APPROXIMATE INERTIAL MANIFOLDS FOR NONLINEAR PARABOLIC EQUATIONS VIA STEADY-STATE DETERMINING MAPPING YUNCHENG

ABSTRACT. For nonlinear parabolic evolution equations, it is proved that, under the assumptions oflocal Lipschitz continuity of nonlinearity and the dissipativity of semiflows, there exist approximate inertial manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the graph of the steady-state determining mapping based on the spectral decomposition. It is also shown that the thickness of the exponentially attracting neighborhood of the AIM converges to zero at a fractional power rate as the dimension of the AIM increases. Applications of the obtained results to Burgers’ equation, higher dimensional reaction-diffusion equations, 2D Ginzburg-Landau equations, and axially symmetric Kuramoto-Sivashinsky equations in annular domains are included.


INTRODUCTION
In the recent decade there have been rapidly expanding research progresses in the global dynamics (that is, the long time behavior of solutions) of nonlinear evolution equations with some sort of dissipativity.The existing theory consists of the existence of absorbing sets, the existence of a compact global attractor of finite Hausdorff and fractal dimensions for many typical parabolic and hyperbolic equations, the existence of inertial manifolds for some parabolic equations and nonlocal beam equations (mostly known only for the one space dimension cases), and the existence of approximate inertial manifolds by a variety of approaches.We refer to Hale and Temam [2] as general references.
Specifically, the methodology in proving that the graph of the steady-state determining wnapping forms AIM is based on the weak mot,otonicity (3.10) (which is actually a consequence of the local Lipschitz continuity combined with the dissipa|ivity, rather than a new condition) and on the localizing and pasting technicality in Section 4 and Section 5.This technicality consists of the intervention of an x- stationary (or semi-stationary) intermediate equation (4.2), the local estimates (5.17) and (5.19) in breaking terms, and finally the subinterval optimization.Hence the methodology is independent of the concrete or unique properties possessed only by Navier-Stokes equation or by some other equations.
The main results in Theorem 5.4 also include the decreasing rote expressions (5.15): J() l/la 3/2 in H and (5.16): N(la) l/la in V of the exponentially attracting neighborhoods of the AIMs.The theory developed in this work is extensively applicable to various dissipative nonlinear parabolic equations, including the 2D Navier-Stokes equations.For different exemplification, in this paper, the applications of this general theory to Burgers' equation, higher dimensional reaction-diffusion equations, 2D Ginzburg- Landau equations, and axially symmetric Kummoto-Sivashinsky equations are presented in detail.It is also worth mentioning that the obtained results can be applied to the class of evolution equations studied independently by Smiley 13].
The rest of the paper is organized as follows.In Section 2 the assumptions, the global existence and regularity of solutions, and the existence of absorbing sets are presented.Section 3 deals with the truncation, the decomposition, the sleady-state equation, and the existence as well as the Lipschtz continuity of the determining mappings.In Section 4 three technical lemmas involving the semi-stationary intermediate equations are proved.The main theorems on the existence and properties of AIM are provided in Section 5. Finally in Section 6, applications of the obtained results to four different types of nonlinear parabolic equations are illustrated.

SOLUTION SEMIGROUP AND ABSORBING SETS
Let H be a real separable Hilbert space and denote its norm and inner-product by I1 and (', ') respectively.We make following assumptions on the linear operator A and the nonlinear part R(u) and f in the abstract evolution equation (1.1).
ASSUMPTION (AI).Assume that A: D(A)(c H)--, H is a closed linear operator, and it is positive definite and self-adjoint, and has compact resolvent..By this assumption the spectrum set o(A) of A consists of countably many positive eigenvalues .i: 1, 2 }, each repeated up to its finite multiplicity and lim ,i -'+ as i---, ,x,.The corresponding complete and normalized eigenvectors of A are denoted by ei: 1,2 which form an orthonormal basis for the space H.For any number s " 0, the fractional power operator As: D(As) H is well defined by the spectral mapping.In particular, we define another real Hilbert space V by V-D(AI/2) with the norm v IAl/:vl and the induced inner-product ((., .)).Denote by .o the minimum eigenvalue of the operator A, ho > 0. Note that due to Assumption (AI), A generates an analytic contraction semigroup denoted by {T(t), :," 0 }.ASSUMPTION (A2).R(.) is a nonlinear mapping from V to H and also from D(A) to V, satisfing the following conditions, R(u) (Co( where : > 0, ' e R, and 0 < I < o are constants, Ci(...), 0, 1, 2, 3, are nonnegative continuous functions increasing with their respective variables.Besides assume that fe V is a time-invadant function.
REMARK 2.1.The above assumption (A2) on the nonlinearity R(u) is only a typical set o! conditions under which one can work out the existence of global solutions over [0,oo) and the existence o| absorbing sets.There are certainly alternative sets of conditions which can replace (A2) in different settings for the same purpose: obtaining the global dissipativity in terms of the absorbing property.After the stage of showing the absorbing property, (A2) plays no more role except the first and third inequality conditions of in (A2).
For any UoE H, the concepts of mild solution, strong solution, and classical solution of the evolution equation I. l) are provided in Pazy 14].We refer to Temam 2 for the concept of absorbing set and absorbing property.Following is the main result in this preliminary section.THEOREM 2.2.For any Uo E V, there exists a unique classical solution u(.) of the evolution equation I. such that u E C([0,oo); V) N CI((0,oo); H) t C((0,oo); D(A)).
(2.1)Moreover, there exists an absorbing set in V for the semiflow generated by (1.I).
PROOF.We shall use the two basic inequalities, cf.Pazy [14], related to the regularity of an analytic semigroup {T(t), > 0 }" AaT(t) IIH._, H s Ma -a e-tot, for all > 0, T(t)x x Bat all Aax II, for all xe D(Aa) and > 0, ( (2.3) where 0s at s 1, Ms and Ba are constants depending on at, and to satisfies 0 < to < )o.Below we sketch the proof of this theorem by several steps.STEP 1.The local existence and uniqueness of a mild solution on some interval [0, r].This can be done by considering the integral equation v(t) A1/2T(t)Uo + A1/2T(t s)[f-R(A-1/2v(s))]ds, for t[0,r], (2.4) and using the contraction mapping and fixed point argument to choose small "r > 0 such that there exists a unique strongly continuous solution v(.) of the equation (2.4), so that u(t) A-l/2v(t), tz [0,r], is the mild solution of (1.1) and u : C([0,r]; V).

STEADY-STATE DETERMINING MAPPINGS
In this section we make some preprations to construct approximate inertial manifolds: truncation, space decomposition, and define the steady-state determining mappings.
Let 0p(r) 0(2 with p being the fixed constant mentioned above.Define a new nonlinear mapping by F(u) Op(ll u 2) R(u), for u V.In the sequal we shall consider the modified equation: By the result of STEP 4 in the proof of Theorem 2.1, specifically (2.23), the ball B(0;13) centered at the origin and of radius 13 is an absorbing set for the semiflow in V generated by the equation (1.1).Therefore this truncation that replacing (1.I) by L.2) preserves the long-time dynamics of the original equation (1.1), since within the absorbing ball B(0;13) both equations are the same.But the new truncated nonlinearity has following uniform properties in which Cl > 0 and c2 > 0 are two uniform constants: IF(u)l c Ilu II, for any u EV, for any u and v in V, IIF(u) F(v)ll c2 IAumvl, for any u and v in D(A).
DECOMPOSITION.For any given positive number la > .o,there exist a nontrivial spectral decomposition of o(A) and a corresponding orthogonal decomposition of the Hilbert spaces H and V: H X + Y, (denoted briefly by X + Y if is relatively fixed), V X + Zt, (denoted briefly by X + Z if la is relatively fixed), where + ola(A) {o(A): 7, la} o (A) {Xo(A): k < ta} XI Span{ei: < n()}, Y CIH Span {ei:i > n(la)}, and Zla CIv Span {ei: > n(/a)}, in which n(t) max{i (integer): .i< la }.Let P: H X be the orthogonal projection and QH IH P. Then X PH, Y QH, and Z QV.Apply the projections P and Q to the equation (3.2), we get two coupled equations: which is called the steady state equation.It can also be decomposed as Ax + PF(x + z) g, (3.7)x Az + QF(x + z) h, (3.7)z with x e Xt and z Z. Suppose that for any given h 6 7_ and any x e Xt, the component equation (3.7)z has a unique solution z, denoted by z p(x,h), (3.8)  then the solution of the steady state equation (3.6) reduces to the solution of the following equation, Ax + PF(x + x,h)) g, in (3.9) which is called, according to Chow and Hale 15 ], a bifurcation equation.The mapping : Xv Z Z defined by (3.8), if exists, is called a steady-state determining mapping.
In order to study the existence and properties of such determining mappings that will be a tool in contructing AIMs, we now emphasize an important consequent property derived from the uniform Lipschiotz continuity of (3.3) as follows.
PROOF.By (3.3) the truncated nonlinearity F(u) possesses the uniform Lipschitz continuity IF(u) F(v)l c2 Ilu vii, for any u and v in V. Thus we have This indicates that (3.10) holds.THEOREM 3.2.For any la > (c2) 2 and the decomposition (3.4) and (3.7) associated with this there exists a Lipschitz continuous mapping q: X Z Z, such that z--x, h) (3.11)   is a unique solution of the equation (3.7)Z.Moreover, this mapping q has the following properties, qxl, hi)-qx2, h2) la-c21a 1/2 [c2 x-x2ll + la-l/21l hi-h2lll, (3.12) p(Xl, hi)-p(x2, h2)II   where Az AIZ (the restriction of A on Z).G is a densely defined operator in QV.We want to show that G is surjective, i.e.Ran (G) QV.For any heQV, the equation Gw h is solvable if and only if the following equation is solvable: w (Az)-l[h-QF(x + w)l for w QV.
Define a mapping J QV QV by Since we have J(w) (Az)-llh QF(x + w)l.J(wl) J(w2)ll (Az)'I IIL(H;V) QF(x + Wl)-QF(x + w2) where II(Az)-IIL(H.V) s In-1/2 can be directly verified by the eigen-expansion.Since p I/2> c2, (3.17) shows that is a contraction mapping so that it has a unique fixed point w for any given pair (x, h) E X Z. Therefore (3.15) is solvable and the equation (3.7) Z has a unique solution z, denoted by (3. ).
We call the mapping q0 obtained in this theorem as the determining mapping associated with p.

TECHNICAL LEMMAS
In this section three lemmas will be established as intermediate steps toward the main result.First we sketch the guiding procedure leadin,-to the construction of AIMs as follows.In fact we look for an AIM in the form of Ply Graph ,0(., Q0 {x + g(x, Q0: for all x: Xla], (4.1)   where p > 0 is to be chosen.Since an AIM needs not to be invariant, for the Bnite dimensional and Lipschitz continuous manifold PIv defined by (4.1), the only thing to be done is the exponential attraction of any orbits to a "thin" neighborhood of this manifold PI..In order to prove this attraction property, naturally we shall try to estimate the distance between the Z-component z(t) of the original solution u(t) and the image p(x(t), Q0 ofthe X-component x(t) ofu(t), since x(t) + p(x(t), Q0 is on the manifold PIv.
Let > 0 be arbitrarily given.Denote by x x(t) and z z(to).We shall estimate the difference between the solutions of following two equations with the same initial data: dt + Az + OF(x(t) + z) h, z(to) z , > , (4.9)

.10)
where A A Z. Note that (4.9) is exactly the second component equation (3.7)z with an initial value condition at o. Also note that the truncated nonlinear mapping F has the uniform bound and the uniform Lipschitz continuity, so that (referring to the proof of Theorem 2.1) the classical solution w(.) of the equation (4.10) exists uniquely for e It , co) and has the same regularity as the solution z(.) of the equation (4.9) has.
We need an auxiliary lemma as follows.LEMMA 4.3.After any solution u(t) of the original equation (I.I) with u(0) Uo enters the absorbing ball B(0;p) c V for ever, the solution is uniformely Holder continuous with the exponent I/2, i.e. there is a constant q > 0, such that for any solution u(t) of l.I), Ilu(t) u(s)ll q Itsl 1/2 for t, s e [to(Uo), co), (4.11)where to(Uo) is the final entering time of the trajectory u(t) started from Uo into the absorbing ball.

APPROXIMATE INERTIAL MANIFOLDS
In this section we shall prove te existence of approximate inertial manifolds for the original equation (1.1) by the approach of construction based on the determining mapping o.First of all, we define an approximate inertial manifold as follows.
DEFINITION 5.1.A subset Yl. c V is called an approximate inertial manifold (AIM) for the semiflow generated by the evolution equation (1.1) in V, if Y'l, satisfies the following conditions a) Y'I, is a finite-dimensional and Lipschitz continuous manifold; b) There are uniform constants at > 0, K > 0, and q > 0, such that for any given bounded set in V, there is a constant to to(), such that for any initial state Uo E , the solution satisfies dist V (u(t), '1,) ": 0t + K exp( -rl0-to)), for [to, co).c) For any trajectory u(t) of the equation (1.1), there is a tracking orbit v(t) (which may not be a trajectory) on the manifold Y'I,, such that after a transient period I0,tol, the error satisfies the same exponential bound as above, i.e.Ilu(t)-v(t)ll < a + K exp( -q(t-Io)), for e [to, oo).DEFINITION 5.2.A sequence of subsets {1'1, k k'-in V is called a regular chain of approximate inertial manifolds if each PI, k is an AIM la the above sense, dim Y'l, k is nondecreasin8, and the thickness a(Pl, k) decreases and converges to zero as k co.
The argument in proving the main theorem is the localizing andpasting technicality as follows.Let A > 0 be arbitrarily given.Denote by tn to + nA, Xn X(tn) Pu(tn), Zn Z(tn) Qu(tt0, h Qf, Y YOU dn z(t") x(t"),Q0 z-n-qxn, Qf)II.
According to Definition 5.1, this set "[.t given by (5. 1) in V is an AIM for the semiflow generated by the evolution equation 1. ).We emphasize specifically the obtained attraction property: By (5.17) with (5.18) and (5.19) with (5.20), any solution u(t) of (1.1), after its entering into the absorbing ball, will be tracked in the sense of(5.13)and(5.14) by an orbit v(t) given below on the AIM , v(t) :: Pu(t) + e0(Pu(t),Qf).
Therefore the condition b) and condition c) in Definition 5.1 are both verified.The proof of Theorem 5.4 is completed.
COROLLARY 5.5.There is a regular chain of approximate inertial manifolds Mk:k [2(c: for the semiflow generated by the evolution equation (1.1).PROOF.This is an immediate consequence of Definition 5.2 and the thickness formulas (5.18) and (5.20) for intergers larger than 2(c2)2.Therefore the assertion is true.The obtained results can be generalized as follows.We replace the Assumption (A2) by the following Assumption (A3), and generalize the property (3.10) to the following Assumption (A4).ASSUMPTION (A3).(dissipation condition) Assume that for any uoV, there exists a unique global strong solution u(t) of the equation (1.1) fort 0 with u(0) Uo.The semiflow in V generated by this equation has the absorbing property.Moreover, after a truncation (3.3) holds.
(5.24) la -c3 2" n 6. APPLICATIONS In this section we present several application examples for which Theorem 5.4 or Theorem 5.7 can be used to assert the existence ofa regular chain of approximate inertial manifolds.
REACTION-DIFFUSION EQUATIONS OF HIGH DIMENSION.Consider a reaction-diffusion equation du + Au + R(u) f, 0, (6.7) dt where A vAu" D(A) (= H2(f)lqHlo([/) )--, H (= L2(I)) ), and t) is an open, bounded and connected of R n with a Lipschitz continuous boundary F, f ZHo(I)) is fixed, and R(s) is a polynomial of odd set degree with a positive leading coefficient, i.e.
2p-1 R(s) bk sk, p > (integer), b2p-I > 0. (6.8) In Temam [2], it was proved that there exists a global attractor with finite fmctal dimension for this equation (6.7) and that for space dimension n and 2 there exist inertial manifolds.In Mallet-Parer and Sell 17 the spatial averaging principle was applied to prove the existence of inertial manifolds for this type of equations with n 3 if (0.2n)3.But for the higher dimensional case (n :" 3) with a general domain , the existence of inertial manifolds is unknown.Here we can apply our result of Theorem 5.4 to the higher dimensional reaction-diffusion equations with a general domain by checking the conditions in (A2).
n Denote by V nlo() with the norm v Y Idv/dxil 2 )1/2, where stands for the H-norm.
i--l Note that the absorbing property in V of (6.7) with (6.8) under the above conditions has been shown in Temam [2], we need only to verify the first conditions in 2), and the condition 3) of(A2).
We have, for any u and veV, Next we check the condition 3) of (A2).Note that by Young's inequality there is a uniform constant D > 0 such that 2p-2 E bk s k+l .b2p-IIsl 2p+ DI, foranys R.
Thus we can conclude that for the equation (6.7) with n 3 or n 4, and 2p- 3 (polynomial (6.8) of degree 3 with b3 > 0), (6.12) there exist the AIMs in the form of (5.11 and have the properties (5.12) through (5.14).
2D GINZBURG-LANDAU EQUATIONS.The complex Ginzburg-Landau equation (with the Dirichlet boundary condition) in the following form: 0u --(1 +iv)Au+(:+iq)lul2u-ru=0, for(t, x)R +, f, (6.13) UlF=0, fort>0, where v, rl, and : are real numbers with K > 0, and l is an open, bounded, and connected set of Rn (n 1), with a boundary F being Lipschitz continuous, can serve as mathematical models for the behavior of superconductors in a magnetic field and near the critical temperature, cf.Gorkov 18], nonlinear instable waves in plane Poiseuille flows, cf.Stewartson and Stuart 19 ], and other applications, cf.Temam [2].Note that the existence of global atmctor for n < 2 and the existence of inertial manifolds only for n have been proved in Doering et al. [20].However, for the space dimension n 2 and the general domain fL the existence of inertial manifolds is an open issue.
Take the viewpoint to visualize a complex function as a vector of two real components, namely u col (Ul, u2), set up H [L2(fl)] 2 with the norm [. [, V [Hol(f)] 2 with the norm II.ll, define an operator (6.14) It is easy to verify that A: D(A)-,.H satisfies (AI).Denote by R(u) B(u) u.Let n =2.Then one can verify that R(u) satisfies (A2).Since the absorbing property was proved, cf.Temam [2], below we only check the third inequality condition and the fifth inequality condition of (A2).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: PROOF.Def'me a mapping G: D(A) Cl QV--QV by G: w Azw + QF(x + w).