Gradient bounds for a thin film epitaxy equation

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1 Available online at ScienceDirect J. Differential Equations 6 (7) Graient bouns for a thin film epitay equation Dong Li a, Zhonghua Qiao b,, Tao Tang c, a Department of Mathematics, University of British Columbia, 984 Mathematics Roa, Vancouver, BC, V6T Z, Canaa b Department of Applie Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong c Department of Mathematics, South University of Science an Technology, Shenzhen, Guangong 5855, China Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong Receive June 6 Available online 8 October 6 Abstract We consier a graient flow moeling the epitaial growth of thin films with slope selection. The surface height profile satisfies a nonlinear iffusion equation with biharmonic issipation. We establish optimal local an global wellposeness for initial ata with critical regularity. To unerstan the mechanism of slope selection an the epenence on the issipation coefficient, we ehibit several lower an upper bouns for the graient of the solution in physical imensions 3. 6 Elsevier Inc. All rights reserve. Keywors: Epitay; Thin film; Maimum principle; Graient boun. Introuction Let ν >. Consier t h = (( h ) h) ν h (.) an the D version * Corresponing author. aresses: li@math.ubc.ca (D. Li), zhonghua.qiao@polyu.eu.hk (Z. Qiao), tangt@sustc.eu.cn (T. Tang). 6 Elsevier Inc. All rights reserve.

2 D. Li et al. / J. Differential Equations 6 (7) h t = (h 3 h ) νh. (.) Eq. (.) is a nonlinear iffusion equation which moels the epitaial growth of thin films. It is pose on the spatial omain which can either be the whole space R, the L-perioic torus (L > is a parameter corresponing to the size of the system) R /LZ, or a finite omain in R with suitable bounary conitions. In this work for simplicity we shall be mainly concerne with the π-perioic case = T = R /πz but our results can be easily generalize to other settings. The function h = h(t, ) : R R represents the scale height of a thin film an ν>is positive parameter which is sometimes calle the iffusion coefficient. Typically in numerical simulations one is intereste in the regime where ν is small so that the nonlinear effects become ominant. The D version (.) is connecte with the Cahn Hilliar equation: t u = (u 3 u) ν u through the ientification u = h. This connection breaks own for imension. Define the energy E(h) = ( 4 ( h ) + ν h ). (.3) The equation (.) can be regare as a graient flow of the energy functional E(h) in L ( ). In fact, it is easy to check that t E(h) = th, (.4) i.e. the energy is always ecreasing in time as far as smooth solutions are concerne. Alternatively one can erive the energy law from (.) by multiplying both sies by t h an integrating by parts. The first term in (.3) moels the Ehrlich Schowoebel effect [3,,3]. Formally speaking it forces the slope of the thin film h. For this reason Eq. (.) is often calle the growth equation with slope selection. On the other han, in the literature there are also moels without slope selection, such as t h = ( + h h) ν h. (.5) Heuristically speaking, if in (.5) the slope h is small, then + h h an one recovers the nonlinearity in (.). However this line of argument seems only reasonable when h which is a typical transient regime an not so appealing physically. Inee the long time interfacial ynamics governe by (.) an (.5) can be quite ifferent, see for eample the iscussion in [5]. The secon term in (.3) correspons to the fourth-orer iffusion in (.). It has a stabilizing effect both theoretically an numerically.

3 7 D. Li et al. / J. Differential Equations 6 (7) Eq. (.) can also be viewe as regularize version of the equation t h = (( h ) h). (.6) The wellposeness of (.6) is a rather subtle issue. In light of recent evelopments [,], one shoul epect generic illposeness although the unerlying mechanism will be ifferent. However as it turns out, if there is a smooth solution to (.6) on some finite time interval, then it must amit some form of a maimum principle. We recor it here as Proposition. (Maimum principle for smooth solutions to (.6)). Let the imension an T = R /πz be the usual π-perioic torus. Let T > an assume h C t C ([, T ] T ) is a classical solution to (.6). Then h(t, ) ma{ h(, ), }, t T. (.7) If the imension =, then a better boun is available: h(t, ) ma{ h(, ), 3 }, t T. (.8) We stress that Proposition. is a conitional result, namely it assumes the eistence of a smooth solution. On the other han the wellposeness of classical solutions to the regularize equation (.) is much easier to obtain thanks to the fourth orer issipation on the right han sie. In the Fourier space, the biharmonic operator seems to offer much stronger issipation an amping effect than the usual Laplacian operator, as can be seen from stuying the linear equations t h = Ah, A = or. Since equation (.) can be viewe as a regularize version of (.6), it is very natural to stipulate that solutions to (.) shoul behave much better than those to (.6) from a general perspective. From this heuristics, it is very tempting to epect that Proposition. also hols for (.). Preliminary numerical eperiments seem to support this, thus Conjecture. Let ν >. For general smooth initial ata h, the corresponing solution h = h(t, ) to (.) satisfies the boun h(t) ma{ h, }, t>. A weaker form of Conjecture is the following: Conjecture. Let ν >. For general smooth initial ata h, the corresponing solution h = h(t, ) to (.) satisfies the boun h(t) ma{ h,α }, t>, where α > is a constant epening only on the imension.

4 D. Li et al. / J. Differential Equations 6 (7) Perhaps a better formulation of Conjecture is that h(t) F( h, ) for some function F inepenent of (ν, ). The main point in both Conjecture an Conjecture is that the constants in the upper bouns of h are inepenent of ν. If true these graient bouns can lea to better stability estimates of numerical algorithms (see [5,,6,4,7 9]). On the other han, it is not so ifficult to etract a ν-epenent upper boun on h, see Corollary. below. Perhaps a bit surprisingly, the goal of this paper is to isprove Conjecture. Conjecture is still open at the time of this writing. However we shall give a lower boun for the constant in Conjecture. Namely, we shall show that α C > for some eplicit constant C epening on the imension. To make the paper self-containe, we first establish local an global wellposeness for (.). For H initial ata in imensions =,, 3, a fairly satisfactory wellposeness theory has been worke out in [5] using energy estimates an Galerkin approimation. By using the metho of mil solutions, our Theorem. below slightly refines this wellposeness result an allows initial ata to be in the critical space H which in particular contains H for 3. Note that although (.) is not scale-invariant, in high frequency approimation, one can regar (.) as t h = ( h h) ν h. (.9) To invoke scaling analysis, one can consier (.9) pose on the whole space R. If h(t, ) is a solution to (.9), then for any λ >, h λ (t, ) = h(λ 4 t,λ) is also a solution. From this one can euce that the critical space for (.9) is L (R ) or H (R ). Thus we have Theorem. (Improve local wellposeness). Let the imension. Consier (.) on the π-perioic torus T with ν >. Let s = /. For any initial ata h H s (T ), there eist T = T(h ) > an a unique local solution h Ct H s with t 4 h Ct C, t 4 h Ct H s +. Moreover h(t) H m for all m, <t<t, where <T is the maimal lifespan of the local solution. In particular h(t) C for all <t<t. If h has mean zero, then h(t) also has mean zero for all <t<t. As is well-known, the long time ynamics is ictate by conserve quantities (or conservation laws). For (.), the energy issipation law (.4) gives a priori H control of the solution with mean zero. Note that if h has mean zero, then h is controlle by h thanks to the Poincaré inequality. Or one can just prove it irectly using the Fourier series. The space H is subcritical in imensions 3 since the corresponing critical space is H. Thus Corollary. (Global wellposeness for 3). Let the imension =,, 3. Consier (.) on the π-perioic torus T with ν >. For any initial ata h H (T ) with mean zero, the corresponing solution h = h(t, ) to (.) obtaine in Theorem. eists globally in time. Remark.. An interesting open problem is to show the global wellposeness of (.) in imension = 4. In that case H is the critical space.

5 74 D. Li et al. / J. Differential Equations 6 (7) The following corollary gives graient bouns on h. For simplicity we assume the initial ata h H (T ) so that the energy is well-efine. By using the smoothing effect one can also treat the case h H (T ) with the help of Theorem.. However the bouns in that case have slightly worse epenence on ν (for initial transient time when the smoothing effect takes place). We shall not well on this subtle issue here an focus instea on the long time bouns. In Corollary. below, we shall only consier the case when the iffusion coefficient ν is not so large (the physically relevant case is ν ), which we enote by the notation <ν. It means <ν ν where ν > is some constant of orer. The numerical value of ν is not so important. For eample one can just take ν =. Corollary. (Graient bouns for 3). Let the imension =,, 3. Consier (.) on the π-perioic torus T with <ν. Assume h H (T ) with mean zero. Let h = h(t, ) be the corresponing global solution to (.). Denote E = T ( ν h + 4 ( h ) ). Then h amits the following bouns: for some absolute constants C, C, C 3 >, sup h(t) C ν 6 E 6 (E 6 + ), if = ; t< sup h(t) C ( E t< ν ) E + log( ), if = ; ν sup h(t) C 3 ν 3 (E + ) 3, if = 3. t< Similarly for some absolute constants C >, C 3 >, sup h(t) e νt h C (E t ν ) E + log( ), if = ; ν sup h(t) e νt h C 3 ν 3 (E + ) 3, if = 3. t Remark.. The above graient boun for = follows trivially from energy law an interpolation inequalities. It oes not use the ynamics at all. On the other han the proof of the bouns for =, 3 uses the mil formulation of the equation together with energy law. In terms of the epenence on ν the bouns here seem not optimal. See for eample Proposition in 5 for more refine results. To isprove Conjecture, we shall use two ifferent methos. The first metho (see Theorem. an Corollary.3 below) gives a weak lower boun approimately of the form + O(ν) (with O(ν) > ). Even though this alreay settles Conjecture in the negative, the obtaine lower boun approaches to as ν ten to zero which is the rawback of the construction. On the other han, the secon metho (see Theorem.3) gives a ν-inepenent lower boun which also

6 D. Li et al. / J. Differential Equations 6 (7) yiels a lower boun for the constant α in Conjecture. It is quite possible that these bouns can be improve further. We now introuce the first construction. To eluciate the main iea, we first state the D version. Theorem.. Consier (.) with ν > an π-perioic bounary conition. There eists a family A of smooth initial ata such that the following hols: () For any h A, we have T h () = an h <. () For any h A, there eists t > (epening on h ) such that the corresponing solution to (.) satisfies h(t, ) >. It is relatively straightforwar to generalize the construction in Theorem. to the equation (.) in all imensions. Corollary.3. Let the imension an T be the usual π-perioic torus. Consier (.) with ν > an on (t, ) [, ) T. There eists a family A of smooth initial ata such that the following hols: () For any h A, we have T h () = an h <. () For any h A, there eists t > (epening on h ) such that the corresponing solution to (.) satisfies h(t, ) >. We now introuce the secon construction. The key iea buils on eamining the linear evolution e νt, an treating the nonlinear part as a correction. Theorem.3. Let the imension an T be the usual π-perioic torus. Consier (.) with ν > an on (t, ) [, ) T. There eists a constant C > epening only on the imension, such that for any ɛ >, there eists h C (T ) for which the following hol: () T h () = an h <. () There eists t > such that the corresponing solution to (.) satisfies Remark.3. Let f() = (π) C = f L (R ) >. R h(t, ) >C ɛ. ξ e 4 e iξ ξ. The constant C in Theorem.3 is given by Remark.4. One can also consier the following version of (.) with fractional issipation: t h = (( h ) h) ν γ h, (.)

7 76 D. Li et al. / J. Differential Equations 6 (7) where γ > controls the orer of issipation. For h : T R, γ can be efine on the Fourier sie as γ h(k) = k γ ĥ(k), k Z. The L -maimum principle hols for the fractional heat propagator e t γ for γ. The behavior of e t γ for γ < an the heat operator e t can be quite ifferent, see for eample [6] for a iscussion in the (Littlewoo Paley) frequency-localize contet. In the wier setting one can even consier operators of the form A = γ / log β (λ + ) (for γ, β an λ > ) an establish a new generalize maimum principle (see [4]) for the rift equation t θ + v θ = Aθ, where v is a given arbitrary eternal velocity fiel transporting the scalar quantity θ. On the other han, in the regime γ >, the L -maimum principle is no longer epecte since the corresponing funamental solution may change signs. Base on this, an analogue of Theorem.3 is epecte to hol for (.) when γ >. In that case the constant C is replace by. Notation an preliminaries C,γ = F (e ξ γ ) L (R ) >. In this section we collect some notation an preliminaries use in this paper. For any = (,, ) R, we use the Japanese bracket notation = We enote by T = R /πz the π-perioic torus. Let = R or T,. For any function f : R, we use f L p = f L p ( ) or sometimes f p to enote the usual Lebesgue L p norm for p. If f = f(, y) : R, we shall enote by f p L L p to enote the mie-norm: y f p L L p y = f(,y) p L y ( ). p L ( ) In a similar way one can efine other mie-norms such as f C t H m etc. For any two quantities X an Y, we enote X Y if X CY for some constant C >. Similarly X Y if X CY for some C >. We enote X Y if X Y an Y X. The epenence of the constant C on other parameters or constants are usually clear from the contet an we will often suppress this epenence. We enote X Z,,Z m Y if X CY where the constant C epens on the parameters Z,, Z m. We aopt the following convention for Fourier transform pair on R : (Ff )(ξ) = f(ξ)= ˆ R f()e i ξ,

8 D. Li et al. / J. Differential Equations 6 (7) f()= (π) R ˆ f(ξ)e i ξ ξ. Sometimes the inverse Fourier transform is enote as F. Also for f : T R, an k Z, we enote the Fourier coefficient f(k)= ˆ f()e ik. Of course (uner suitable conitions) f can be recovere from the Fourier series: T f()= ˆ (π) f(k)e ik. k Z Note that if we regar f as a perioic function on R, then (Ff )(ξ) = k Z ˆ f(k)δ(ξ k), (.) where δ is the usual Dirac elta istribution on R. For f : T R an s, we efine the H s -norm an H s -norm of f as ( f H s = ( + k s ) f(k) ˆ ), k Z ( f H s = k s f(k) ˆ ), k Z provie of course the above sums are finite. If f has mean zero, then ˆ f() = an in this case ( f H s k s f(k) ˆ ). k Z Occasionally we will nee to use the Littlewoo Paley (LP) frequency projection operators. To fi the notation, let φ C c (R ) an satisfy φ, φ (ξ) = for ξ, φ (ξ) = for ξ. Let φ(ξ) := φ (ξ) φ (ξ) which is supporte in / ξ. For any f S (R ), j Z, efine j f(ξ)= φ( j ξ) ˆ f(ξ), Ŝ j f(ξ)= φ ( j ξ) ˆ f(ξ), ξ R. We recall the Bernstein estimates/inequalities: for p q, s j f L p (R ) js j f L p (R ), s R; S j f L q (R ) + j f L q (R ) j( p q ) f L p (R ).

9 78 D. Li et al. / J. Differential Equations 6 (7) We also nee the Bernstein inequalities for perioic functions. Let f : T R be a smooth function an lift f to be a perioic function on R. Then in this way f S (R ) an one can efine j f for any j Z. By epressing j f in terms of a convolution integral, it is easy to check that j f is also a perioic function on R an thus can be ientifie as a function on T. Amore irect way is just to use (.) an recognize j f as (on the Fourier sie) the partial sum of δ-istributions in a yaic block. It is then natural to epect that the following Bernstein -type inequalities hol (note that the norms are evaluate on T ): for any p q, s j f L p (T ) js j f L p (T ), s R; (.) j f L q (T ) j( p q ) f L p (T ), j Z; (.3) S j f L q (T ) j( p q ) f L p (T ), j. (.4) If f has mean zero (so that ˆ f() = ), then one oes not nee the conition j (since S j f = for j < ). Although these inequalities are stanar, we inclue the proof here for the sake of completeness. Proof of (.) (.4). We shall only prove (.) (.3). The proof of (.4) is similar to (.3). First we eal with (.). For some Schwartz function ψ (ψ = F ( ξ s φ(ξ))), we have ( s j f )() = js j ψ( j ( y))f (y)y R = js k Z = js T T j ψ( j ( y + πk))f (y)y ψ j ( y)f(y)y, where ψ j (z) = k Z j ψ( j (z + πk)) is a perioic function on R (an thus can be ientifie as a function on T ). By using Young s inequality on T, we get Easy to check that Therefore s j f L p (T ) js ψ j L (T ) f L p (T ). ψ j L (T ) j ψ( j z) L z (R ) = ψ L (R ). s j f L p (T ) js f L p (T ). By using a fattene projection j = l= j l (an noting that j f = j j f ), one can then erive (.).

10 D. Li et al. / J. Differential Equations 6 (7) Net we erive (.3). By Young s inequality, we have j f L q (T ) ψ j L r (T ) f L p (T ), where r = + q p. By (.) an ˆ f() =, easy to check that j f = if j <. Therefore we may assume without loss of generality that j. Then by using the fact that ψ is Schwartz, we get k Z j ψ( j (z + πk)) L r z (T ) k j ψ( j (z + πk)) L r z (T ) + j ψ( j z) L r z (R ) + j j r. k > j j k Thus (.3) is prove. 3. Proof of Proposition. For t T, consier f(t, ) = h(t, ). Note that Clearly t h = ( h) f + (f ) h + Therefore tf = h t h t h = (f ) h + f h. j h j f + j= = h( h f)+ (f )( h) ( h) + + j h( h j f) j= j h j f. j= ( h j h) j f j= = h( h f)+ (f )( h) ( h) + f + By efinition, it is easy to check that Therefore h h = f f = h h + ( k j h). k,j= ( k j h). k,j= j h k h jk f. (3.5) j,k=

11 73 D. Li et al. / J. Differential Equations 6 (7) Plugging this epression into (3.5), we then obtain tf = (f ) f (f ) + f + k,j= j h k h j k f. k,j= ( k j h) + h( h f) Now let ɛ >be a small parameter which will ten to zero later. Consier the auiliary function Note the equation for f ɛ reas as f ɛ (t, ) = f(t,) ɛt, t T, T. tf ɛ = ɛ + (f ɛ + ɛt ) f ɛ (f ɛ + ɛt ) + h( h f ɛ ) + f ɛ + ( k j h) k,j= j h k h j k f ɛ. (3.6) k,j= Since f ɛ is a continuous function on the compact omain [, T ] T, it must achieve its maimum at some point (t, ), i.e. ma f ɛ (t, ) = f ɛ (t, ) =: M ɛ. t T, T We iscuss several cases. Case. <t T an M ɛ >. In this case observe that f ɛ (t, ) =, f ɛ (t, ), c j c k ( j k f ɛ )(t, ), for any (c,,c ) R. k,j= Therefore by (3.6) an the fact that M ɛ >, we have ( tf ɛ )(t, ) (t, ) ɛ + (M ɛ + ɛt )( f ɛ )(t, ) (M ɛ + ɛt ) ɛ<. ( k j h) k,j=

12 D. Li et al. / J. Differential Equations 6 (7) This obviously contraicts to the fact that <t T an (t, ) is a maimum. Hence Case is impossible. Case. <t T an M ɛ. In this case we obtain the boun Case 3. t =. Clearly then ma f(t,) ɛt +. t T, T ma t T, T f(t,) ma f(,)+ ɛt. Concluing from all cases an sening ɛ to zero, we obtain (.7). In the case imension =, the proof of (.8) is similar. Set g = h. Note that T t g = (g 3 g) = (3g )g + 6g(g ). Clearly (3g )g is elliptic when 3g >, whence g(t) ma{ g(), 3 }, t. 4. Proof of Theorem. Lemma 4.. Let ν > an L = ν. Then for any integer m an any t >, we have similarly for any integer m an any t >, D m e tl f L (T ) ν,,m ( + t m 4 ) f H / (T ) ; (4.7) D m e tl f L (T ) ν,,m t m 4 f L (T ), (4.8) D m e tl f L (T ) ν,,m ( + t m 4 ) f L (T ). (4.9) In the above D m enotes any ifferential operator of orer m. For eample D can be any one of the operators i j, i, j. If f has mean zero, then (4.7) an (4.9) can be improve as: D m e tl f ν,,m t m 4 f H, m, t>, (4.) D m e tl f ν,,m t m 4 f, m, t>. (4.) Proof. We first show (4.7). Define =. Clearly D m e tl f = D m e tl f = K ( f) where enotes the usual convolution an K is the kernel corresponing to D m e tl. Then

13 73 D. Li et al. / J. Differential Equations 6 (7) Now since m, D m e tl f L (T ) K L (T ) f. H (T ) K L e νt k 4 k m k + e νt k 4 k m + t m. k Z k Thus (4.7) follows easily. For (4.8), we can regar f as a perioic function on R. Then using the fact that for any multi-ine α with α = m, F (ξ α e t ξ 4 ) L (R ) t m 4, we get D m e tl f L (T ) = Dm e tl f L (R ) t m 4 f L (R ) t m 4 f L (T ). Similarly one can prove (4.9) by computing everything on the Fourier sie. In the case f has mean zero, we note that ˆ f() =, an (4.) (4.) follows easily. Proof of Theorem.. This is more or less a stanar application of the theory of mil solutions. Therefore we shall only sketch the etails. We recast (.) into the mil form (alternatively one can also construct the mil solution by consiering L = ν as the linear part an taking e tl as the linear propagator): h(t) = e tν h + j= =: e tν h + (h)(t). j e (t s)ν (( h ) j h)(s)s Fi h H / (T ). Define h () = e tν h, an for j, For T >, introuce the Banach space with the norm h (j) (t) = e tν h + (h (j ) )(t). { X T = h Ct H ([,T] T ) : t 4 h C t C,t 4 h C t H + } For convenience enote the seminorm h XT = h Ct + t 4 h L H t, + t 4 h Ct H +. h YT = t 4 h L t, + t 4 h Ct H +. We shall show that for sufficiently small T > (epening on the profile of h ), the iterates h (j), j form a Cauchy sequence in the set

14 D. Li et al. / J. Differential Equations 6 (7) B T ={h X T : h XT h H (T ), h Y T ɛ h H (T ) }, where ɛ > is a sufficiently small constant epening only on (ν, ) an h H. We shall only verify that h (j) B T an omit the contraction argument since it is quite similar. Consier first j =. For h H (T ), obviously e ν t h C t H h H. By Lemma 4. an a ensity argument, we have for h H, lim t 4 e νt h L t + =, lim t 4 e νt h t + H Thus for T > sufficiently small, h () XT 3 h H, h () YT ɛ h H, where ɛ will be taken sufficiently small (epening on (ν, ) an h H ) later when we verify + =. the estimates for h (j), j. Now inuctively assume h (j ) B T. To show h (j) B T, it suffices for us to check (h (j ) ) XT ɛ h H. To simplify notation, in the computation below we shall rop the superscript (j ) an write (h (j ) ) simply as (h). We also write ν, simply as. Note that without loss of generality we can assume t, so that when applying Lemma 4., we have + t m 4 t m 4 (i.e. the constant is not neee). Now by Lemma 4., we have (h)(t) H + + e (t s)ν ( h ) h ) (s)s (t s) 4 h(s) s (t s) 4 ( h(s) h(s) ) s (t s) 4 s 4 s s 4 h(s) C s H + (t s) 4 s 3 4 s s 4 h(s) C s H + s 4 h(s) L s L

15 734 D. Li et al. / J. Differential Equations 6 (7) t s 4 h(s) C s H + t h H + h 3 Y t. + s 4 h(s) C s H + s 4 h(s) L s L Thus for T > sufficiently small an ɛ sufficiently small, Similarly easy to check that Thus (h) Ct ɛ H ([,T ] T ) h. H t 4 (h)(t) Ct H + + t 4 (h)(t) L ([,T ] T t, ([,T ] T ) ɛ ) 5 h. H (h) XT ɛ h H. We have finishe the proof of eistence an uniqueness of a solution in the Banach space X T. The smoothing estimate of h(t) for t >is utterly stanar. For eample if we know h L t H m ([t, t ] T ) on some time interval [t, t ], then for t (t, t ], D m+ t t t e (t s)ν (( h ) h)(s)s (t s) 3 4 ( h(s) ) h(s) H m s (t s) 3 4 s h L s H m + t (t s) This shows that h has higher regularity H m+ 3 4 s s h L s H m s 4 h L s L. on (t, t ] (the linear part only for t (t, t ]). We omit further etails. e (t t )ν h(t ) H m+

16 D. Li et al. / J. Differential Equations 6 (7) Proof of Corollary. an Corollary. Proof of Corollary.. Let the imension 3. We first assume that the initial ata h H 4 (T ) with mean zero. Denote the corresponing solution obtaine by Theorem. as h. To boun t h, we nee to control h h h h h h H 4. The H 4 regularity is use to control h. It is then easy to check that h C t H 4 C t L an where t E = th, (5.) E(t) = ν h(t) + ( h(t) ). 4 T Alternatively to avoi the issue of ifferentiability, one can interpret (5.) as the integral formulation: E(t ) = E(t ) t h t for any t <t. t From energy conservation we get h(t) H h H for any t>. Now for H initial ata (recall the critical space in Theorem. is H / an / < for 3), the lifespan of the local solution epens on the H -norm of the initial ata. Thanks to this fact an the estimate h(t) H h H, the corresponing local solution can be continue for all time by a stanar argument. This conclues the proof of global wellposeness uner the assumption that h H 4. Now let h H (T ) with mean zero. By Theorem., there eists a local solution h on [, T ] for some T > epening on h. Let h = h(t /). By Theorem., h H m for all m. In particular h H 4. Now with h as initial ata, the corresponing solution can be enote as h(t) = h(t + T /). One can then repeat the argument escribe in the previous paragraph to obtain global wellposeness. Proof of Corollary.. The D case. Note that by energy law we have E(t) E. Thus h(t) ν E, h(t) 4 E 4 +. By using the Gagliaro Nirenberg interpolation inequality, we have Therefore h h 3 4 h 3. h(t) ν 6 E 6 (E 6 + ). The D case. We first perform a short time estimate. Let <ɛ< which will be taken sufficiently small. Consier

17 736 D. Li et al. / J. Differential Equations 6 (7) h(t) = e νt h + e ν(t s) ( h ) h(s)s. Easy to check that in D, + ɛ h +ɛ h ɛ (recall h has mean zero). Then +ɛ h(t) ɛ ɛ e νt h ɛ + +ɛ e ν(t s) (( h ) h)(s)s ɛ s (νt) ɛ h H + (ν(t s)) 3+ɛ 4 ( h(s) 3 H + h(s) H )s (νt) ɛ ( E ν ) + ν 3+ɛ 4 t ɛ 4 (( E ν ) + ( E ν ) 3 ). In the above when bouning the nonlinearity, we use the estimate h h ɛ h h ɛ ɛ h 3 H. Thus for t an <ν, we get + ɛ h(t) ( E + ). ν By repeating the same analysis with t an h replace by h(t ) (note that only h H enters the analysis), we get for all t + ɛ h(t) ( E + ). ν Now note that h(t) H ( E ν ). Using Littlewoo Paley ecomposition (note that S h = ), we get Optimizing in j, we get h(t) L (T ) j h L (T ) + j h L (T ) j j j>j (j + 3) h H + j + ɛ ɛ h (j + 3)( E ν ) ɛ + j E + ( ). ν sup h(t) ( E t< ν ) E + log( ). ν

18 D. Li et al. / J. Differential Equations 6 (7) Now to obtain the estimate for t, we simply note that for t, by repeating the analysis before, On the other han, + ɛ (h(t) e νt h ) ( ) E +. h(t) e νt h H h H + h H ( E ν ). Thus we obtain the same boun for h(t) e νt h. This finishes the estimate for the D case. The 3D case. We shall again perform a short time estimate. Write It is easy to check that We then get for t, h(t) = e νt h + h(t) (νt) 8 h H + e ν(t s) (( h ) h)(s)s. e ν t h L (T 3 ) (νt) 8 h H (T 3 ). t 8 ν 5 8 E + ν 7 8 t 8 (ν 3 E 3 + ). ν (ν(t s)) 7 8 ( h(s) h(s) )s Choosing t ν 7 then yiels h(t) ν 3 3 (E + ). For general t ν 7, we can replace h by h(t ν 7 ) an repeat the above analysis. This ens the estimate for the 3D case. The following proposition shows that in D, there eists initial ata such that the corresponing solution obeys uniform in time graient bouns which are inepenent of ν. Proposition 5.. Let the imension =. Consier (.) on the π-perioic torus T with < ν. Assume h H (T) with mean zero an let h = h(t, ) be the corresponing global solution to (.). Denote ( E = ν h + 4 ( h ) ). Then for all t > an some absolute constant C >, T h(t) C ma{,ν 6 E 3 }. (5.3)

19 738 D. Li et al. / J. Differential Equations 6 (7) For each < ν, there eists a family A ν of initial ata, such that if h A ν, then E ν, an the corresponing solution satisfies h(t) B, t, where B > is an absolute constant. (In particular, it is inepenent of ν.) Proof of Proposition 5.. We first show (5.3). Denote h = A an g = h. If A we are one. Now assume A >, then obviously A g. Now by Gagliaro Nirenberg interpolation, we get Thus A g g g g h h g h A. A g 3 h 3 E 6 ( E ν ) 6 ν 6 E 3. We now show that there eists initial ata h such that E ν. The iea is to mollify the sawtooth -type profile an a a δ-cap (δ ν) aroun each tips of the sawtooth. To this en, let L 3be an integer an efine g () = sgn(sin(l τ))τ, [ π,π], where sgn is the usual sign function:, z>, sgn(z) =, z=,, z<. The value of L is not important as long as it is inepenent of ν. Now aroun each local maima or minima of g, easy to check that g change its sign from to, or to. At the maima (minima), g is unefine. One can then mollify g therein within a δ-neighborhoo. Denote the mollifie function as g δ. Then ( E(g δ ) = ν g δ + 4 ( g δ ) ) L ν δ δ + δ. T Choosing δ ν then yiels E(g δ ) L ν. Proposition 5.. Let the imension =. Consier (.) on the π-perioic torus T with < ν. Assume h H (T) with mean zero an let h = h(t, ) be the corresponing global solution to (.). Then lim sup h(t) K, (5.4) t

20 D. Li et al. / J. Differential Equations 6 (7) where K is a constant epening only on the initial ata h. If in aitional h is even in, then (5.4) can be improve to lim sup h(t). (5.5) t Remark 5.. Recall that in the D case, the equation (.) can be transforme into the usual Cahn Hilliar equation via the change of variable u = h. The convergence to steay states (an consequently graient bouns) can be obtaine using the Łojasiewicz Simon inequality (cf. []). Our proof below however oes not appeal to this theory an gives an alternative approach. Proof of Proposition 5.. First observe that by using Theorem. an a shift in time we may assume h H (T). By using the Duhamel formula h(t) = e νt 4 h + e ν(t s) 4 ((h )h )(s)s, the energy law, an the eponential (in time) ecay of the propagator e ν(t s) 4 (acting on meanzero functions), it is not ifficult to erive that sup h(t) H (T) ν,e. (5.6) t This estimate will be use below. Step : we show that lim t t h =. Denote g = t h, then g satisfies the equation t g = ((3h )g ) ν 4 g. Consier t >t, where t will be picke later. We have g(t) = e ν(t t ) 4 g(t ) + t = e ν(t t ) 4 g(t ) + t t e ν(t s) 4 ((3h )g )(s)s e ν(t s) 4 ((3h )g)(s)s e ν(t s) 4 (6h h g)(s)s. (5.7) Now note that for any function g : T R (not necessarily having mean zero), one has for m, m e νt 4 g m,ν e νt/ t m 4 g. Here the point is that since m, g can be replace by g g ( g enotes the mean of g) an g g.

21 74 D. Li et al. / J. Differential Equations 6 (7) Now continuing from (5.7), we get (by using (5.6)) g(t) ν,e g(t ) + + t t (t s) e ν(t s)/ g(s) s (t s) 4 e ν(t s)/ g(s) s. (5.8) By using the energy law, we have g(s) s <. Thus one can fin t sufficiently large such that g(t ) an also t g(s) s. By (5.6), we also have sup s g(s). These estimates with (5.8) an an ɛ-δ argument (One nees to split the time interval in (5.8). For s close to t, we use the smallness of the time interval an the estimate g(s). For s away from t, use t g(s) s.) then easily yiel lim g(t) =. t Interpolating the above estimate with (5.6) (recall g(t) = t h = (h 3 h ) ν 4 h), we get lim th =. (5.9) t Step : we show (5.5). Easy to check that the even symmetry is propagate in time. Denote f = h. Then ( f 3 f νf ) = t h. In view of the even symmetry of h, we have f(t, = ), f(t, = ). Thus (f )f ν f = ( t h)(t, y)y. A simple maimum principle argument together with (5.9) then yiel (5.5). Finally the proof of (5.4) is similar. In the general case, observe that (since f = h) π T (f 3 f νf (t, )) = π f 3 (t, ). } T {{ } :=m(t) By the Mean Value Theorem, there eists [ π, π] such that We then have f 3 (t, ) f(t, ) νf (t, ) = m(t).

22 D. Li et al. / J. Differential Equations 6 (7) f 3 f νf = ( t h)(t, y)y + m(t). Now observe that m(t) h(t) T (h ) + E, where E is the initial energy. The boun (5.4) then again follows from a maimum principle argument using this estimate. 6. Proof of Theorem. an Corollary.3 The following perturbation lemma is more or less stanar. It follows from the local theory an we omit the proof. Proposition 6. (Finite time stability of solutions). Let ν > in (.). Let u H k, k >/ an u be the corresponing solution. Let T > be given an assume u has lifespan bigger than [, T ]. Then for any ɛ >, there eists δ> such that the following hols: For any v H k, k>/ with v u H k <δ, there eists a solution v to (.) corresponing to the initial ata v an has lifespan containing [, T ]. Furthermore we have ma v(t) u(t) H t T k <ɛ. In particular by shrinking δ further if necessary, we have We now complete the proof of Theorem.. ma v(t) u(t) <ɛ. t T Proof of Theorem.. Step. We first show that there eists a smooth solution w to (.) with initial ata w such that w = an for some t >, C > w(t ) >C >. (6.3) Let η >be sufficiently small an w be a smooth π-perioic function with mean zero (Here one can choose w such that it is o in when regare as a function on R. This in turn easily implies that w has mean zero on [ π, π].) such that w () = η 5, <η, w () <, η ξ π. (6.3) Denote by w = w(t, ) the corresponing solution to (.). Observe that w () = 5η4, for <η.

23 74 D. Li et al. / J. Differential Equations 6 (7) Obviously it follows that w () with equality holing only at = (an its π-perioic images). By a irect calculation, we have for <η, Clearly it hols that Now since ( w ) 3 w = ( 5η 4 ) 3 ( 5η 4 ) = O( 4 ). (( w ) 3 w ) = =. t (w ) = (w 3 w ) ν 5 w, we have ( t w)(, ) = (( w ) 3 = w ) ν 5 w = νη >. = Since A(t) = ( w)(t, ) is a continuously ifferentiable function of t with A() =, A ()>, obviously (6.3) hols. Step. The perturbation argument. Let φ C c ({ : <η}) be a fie smooth cut-off function with φ() = for < η. Let φ be even in an let v δ () = w () δφ(). Note that v δ is o in an still has mean zero. Clearly an can be mae arbitrarily small. On the other han for <η/, v δ w H δ φ() H const δ (6.3) v δ () = w () δ = 5η 4 δ δ. For η/ π, since by construction we have w () β, for some constant β >. Obviously by choosing δ> sufficiently small we can have Therefore we have shown v δ () β, η/ π. v δ <.

24 D. Li et al. / J. Differential Equations 6 (7) Now let v δ be the solution to (.) corresponing to initial ata v δ. By (6.3), (6.3) an Proposition 6., for δ> sufficiently small, we have v δ (t ) >C >, where C is another constant. Define A ={v δ : δ is sufficiently small}. This conclues our construction. Proof of Corollary.3. The essential ieas are alreay in the proof of Theorem.. Therefore we only sketch the necessary notational moifications. Take η> sufficiently small an a = (,, ) T (here is the imension). Note that by efinition a =. We efine a smooth function w C (T ) such that w () = a η j 5, j= for <η. Let D =[ π, π] be the funamental omain of the torus T. For η, D, we simply require an w () <. Take a raial φ C c ({ R : <η}) such that φ() for η/. For δ> sufficiently small, efine v δ = w () δ (a ) φ() A ={v δ : δ> is sufficiently small}. The set A is the esire family of initial ata. 7. Proof of Theorem.3 In this section we give the proof of Theorem.3. Proof of Theorem.3. Without loss of generality we assume the imension =. The case can be prove with suitable moifications. Fi ɛ >. Let f()= e ξ 4 e iξ ξ. π R Define C = f L (R), A = f L (R).

25 744 D. Li et al. / J. Differential Equations 6 (7) Define t > such that C 3 A ν t = ɛ 3. (7.33) Step : We show that there eist t > with t t an h C (T) with mean zero such that h < an e νt h >C ɛ 3. (7.34) To show this, we first choose F(t, ) to be an o function of which is π-perioic, an such that sgn(f (s/(νt) 4 ))s, t 5 ; F(t,)=, t sgn(f (s/(νt) 4 ))s π; linear interpolation, t 5 t sgn(f (s/(νt) 4 ))s. Easy to check that for t / the function F(t, ) is well-efine. Furthermore F(t,)= sgn(f (/(νt) 4 )), a.e. t 5 ; an F. Define ( ) G(t, ) = e νt ( F(t, )) (t, ). Then clearly if t is sufficiently small, then G(t, ) f( ) (νt) (νt) 4 f( ) (νt) 4 (νt) 4 4 t 5 = f L (R) >t 5 = C f() >ν 4 t >t 5 f( ) (νt) (νt) 4 4 >C ɛ 4. In the last inequality above, we use the fact that f is a Schwartz function an the tail contribution to the integral can be mae arbitrarily small (by taking t small). Now take an even function ψ Cc (R) such that ψ, ψ() = for an ψ =. Define ψδ () = δ ψ(/δ) an

26 D. Li et al. / J. Differential Equations 6 (7) ( ) F δ (t, ) = ( δ) ψ δ F(t, ) (t, ), where is the usual convolution on R. Easy to check that F δ <, F δ is π-perioic, o in an has mean zero. Define ( ) G δ (t, ) = e νt ( F δ (t, )) (t, ). Obviously for δ sufficiently small, we have G δ (t, ) >C ɛ 3. Thus (7.34) is achieve with h () = F δ (t, ). Step : Control of the nonlinear solution. We shall fi t an h from Step. With h as initial ata, let h be the corresponing solution to (.). We argue by contraiction an assume that Then Now since we get sup h(t, ) C ɛ. (7.35) t t h 3 h C 3, <t t. h(t) = e νt h + h(t) e νt h ( ) e νs (h 3 h )(t s) s, e νs ((h 3 h )(t s)) s. Regar (h 3 h ) as a π-perioic function on R. Recall that f () = F ( ξ e ξ 4 ). Then e νs ((h 3 h )) L (T) = e νs ((h 3 h )) L (R) F ( ξ e νs ξ 4 ) L (R) h3 h L (R) f L (R) (νs) C 3 = A (νs) C 3.

27 746 D. Li et al. / J. Differential Equations 6 (7) Thus we obtain for <t t, h(t) e νt h A ν t C 3. Since t t, by (7.33) an Step, we get h(t ) >C ɛ 3 ɛ 3 = C ɛ 3 which is an obvious contraiction to (7.35). Acknowlegments D. Li was supporte in part by an Nserc iscovery grant. The research of Z. Qiao is partially supporte by Hong Kong Research Council GRF grant PolyU 534, an NSFC/RGC Joint Research Scheme N_HKBU4/. The research of T. Tang is mainly supporte by Hong Kong Research Council GRF Grants an Hong Kong Baptist University FRG grants. References [] J. Bourgain, D. Li, Strong ill-poseness of the incompressible Euler equation in borerline Sobolev spaces, Invent. Math. () (5) [] J. Bourgain, D. Li, Strong illposeness of the incompressible Euler equation in integer C m spaces, Geom. Funct. Anal. 5 () (5) 86. [3] G. Ehrlich, F.G. Hua, Atomic view of surface iffusion: tungsten on tungsten, J. Chem. Phys. 44 (966) 36. [4] H. Dong, D. Li, On a generalize maimum principle for a transport-iffusion moel with log-moulate fractional issipation, Discrete Contin. Dyn. Syst. 34 (9) (4) [5] B. Li, J.G. Liu, Thin film epitay with or without slope selection, Eur. J. Appl. Math. 4 (3) [6] D. Li, On a frequency localize Bernstein inequality an some generalize Poincaré-type inequalities, Math. Res. Lett. (5) (3) [7] D. Li, Z. Qiao, T. Tang, Characterizing the stabilization size for semi-implicit Fourier-spectral metho to phase fiel equations, SIAM J. Numer. Anal. 54 (3) (6) [8] D. Li, Z. Qiao, On secon orer semi-implicit Fourier spectral methos for D Cahn Hilliar equations, J. Sci. Comput. (6), [9] D. Li, Z. Qiao, On the stabilization size of semi-implicit Fourier-spectral methos for 3D Cahn Hilliar equations, accepte by Commun. Math. Sci. (6). [] Z. Qiao, Z. Zhang, T. Tang, An aaptive time-stepping strategy for the molecular beam epitay moels, SIAM J. Sci. Comput. 33 (3) () [] P. Rybka, K. Hoffmann, Convergence of solutions to Cahn Hilliar equation, Comm. Partial Differential Equations 4 (5 6) (999) [] R.L. Schwoebel, E.J. Shipsey Step, Motion on crystal surfaces, J. Appl. Phys. 37 (966) 368. [3] R.L. Schwoebel, Step motion on crystal surfaces II, J. Appl. Phys. 4 (969) 64. [4] J. Shen, C. Wang, X. Wang, S.M. Wise, Secon-orer conve splitting schemes for graient flows with Ehrlich Schwoebel type energy: application to thin film epitay, SIAM J. Numer. Anal. 5 () () 5 5. [5] C. Xu, T. Tang, Stability analysis of large time-stepping methos for epitaial growth moels, SIAM J. Numer. Anal. 44 (4) (6) [6] C. Wang, S. Wang, S.M. Wise, Unconitionally stable schemes for equations of thin film epitay, Discrete Contin. Dyn. Syst. Ser. A 8 ()