for u ∈ X σ . Moreover, the proof of Theorem 3.3.6 in [5] can be used to derive the estimate Φ 1 u σ ≤ L(r) u α on the balls B r ⊂ X α . This implies that F ∈ BC ν (X σ , X) if F ∈ BC ν (X α , X) and the X α -dissipativity implies the X σ -dissipativity.
Thus, in all constructions related to SPE (2.1), one can replace the nonlinearity index α with any value σ ∈ (α, 1). The linear operator A : X ϑ+1 → X ϑ is positive definite in X ϑ with ϑ > 0. If F ∈ BC 2 (X ϑ+α , X ϑ ), then one can consider (2.1) in the pair of spaces (X ϑ , X ϑ+α ) instead of (X, X α ). In this case, the phase dynamics preserves all its properties listed above.
We say that the phase dynamics of (2.1) is asymptotically finite-dimensional if there exists an inertial manifold, i.e., a smooth finite-dimensional invariant surface M ⊂ X α containing the attractor and exponentially attracting (with asymptotic phase) all solutions u(t) at large times. Such a manifold is usually [13,16,19] a Lipschitz graph over the highest modes of the operator A. The restriction of SPE (2.1) to M is an ODE in R N , N = dim M which completely describes the final dynamics of the original evolution system.
A less rigorous approach to the problem of finite-dimensional limit dynamic of SPE was proposed in [14,15]. So the dynamics of (2.1) on the attractor is finite-dimensional there exists an invariant compact set K ⊂ R N such that the dynamical systems {Φ t } on A and {Θ t } on K are Lipschitz conjugate for t ≥ 0. The properties of the dynamics to be asymptotically finite-dimensional and to be finite-dimensional on the attractor have not yet been separated; there is a hypothesis [19] that they are equivalent.
Here are two criteria for the dynamics to be finite-dimensional on the attractor [14] under the assumption that F ∈ BC 2 (X α , X).
(Fl) The phase semiflow on A can be extended to the Lipschitz flow: where M > 0 and κ ≥ 0 depend only on A.
(GrF) The attractor is a Lipschitz graph over the highest Fourier modes: for some finite-dimensional spectral projection P ∈ L(X α ) of the operator A and all Property (GrF) was established for scalar equation (1.1) in [9] independently of the results obtained in [14,15]. We shall further use other sufficient conditions for the dynamics to be finite-dimensional on the attractor, which were obtained in [15] The smoothness of the semiflow {Φ t } and the invariancy of the compact set A ⊂ X α imply the regularity of the identical embedding N → X α × X α and hence the regularity of any field Π : N → Y that can be continued to a C 1 -mapping into the (X α × X α )neighborhood of the set N . In this situation, which is also multiplicative if Y is a Banach algebra. In the last case, if the elements of We start from the decomposition of the vector field G(u) on A, where T 0 ∈ L(X α ) and T ∈ L(X 1 , X) are unbounded linear sectorial operators in X similar to normal ones. We write for a > ξ > 0 and assume that, for some c > 0, θ ∈ [0, 1], the total spectrum is localized in the domain The statement of this theorem is slightly changed as compared to [15].

Parabolic systems
Now we consider the system of equations (1.
and periodic boundary conditions We assume that the matrix function f = f (x, u) and the vector function g = g(x, u) belong to the smoothness class C ∞ on J ×R m and write system (1.1) in the abstract form (2.1) with X = L 2 (J, R m ), positive definite operator Au = u−Du xx , and nonlinearity F : which is fixed below.
We shall generalize the conclusions of [15, pp. 991-992] about the smoothness of the nonlinear function F and the phase dynamics of (1.1) to the case m > 1. We let the symbol ֒→ denote linear continuous embeddings of function spaces and shall use necessary results obtained in [5,17]. For an arbitrary C ∞ -function z : In the case of finite functions f = f (u), g = g(u), the dissipativity of system (1.1) with phase space X 1/2 , and hence also with X α , 3/4 < α < 1, was proved in [8, Theo- that are finite in u and can also be generalized in other directions. Anyway, we further assume that system (1.1) is dissipative in X α and there exists the global attractor A ⊂ X α . Using the above-listed properties of nonlinearity F and following the reasoning in [15, p. 992], we formulate the following remark.
Our goal is to apply Theorem 2.2 to system (1.1) and to prove that the final dynamics is finite-dimensional. Let be the vector field of system (1.1), and let N = A × A ⊂ X α × X α . The main idea, as in [15], is related to the change of variable in the linear differential expression with respect to x ∈ J for the difference G(u) − G(v) for a fixed (u, v) ∈ N , which allows one to eliminate the dependence on ∂ x h, h(x) = u(x) − v(x). Along with the convection matrix f = {f ij } we consider the m × m function matrices We put where E is the unit m × m matrix, and Proceeding as in [6], we apply the transformation h = Uη to the differential expression Similar problems are considered in the monograph [2]. We often write B 0 , B, and U omitting the dependence on u and v and sometimes on x. Taking into account the fact that where the numerical parameter ω > 0 will be chosen later.
Everywhere below, I .
is a solution of the Cauchy problem Now we prove regularity in the sense of Definition 2.1 of some vector fields on the compact set N ⊂ X α × X α . If Y ֒→ Y 1 for the function spaces Y and Y 1 , then the regularity of the field Π : N → Y implies the regularity of Π : N → Y 1 .
Lemma 3.3. The field of operators T 0 on N is bounded ranging in L(X α ) and regularly ranging in L(X α , X).
It remains to verify that the fields Π : N → L 2 (J, M m ) are continuous and to use is continuous in the X α -metric; the same holds for the mappings u → u xx and u → u x of the set co A ⊂ X α into X for u ∈ co A ⊂ X 1 . In the relation (f (x, u)) Proof. We consider (3.3) for arbitrary u, v ∈ X α as the non-autonomous evolution in the Banach algebra L(X) with identically zero sectorial linear part and the parameter is Lipschitz in x ∈ J, linear in U ∈ L(X) and of class C 1 with respect to the parameter (u, v). Under these conditions, by [5,Theorem 3.4.4], the mapping (u, v) → U(x; (u, v)), X α × X α → L(X) is continuously differentiable, and hence the operator field U : N → L(X) is regular. Now we formulate an important condition on the diffusion matrix D and the convection matrix f p of system (1.1).
For the scalar diffusion matrix D = dE, this assumption is satisfied automatically. In the case of m distinct diffusion coefficients d j , Assumption 3.5 holds under the condition that the matrix f is diagonal on co A, and in the case of s distinct diffusion coefficients, 1 < s < m, it holds under the condition that the matrix f on co A inherits the block structure (with respect to the same d j ) of the matrix D = diag{d j }.

Main results
The conditions for the dynamics to be finite-dimensional on the attractor will depend on the structure of the diffusion matrix D and the nonlinear function f in (1.1). By and we have (1), So the periodic boundary conditions h(1) = h(0), h x (1) = h x (0) become where V (1) = E in general. Further, we use the notation The following assertion plays the key role. Proof. By the condition of the lemma, we have V 1 = C −1 VC for positive definite For fixed u, v ∈ A, we let ϕ j ∈ R m and µ j > 0 denote orthonormal eigenvectors and eigenvalues of the operator V with j ∈ 1, m. We assume that H 0 = H 0 (u, v) = ωI − D∂ xx with boundary conditions (4.2) for some ω > 0. We also assume that χ k,j (x) = e 2πkix · ϕ j , x ∈ J, k ∈ Z, j ∈ 1, m.
As we see, ψ k,j are eigenfunctions of the operator H 0 with eigenvalues The operators V 1 (u, v) continuously depend on (u, v) ∈ N , and hence this also holds for their spectrum. By the compactness of N ⊂ X α × X α , we have 0 < c 1 ≤ µ j ≤ c 2 , j ∈ 1, m, for some c 1 (A), c 2 (A). Thus, the values | ln µ j | are uniformly bounded in j ∈ 1, m and u, v ∈ A. We put for x ∈ J and H = S 0 H 0 S −1 0 . Then S 0 ψ k,j = χ k,j and Hχ k,j = S 0 H 0 ψ k,j = λ k,j S 0 ψ k,j = λ k,j χ k,j .
Since the system of functions {χ k,j } is complete and orthonormal in X = L 2 (J, R m ), it follows that the operators H = H(u, v) are normal in X for u, v ∈ A. Let S = . We use Lemma 3.6 to write decomposition (2.4) of the vector field (3.1) on the attractor A with and operators T 0 (u, v) of the form (3.4.1). We see that S −1 = U(x)S −1 0 . By Lemma 3.4, the operator field U on N ranging in the Banach algebra L(X) is regular, and hence the field of inverse operators V : N → L(X) is regular. Since Acting as in the proof of Lemma 3.4, we can prove the regularity of the operator field Let Σ H = Σ T be the total spectrum of the field of operators H(u, v) on N . Using    Indeed, in this case, (4.4) holds with C = E.
We shall give two more arguments ensuring that the final dynamics is finite-dimensional.
Theorem 4.5. Assume that system (1.1) is dissipative in X α with α ∈ (3/4, 1) and Assumption 3.5 holds. Then the phase dynamics is finite-dimensional on the attractor if the following two conditions are satisfied : (i) the numerical matrices f (x, u(x)) have m distinct real eigenvalues for each (x, u) ∈ J × co A; (ii) the matrices f (x, u) commute with each other for any (x, u) ∈ J × co A.
Proof. Condition (ii) and Assumption 3.5 imply that the matrices D −1 f (x, u(x)) commute with each other on J × co A. It is known [11,Theorem 8.6.1] that two simple  Proof. The conditions of the theorem guarantee that the numerical matrices ) commute with each other on J for each u ∈ co A. As in the proof of In contrast to Theorem 4.5, we here admit the multiplicity of eigenvalues of the numerical matrices f (x, u(x)), but we assume that these matrices are symmetric.

Some applications
Here we consider several examples illustrating the above-described theory in terms of properties of the convection matrix f . We assume that system (1.1) is dissipative in the phase space X α with α ∈ (3/4, 1). We assume that all the conditions assumed below on f = f (x, u) also hold for x ∈ J and u = u(x), u ∈ co A. (i) the matrix Q has m distinct real eigenvalues and f 1 (x, u(x)) = 0 for x ∈ J and u ∈ co A; (ii) the matrix Q is symmetric.
Proof. The numerical matrices f = f 1 (x, u(x))Q are commutative. In the case of (i), each of these matrices has distinct real eigenvalues λ j f 1 (x, u(x)), where λ 1 , . . . , λ m are eigenvalues of Q, and Theorem 4.5 can be applied. In the case of (ii), the fact that the dynamics is finite-dimensional on the attractor is a direct consequence of Theorem 4.6. In the case of scalar diffusion matrix D = dE, the conditions of Proposition 5.5 can be reduced to the relation Q t (x) = Q(1 − x) for x ∈ J.

Other possible approaches
The above presentation is based on Theorem 2.2, which means the verification of regularity (in the sense of Definition 2.1) of operator vector fields on the attractor.  where H = H(u 1 , u 2 ) ∈ L(X 1 , X) are normal (or uniformly with respect to (u 1 , u 2 ) similar to normal) sectorial operators and R = R(· ; u 1 , u 2 ) : R → L(X) is a continuous operator function. Assume that the norms of the operators S, S −1 , and R are uniformly bounded in the parameter (u 1 , u 2 ). In this case, if the spectrum Σ H combined over u 1 , u 2 ∈ A is "sufficiently rare", then using the technique give in [19], one can verify that the phase flow is Lipschitzian on the attractor and then apply criterion (Fl). In contrast to the preceding presentation, we here have to deal with second-order linear differential expression in t ∈ R and not in x ∈ J. The assertions of Section 4 can be obtained in this way after the change v = V −x 1 V (x)h, where V 1 = V (1) is the monodromy operator of problem (4.1).
Another possible approach to the problem of finite-dimension of the final dynamics is related to the verification of criterion (GrF). In [10], a scalar parabolic equation of the form (1.1) is considered in a rectangle with Dirichlet boundary condition. The authors present conditions under which the attractor is a Lipschitz graph over finitely many first modes of the Laplace operator. And they use the cone condition well-known in the literature [13,16,19]. In this connection, it seems to be very perspective to study the problem of finite-dimensional reduction of systems of equations of the form (1.1) on the two-dimensional torus T 2 .