Global solutions of the two-dimensional Kuramoto-Sivashinsky equation with a linearly growing mode in each direction
aa r X i v : . [ m a t h . A P ] F e b GLOBAL SOLUTIONS OF THE TWO-DIMENSIONALKURAMOTO-SIVASHINSKY EQUATION WITH ALINEARLY GROWING MODE IN EACH DIRECTION
DAVID M. AMBROSE AND ANNA L. MAZZUCATO
Abstract.
In two spatial dimensions, there are very few global exis-tence results for the Kuramoto-Sivashinsky equation. The majority ofthe few results in the literature are strongly anisotropic, i.e. are resultsof thin-domain type. In the spatially periodic case, the dynamics of theKuramoto-Sivashinsky equation are in part governed by the size of thedomain, as this determines how many linearly growing Fourier modes arepresent. The strongly anisotropic results allow linearly growing Fouriermodes in only one of the spatial directions. We provide here the firstproof of global solutions for the two-dimensional Kuramoto-Sivashinskyequation with a linearly growing mode in both spatial directions. Wedevelop a new method to this end, categorizing wavenumbers as low(linearly growing modes), intermediate (linearly decaying modes whichserve as energy sinks for the low modes), and high (strongly linearlydecaying modes). The low and intermediate modes are controlled bymeans of a Lyapunov function, while the high modes are controlled withoperator estimates in function spaces based on the Wiener algebra. Introduction
We study the Kuramoto-Sivashinsky equation,(1) ψ t = − ∆ ψ − ∆ ψ − |∇ ψ | . The Kuramoto-Sivashinsky equation is a well-known model of flame frontpropagation and was first derived in [19], [28]. We consider this on a rect-angular domain [0 , L ] × [0 , L ] with periodic boundary conditions, andwill prove a global existence theorem for solutions with sufficiently smalldata. There are a number of global existence theorems for the Kuramoto-Sivashinsky equation in one spatial dimension [17], [24], [29], and detailedstudies of the asymptotics of these solutions [10], [15], [16], [25]. Theseone-dimensional results rely on structure of the nonlinearity which is notpresent in two spatial dimensions, and thus there are far fewer global resultsavailable in the two-dimensional case.In two spatial dimensions, most prior global existence results are inher-ently anisotropic, i.e., are results of thin-domain type, in which the two-dimensional solution in a thin domain is shown to remain close to the one-dimensional solution. Such studies were initiated by Sell and Taboada [27].Other anisotropic global existence theorems are the works [9], [23]. Then there are global existence and singularity formation results for modifiedequations. The fourth-order nature of the parabolic evolution (1) of courseimplies the absence of a maximum principle. Other authors have shownthat related systems with maximum principles do have global solutions [20],[22], or have also modified the nonlinear term, showing that related equa-tions have finite-time singularities [8] or global solutions [11], [26]; see also[30] for a numerical study of a modified equation. The second author andFeng have shown that a modification of (1) with additional advection alsohas global solutions [13]. Rather than modifying the equation or relyingon anisotropy, the authors have previously given a global existence theo-rem for the two-dimensional Kuramoto-Sivashinsky equation, but under therequirement that the domain size be sufficiently small [7].The linear operators − ∆ and − ∆ on the right-hand side of (1) may beviewed as being in competition with each other; these represent a higher-order forward parabolic effect and a lower-order backward parabolic effect.That the forward parabolic term is higher-order than the backward parabolicterm implies that there are at most finitely many linearly growing Fouriermodes; the discreteness of the Fourier variable in the spatially periodic caseimplies that for L and L small enough, there are in fact no linearly growingFourier modes. To be precise, if L and L are each in the interval (0 , π ) , then there are no linearly growing Fourier modes in (1), and this is thecase studied in [7]. In the current study, by taking each of L and L slightly larger than 2 π, we ensure that there is exactly one linearly growingmode in each of the x -direction and the y -direction. In all previous globalexistence results for the two-dimensional Kuramoto-Sivashinsky equation,either there were no linearly growing modes at all [7], or (in the stronglyanistropic works [9], [23], [27]) the linearly growing modes were only in onedirection. The current work is therefore the first global existence theoremfor the two-dimensional Kuramoto-Sivashinsky equation to allow a growingmode in each spatial direction. We note that the interested reader mightsee [18] for a detailed numerical study of the dependence of the dynamics ofsolutions on the size of the spatial domain/the number of linearly growingFourier modes present.The method of proof primarily combines ideas from the prior work ofthe authors [7] and from the one-dimensional global existence theorem ofGoodman [17]. We will now describe the formulation of the problem to beused and how these ideas come into play.We immediately notice that, while the mean of ψ , ¯ ψ, is not preservedunder the time evolution, its growth is governed by the L -norm of thegradient of ψ , which does not depend on the mean itself. As a matter offact, if we define φ = P ψ, where P is the projection which removes themean of a periodic function, the equation satisfied by φ is(2) φ t = − ∆ φ − ∆ φ − P |∇ φ | . HE 2D KURAMOTO-SIVASHINSKY EQUATION 3
The evolution equation for ¯ ψ is then(3) ¯ ψ t = − L L Z L Z L |∇ φ | dxdy. We therefore see that the mean of ψ exists and is finite at time T as longas φ ∈ L ([0 , T ]; ˙ H ), where ˙ H denotes the homogeneous L -Sobolev spaceof order 1. For simplicity, as in Nicolaenko, Scheurer, and Temam [24], weconsider symmetric solutions: φ ( x, y, t ) = X k,j ≥ a k,j ( t ) cos (cid:18) πkxL (cid:19) cos (cid:18) πjyL (cid:19) , We introduce a decomposition of the Fourier modes into three categories.With our choice that L and L are each slightly larger than 2 π, we have twolinearly growing Fourier modes, a , and a , ; these linearly growing modesare the first type which we treat specially. We next take two intermediatemodes, which are the a , and a , modes; these are linearly decaying modeswhich we use to absorb energy from the lowest modes. Finally, our thirdcategory consists of all remaining Fourier modes; we consider these to bestrongly decaying.We therefore need evolution equations for five components of the solution:the coefficients of the growing modes a , and a , , the coefficients of the nextmodes in each direction, a , and a , , and then the remainder of the solution,which we will call w. We let P be the projection onto the complement ofthe span of the 4 modes introduced above. We may then write φ as(4) φ ( x, y, t ) = a , ( t ) cos (cid:18) πxL (cid:19) + a , ( t ) cos (cid:18) πxL (cid:19) + a , ( t ) cos (cid:18) πyL (cid:19) + a , ( t ) cos (cid:18) πyL (cid:19) + w ( x, y, t ) , where w = P φ. The KSE is equivalent, at least formally, to a coupledsystem of 5 equations, 4 ODEs for the modes a , , a , , a , , a , , and aPDE for w. Our first goal is to derive this coupled system. Throughtout, forease of notation, we will denote derivatives as subscripts, so a , t = ddt a , .For the modes we are treating specially, we will bring out the quadraticinteractions between these modes, and consider the rest of the nonlinearityto be a smaller remainder. The evolution equation for a , is then a , t = − (cid:18) πL (cid:19) + (cid:18) πL (cid:19) ! a , + 8 π L a , a , + F , ,x + F , ,y , where F , ,x and F , ,y arise from the projection onto the (1 , φ x + φ y , where F , ,x corresponds to φ x and F , ,x corresponds to φ y . Formulas for F , ,x and F , ,y can be written explicitly, and these are contained in Section 1.1. DAVID M. AMBROSE AND ANNA L. MAZZUCATO
The evolution equation for a , is a , t = − (cid:18) πL (cid:19) + (cid:18) πL (cid:19) ! a , − π L a , + F , ,x + F , ,y , where F , ,x and F , ,y play the same role as F , ,x and F , ,y do for a , .Again, we mention that the formulas for F , ,x and F , ,y are contained inSection 1.1. We have the corresponding evolution equations for a , and a , , a , t = − (cid:18) πL (cid:19) + (cid:18) πL (cid:19) ! a , + 8 π L a , a , + F , ,x + F , ,y ,a , t = − (cid:18) πL (cid:19) + (cid:18) πL (cid:19) ! − π L a , + F , ,x + F , ,y . Formulas for F , ,x , F , ,y , F , ,x , and F , ,y will be shown in Section 1.1below.Finally, we may write the evolution equation for w simply as(5) w t = − ∆ w − ∆ w + P (cid:0) ( ∂ x φ ) + ( ∂ y φ ) (cid:1) , where φ and w are related through (4).In our system, a , and a , are the only linearly growing Fourier modes,which exist because of the specified size of the spatial domain. The modes a , and a , are linearly decaying modes which we treat specially becausethey allow energy to transfer from the linearly growing modes; i.e., these areenergy sinks for the lowest modes (whereas the lowest modes can be viewedas functioning as energy sources). Then, w contains all the remaining decay-ing modes; we view these modes as forcing the four modes we treat specially.Goodman introduced a toy model [17] to show how energy transfers betweena 1 − mode and a 2 − mode, in the absence of further forcing. In Goodman’scase, energy was conserved by the nonlinear terms. The conservation ofenergy (i.e., a conserved L norm) does not hold for the nonlinearity intwo spatial dimensions. We observe however that Goodman’s argument ismore robust than this, and can be modified to handle the presence of smallforcing.We introduce some notation for the coefficients of the linear terms in themodes which we treat specially: ε i = − (cid:18) πL i (cid:19) + (cid:18) πL i (cid:19) , (6) B i = (cid:18) πL i (cid:19) − (cid:18) πL i (cid:19) , where i ∈ { , } . We will sometimes denote ε = max { ε , ε } as well. Ofcourse, since 2 π < L i < π we have that all of these coefficients are positive.Furthermore, by taking L i only slightly larger than 2 π we can make ε i HE 2D KURAMOTO-SIVASHINSKY EQUATION 5 arbitrarily small. Using this notation, our equations for the first two modesin the x -direction become a , t = ε a , + 8 π L a , a , + F , ,x + F , ,y ,a , t = − B a , − π L a , + F , ,x + F , ,y , Similarly, the equations for the first two modes in the y -direction are a , t = ε a , + 8 π L a , a , + F , ,x + F , ,y ,a , t = − B a , − π L a , + F , ,x + F , ,y . Our evolution equation for w, namely (5), is unchanged.As we have indicated already, the four special modes will be treated witha Lyapunov function, generalizing Goodman’s result for a toy model. ThisLyapunov function argument will show that these modes remains of size ε / , if initially of that size. For the 2-modes a , and a , , this bound canbe improved. We will find that the size of each of a , and a , is then atmost proportional to ε , if initially of that size. The norm of w (in a functionspace related to the Wiener algebra) will be shown to be bounded by ε / ,if initially of that size. This method using the Wiener algebra was usedpreviously by the authors in [7], and is inspired by the work of Duchon andRobert on vortex sheets [12]. The first author and his collaborators haveadditionally developed and used the technique in [2], [3], [4], [5], [21].The following is the (non-technical version of) our main theorem: Theorem 1.
There exists ε ∗ > such that for any ε ∈ (0 , ε ∗ ) , if a , (0) ∼ ε / , a , (0) ∼ ε, (7) a , (0) ∼ ε / , a , (0) ∼ ε, w ∼ ε / , then the 2D Kuramoto-Sivashinsky equation with these data has a solutionon an arbitrary time interval [0 , T ] . We will give more precise bounds on the initial data and will state atechnical version of the theorem later in Theorem 2 in Section 5.The plan of the rest of the paper is as follows. In Section 1.1, we completethe specification of our decomposed evolution equations by detailing formu-las for the forcing functions. In Section 2, we set up an iterative scheme,and we prepare to make estimates which will be uniform in the iterationparameter. We develop propositions which give these uniform estimates on a , , a , , a , , and a , in Section 3. We then develop tools which will giveuniform bounds on w in Section 4. The uniform bounds are established,and the limit of the iterates is taken, in Section 5. We then make someconcluding remarks on future directions in Section 6. DAVID M. AMBROSE AND ANNA L. MAZZUCATO
The first author is grateful to the National Science Foundation for supportthrough grant DMS-1907684. The second author is grateful to the NationalScience Foundation for support through grant DMS-1909103.1.1.
Formulas for the forcing functions.
We let P , be the projectiononto the (1 ,
0) Fourier mode and we let P , be the projection onto the (2 , P , be the projection onto the (0 ,
1) Fouriermode and we let P , be the projection onto the (0 ,
2) Fourier mode.We will determine F , ,x and F , ,y by projecting the nonlinear term ontothe (1 ,
0) mode and then separating out certain quadratic interactions: P , (cid:16) ( φ x ) (cid:17) = (cid:20) − π L a , a , + F , ,x (cid:21) cos (cid:18) πxL (cid:19) , (8) P , (cid:16) ( φ y ) (cid:17) = F , ,y cos (cid:18) πxL (cid:19) . To understand this decomposition better, we begin by calculating φ x : φ x = − πa , L sin (cid:18) πxL (cid:19) − πa , L sin (cid:18) πxL (cid:19) + w x . We square this expression to find ( φ x ) , and we decompose it as follows:( φ x ) = X i =1 Ψ xi , with the terms Ψ i defined asΨ x = 4 π ( a , ) L sin (cid:18) πxL (cid:19) , Ψ x = 16 π ( a , ) L sin (cid:18) πxL (cid:19) , Ψ x = ( w x ) , Ψ x = 16 π a , a , L sin (cid:18) πxL (cid:19) sin (cid:18) πxL (cid:19) , Ψ x = − πa , L w x sin (cid:18) πxL (cid:19) , Ψ x = − πa , L w x sin (cid:18) πxL (cid:19) . Using trigonometric formulas, we see immediately that P , Ψ x = P , Ψ x =0 . Also we may calculate that P , Ψ x = 8 π L a , a , cos (cid:18) πxL (cid:19) , and this is the term explicitly accounted for in (8), i.e., Ψ x makes no con-tribution to F , ,x because this is the term we have separated out already. HE 2D KURAMOTO-SIVASHINSKY EQUATION 7
We have the following equation, then, which defines F , ,x : F , ,x cos (cid:18) πxL (cid:19) = P , [Ψ x + Ψ x + Ψ x ] . To compute F , ,y , we need the corresponding decomposition of φ y . Tobegin, we have the formula for φ y ,φ y = − πL a , sin (cid:18) πyL (cid:19) − πL a , sin (cid:18) πyL (cid:19) + w y . We square this expression, and we decompose the result as φ y = X i =1 Ψ yi , with the terms Ψ yi defined asΨ y = 4 π L a , sin (cid:18) πyL (cid:19) , Ψ y = 16 π L a , sin (cid:18) πyL (cid:19) , Ψ y = w y , Ψ y = 16 π L a , a , sin (cid:18) πyL (cid:19) sin (cid:18) πyL (cid:19) , Ψ ,y = − πa , L w y sin (cid:18) πyL (cid:19) , Ψ ,y = − πa , L w y sin (cid:18) πyL (cid:19) . Again, it is clear that P , Ψ y = P , Ψ y = P , Ψ y = 0 . Our equation for F , ,y is then F , ,y cos (cid:18) πxL (cid:19) = P , [Ψ y + Ψ y + Ψ y ] . We treat F , ,x and F , ,y in the same way. The analogue of (8) is(9) P , (( φ x ) ) = − (cid:20) π L ( a , ) + F , ,x (cid:21) cos (cid:18) πxL (cid:19) . Then we also have P , (( φ y ) ) = F , ,y cos (cid:18) πxL (cid:19) . We see that P , Ψ x = P , Ψ x = 0 , and we have P , Ψ x = − π L ( a , ) cos (cid:18) πxL (cid:19) . DAVID M. AMBROSE AND ANNA L. MAZZUCATO
This is the term we separate out in (9) and we do not include in F , ,x . Wethen have F , ,x cos (cid:18) πxL (cid:19) = P , [Ψ x + Ψ x + Ψ x ] . Similarly, since P , Ψ y = P , Ψ y = P , Ψ y = 0 , we conclude that F , ,y cos (cid:18) πxL (cid:19) = P , [Ψ y + Ψ y + Ψ y ] . Omitting further details, we also give the formulas for the remaining forc-ing functions, which are completely analogous to those we have alreadyshown. These remaining formulas are: F , ,x cos (cid:18) πyL (cid:19) = P , [Ψ x + Ψ x + Ψ x ] ,F , ,y cos (cid:18) πyL (cid:19) = P , [Ψ y + Ψ y + Ψ y ] ,F , ,x cos (cid:18) πyL (cid:19) = P , [Ψ x + Ψ x + Ψ x ] ,F , ,y cos (cid:18) πyL (cid:19) = P , [Ψ y + Ψ y + Ψ y ] . Iterative scheme
We will solve the coupled system of ODEs for the 4 specialized modesand the PDE for the remainder w via an iterative scheme for φ n , where φ n is defined by:(10) φ n ( x, y, t ) = a n , ( t ) cos (cid:18) πxL (cid:19) + a n , ( t ) cos (cid:18) πxL (cid:19) + a n , ( t ) cos (cid:18) πyL (cid:19) + a n , ( t ) cos (cid:18) πyL (cid:19) + w n ( x, y, t ) . In the scheme, the forcing terms are given by formulas corresponding tothose in Section 1.1 in a straightforward way.We start by giving the equations for the a n +1 coefficients: a n +11 , t = ε a n +11 , + 8 π L a n +11 , a n +12 , + F n , ,x + F n , ,y ,a n +12 , t = − B a n +12 , − π L ( a n +11 , ) + F n , ,x + F n , ,y ,a n +10 , t = ε a n +10 , + 8 π L a n +10 , a n +10 , + F n , ,x + F n , ,y ,a n +10 , t = − B a n +10 , − π L ( a n +10 , ) + F n , ,x + F n , ,y , HE 2D KURAMOTO-SIVASHINSKY EQUATION 9
To complete the scheme, we also give the iterated version of (5) for w n :(11) w n +1 t = − ∆ w n +1 − ∆ w n +1 + P (cid:0) ( ∂ x φ n ) + ( ∂ y φ n ) (cid:1) , The iterated system is taken with initial data that do not depend on n, namely, a n +11 , ( t ) = a , (0) , a n +12 , ( t ) = a , (0) ,a n +10 , ( t ) = a , (0) , a n +10 , ( t ) = a , (0) , w n +1 = w . List of constants.
For convenience, we label some combinations ofconstants that will appear in ensuing calculations. We first introduce M , and M , , which will be used in the bounds for a n , and a n , : M , = 12 B L π , M , = 12 B L π . The following constants will be used in the bounds for a n , and a n , : M , = 8 π M , L . M , = 8 π M , L . The constant M will be used in the bound for w n : M = max n K (cid:16) M / , M , K (cid:17) , K (cid:16) M / , M , K (cid:17)o . The formula above for M involves two other constants, K and K . Ofthese, K is a bound for the operator norm of an integral term in the mildformulation of the equation for w n ; this formulation will be developed inSection 4 below. To specify the constant K we need to specify a set, A, ofspecial wavenumber pairs: A = { (0 , , (1 , , (2 , , (0 , , (0 , } . Then K is given by(12) K = sup ( k,j ) ∈ Z \ A | k | + | j |− σ ( k, j ) , where σ is the symbol of the linearized KSE operator − ∆ − ∆,(13) σ ( k, j ) = − (cid:18) πkL (cid:19) + (cid:18) πjL (cid:19) ! + (cid:18) πkL (cid:19) + (cid:18) πjL (cid:19) . We notice that the denominator in (12) is quartic with respect to k and j ,while the numerator is linear. Also, the denominator is always positive, asthe only pairs for which the denominator is nonpositive are ( k, j ) = (0 , , ( k, j ) = (1 , , and ( k, j ) = (0 , , and these three pairs are excluded fromthe set A. Thus, the supremum in (12) is finite and positive.
We let K be an upper bound on the norm of some particular functionsin a certain space, denoted B ρ and defined in Section 4 below, that will beused for the analysis of the w n equation: (cid:13)(cid:13)(cid:13)(cid:13) π L sin (cid:18) πxL (cid:19) sin (cid:18) πxL (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) B ρ ≤ K , (cid:13)(cid:13)(cid:13)(cid:13) π L sin (cid:18) πxL (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) B ρ ≤ K , (14) (cid:13)(cid:13)(cid:13)(cid:13) πL sin (cid:18) πxL (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) B ρ ≤ K , (cid:13)(cid:13)(cid:13)(cid:13) πL sin (cid:18) πxL (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) B ρ ≤ K , (cid:13)(cid:13)(cid:13)(cid:13) π L sin (cid:18) πyL (cid:19) sin (cid:18) πyL (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) B ρ ≤ K , (cid:13)(cid:13)(cid:13)(cid:13) π L sin (cid:18) πyL (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) B ρ ≤ K , (cid:13)(cid:13)(cid:13)(cid:13) πL sin (cid:18) πyL (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) B ρ ≤ K , (cid:13)(cid:13)(cid:13)(cid:13) πL sin (cid:18) πyL (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) B ρ ≤ K . Finally, we introduce a constant K that will be used in the bound on theforcing terms: K = max ( M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πxL (cid:17)(cid:13)(cid:13)(cid:13) B ρ , M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πxL (cid:17)(cid:13)(cid:13)(cid:13) B ρ , M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πyL (cid:17)(cid:13)(cid:13)(cid:13) B ρ , M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πyL (cid:17)(cid:13)(cid:13)(cid:13) B ρ , M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πxL (cid:17)(cid:13)(cid:13)(cid:13) B ρ , M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πxL (cid:17)(cid:13)(cid:13)(cid:13) B ρ , M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πyL (cid:17)(cid:13)(cid:13)(cid:13) B ρ , M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πyL (cid:17)(cid:13)(cid:13)(cid:13) B ρ ) . Goodman’s toy model with added forcing
In [17], Goodman proved that small solutions of the one-dimensionalKuramoto-Sivashinsky equation exist and stay small for all time, using aLyapunov function argument. In his proof, the domain can be of arbitrarysize, and hence there can be any number of linearly growing modes. First,however, he motivated the argument with a toy model, which was con-structed by considering the case in which there was only one growing mode,and neglecting contributions to the evolution from Fourier modes other thanthe first and second modes. The toy model demonstrated how energy trans-fers between a growing mode and a decaying mode, achieving balance. Wemake two modifications to Goodman’s toy model: we have a small param-eter in front of the exponential growth term in the evolution equation forthe growing mode (this growth term was of unit size in [17]), and we allowgiven forcing as well. In this section, we develop bounds in Proposition 1and Proposition 2 that will be utilized in the induction argument in Section5 below.We study the following system(15) a t = ε i a + 8 π L i ab + Q , HE 2D KURAMOTO-SIVASHINSKY EQUATION 11 (16) b t = − B i b − π L i a + Q , for i ∈ { , } , where Q and Q are given function in time. For the remainderof the section, we fix a choice for i ∈ { , } . We will assume the followingbounds for Q and Q :(17) sup t ∈ [0 , ∞ ) | Q | ≤ Kε i , sup t ∈ [0 , ∞ ) | Q | ≤ Kε i . Proposition 1.
Assume (17) holds and let a and b solve (15) - (16) . Thereexists ε ∗ > such that for any value of ε i > satisfying ε i ∈ (0 , ε ∗ ) , if a (0) + b (0) ≤ M ,i ε i / , then a ( t ) + b ( t ) ≤ M ,i ε i for all t > .Proof. We define a Lyapunov function G ( a, b ) = 12 a + 2 b + L i ε i π b. Let us assume that G ( a, b ) ≥ M ,i ε i . Then we have that(18) 12 a + 2 b ≥ M ,i ε i − (cid:12)(cid:12)(cid:12)(cid:12) L i ε i bπ (cid:12)(cid:12)(cid:12)(cid:12) ≥ M ,i ε − b − L i ε i π ≥ M ,i ε i − b . For the first inequality, we have used that, by Young’s inequality,(19) (cid:12)(cid:12)(cid:12)(cid:12) L i ε i bπ (cid:12)(cid:12)(cid:12)(cid:12) ≤ b + L i ε i π , while for the last inequality we have used that it is possible to choose ε i small enough so that L i ε i π ≤ M ,i ε i . It then follows from (18) that12 a + 3 b ≥ M ,i ε i , from which we conclude that(20) a + b ≥ M ,i ε i . We next take the derivative of G with respect to time and use (15)-(16): G t = − ε i a − B i b − L i ε i B i bπ + aQ + 4 bQ + L i ε i π Q . We rewrite this expression as G t = Υ + Υ , where Υ and Υ are given byΥ = − ε i a − B i b − L i ε i B i bπ , Υ = − ε i a − B i b + aQ + 4 bQ + L i ε i Q π . We will show that Υ and Υ are nonpositive when a and b satisfy (20),at least for sufficiently small values of ε i . For Υ , it is enough to considerthe case b >
0, as Υ < b ≤
0. Next, we observe that if b > L i ε i π , then − B i b − L i ε i B i bπ < , and thus Υ < . The remaining case to consider is(21) 0 < b < L i ε i π . For ε i small enough, (20) and (21) together imply a ≥ M ,i ε i . Hence, if (21) holds, we may conclude the following bounds: − ε i a ≤ − M ,i ε i ,L i ε i B i bπ ≤ L i ε i B i π . Using that M ,i = B i L i π by definition, we have − ε i a − L i ε i B i bπ ≤ − M ,i ε i
24 + L i ε i B i π = 0 . We conclude that Υ < . We have shown then that Υ < . We estimate the terms containing Q and Q asfollows. By Young’s inequality, | aQ | ≤ ε i a Q ε i , which, combined with (17), gives | aQ | ≤ ε i a K ε i ε i = ε i a K ε i . We similarly bound 4 bQ as | bQ | ≤ B i b + 4 Q B i ≤ B i b + 16 K ε i B i . Again using (17), we bound the final term in Υ as (cid:12)(cid:12)(cid:12)(cid:12) L i ε i Q π (cid:12)(cid:12)(cid:12)(cid:12) ≤ L i Kε i π . These estimates in turn give the following bound on Υ :(22) Υ ≤ − ε i a − B i b + (cid:20) K ε i + 16 K ε i B i + 2 L i Kε i π (cid:21) . HE 2D KURAMOTO-SIVASHINSKY EQUATION 13
But we assumed that G ≥ M ,i ε i , which implies a + b ≥ M ,i ε i / − ε i a − B i b ≤ − ε i a − ε i b ≤ − M ,i ε i . Therefore, Υ < − M ,i ε i
24 + (cid:20) K ε i + 16 K ε i B i + 2 L i Kε i π (cid:21) . We can take ε i small enough so that (cid:12)(cid:12)(cid:12)(cid:12) K ε i + 16 K ε i B i + 2 L i Kε i π (cid:12)(cid:12)(cid:12)(cid:12) ≤ M ,i ε i . For such values of ε i , then, we have Υ < . We have concluded that G ≥ M ,i ε i implies G t < . Hence, if G is initiallyless than M ,i ε i , then necessarily G < M ,i ε i for all t > G < M ,i ε i initially. We observethat, from the definition of G and (19), G ≤ a + 2 b + b + L i ε i π ≤ a + b ) + L i ε i π . Consequently, G (0) < M ,i ε provided a (0) + b (0) ≤ M ,i ε i (which holds byhypothesis) and proveded ε i is taken small enough so that L i ε i π < M ,i ε i .Assuming then G ( t ) < M ,i ε i for all t >
0, we ask what can we say about a ( t ) + b ( t ). We again use the definition of G together with (19), nowfinding that M ,i ε i > a + 2 b + L i ε i bπ ≥ a + 2 b − (cid:12)(cid:12)(cid:12)(cid:12) L i ε i bπ (cid:12)(cid:12)(cid:12)(cid:12) ≥ a + 2 b − b − L i ε i π ≥ (cid:0) a + b (cid:1) − L i ε i π . Rearranging the left-hand and right-hand sides of this expression gives12 (cid:0) a + b (cid:1) < M ,i ε i + L i ε i π . We then take ε i small enough so that L i ε i π ≤ M ,i ε i . Finally, we conclude a + b < M ,i ε i . This completes the proof. (cid:3)
Proposition 2.
Under the hypotheses of Proposition 1, if also b (0) ≤ M ,i ε i / , then there exists ε ∗ > such that for any value of ε i ∈ (0 , ε ∗ ) , | b ( t ) | ≤ M ,i ε i for all t > . Proof.
From Proposition 1, we have ( a ( t )) ≤ M ,i ε i for all t >
0. From(17), we also have | Q ( t ) | ≤ Kε i for all t >
0. Using Duhamel’s Formula,we rewrite the equation for b in integral form: b ( t ) = e − B i t b (0) + e − B i t Z t e B i s (cid:20) π L i a ( s ) + Q ( s ) (cid:21) ds. We recall that M ,i = π M ,i L i , so that | b ( t ) | ≤ e − B i t | b (0) | + (cid:0) M ,i ε i + 2 Kε i (cid:1) e − B i t Z t e B i s ds. We evaluate the integral and bound the result as | b ( t ) | ≤ e − B i t | b (0) | + 1 B i (cid:0) M ,i ε i + 2 Kε i (cid:1) . Now B i is approximately equal to 12, since we are taking L i close to 2 π ; wemay thus say B i > . Lastly, by again taking ε i sufficiently small and fromthe hypothesis | b (0) | ≤ M ,i ε i , it follows that | b ( t ) | ≤ M ,i ε i , for all t > . (cid:3) The Duchon-Robert framework
In this section we develop the estimates we will use for the iterates w n . Wewill assume that w n belongs to suitable function spaces of analytic functionsin time based on the Wiener algebra. These spaces are Banach algebras andare well adapted to the inductive argument carried out in Section 5. Thebounds on w n follows from estimates on the semigroup generated by thelinearized operator and by estimating the integral in the mild formulationof the PDE, exploiting the algebra structure to control the nonlinearity.These spaces and similar bounds were used by Duchon and Robert [12] toprove the global existence of vortex sheet solutions in incompressible two-dimensional fluid flow.For m ∈ N and ρ ≥ , we define the space B mρ to be the space of distribu-tions on the torus for for which the following weighted sum of their Fouriercoefficients is finite: f ∈ B mρ ⇐⇒ k f k B mρ = X ( k,j ) ∈ Z e ρ ( | k | + | j | ) (1 + | k | + | j | ) m | f k,j | < ∞ . We also have a space-time version of this space, which we call B mρ , definedas the space of distributions on [0 , ∞ ) × T such that k g k B mρ = X ( k,j ) ∈ Z e ρ ( | k | + | j | ) (1 + | k | + | j | ) m sup t ∈ [0 , ∞ ) | g k,j ( t ) | < ∞ . We observe that elements of both B mρ and B mρ are actually functions thatare analytic in space for ρ > HE 2D KURAMOTO-SIVASHINSKY EQUATION 15
The spaces B ρ and B ρ are Banach algebras; indeed, B is exactly theWiener algebra. If f and g are both in B ρ , we have k f g k B ρ ≤ X ( k,j ) ∈ Z X ( ℓ,n ) ∈ Z e ρ ( | k − ℓ | + | j − n | ) e ρ ( | ℓ | + | n | ) sup t ∈ [0 , ∞ ) | f k − ℓ,j − n ( t ) | ! sup t ∈ [0 , ∞ ) | g ℓ,n ( t ) | ! ≤ k f k B ρ k g k B ρ . The analogous estimate for B ρ follows immediately by observing that B mρ consists precisely of the elements of B mρ that are constant in time. Then wemay conclude (simply by the product rule) that the spaces B ρ and B ρ arealso Banach algebras. Indeed, a function f is in B ρ if and only if f and itspartial derivatives ∂ x f and ∂ y f are all in B ρ . We note that we will not use the spaces B mρ or B mρ for m > , althoughthese are Banach algebras as well (for the same reasons).We define the operator I + by I + h ( · , t ) = P Z t e − (∆ +∆)( t − s ) h ( · , s ) ds, where the integral is intended in the Bochner sense and e − t (∆ +∆) denotesthe C (unbounded) semigroup generated by the linearized KSE operatoron B mρ . We will show that I + is bounded from B ρ to B ρ . (This is the onlyfact needed for our purposes, but the integral is actually bounded from B ρ to B ρ . ) Let h ∈ B ρ be given. Then the norm of I + h is given by k I + h k B ρ = X ( k,j ) / ∈ A e ρ ( | k | + | j | ) (1 + | k | + | j | ) sup t ∈ [0 , ∞ ) (cid:12)(cid:12)(cid:12)(cid:12)Z t exp { σ ( k, j )( t − s ) } h k,j ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) , where σ is defined in (13). The triangle inequality then implies k I + h k B ρ ≤ X ( k,j ) / ∈ A e ρ ( | k | + | j | ) (1 + | k | + | j | ) sup t ∈ [0 , ∞ ) exp { σ ( k, j ) t } ·· Z t exp {− σ ( k, j ) s } | h k,j ( s ) | ds. We take the supremum of | h k,j ( s ) | in s , which we can then pull out to obtain: k I + h k B ρ ≤ X ( k,j ) / ∈ A e ρ ( | k | + | j | ) sup t ∈ [0 , ∞ ) | h k,j ( t ) | " sup t ∈ [0 , ∞ ) sup ( k,j ) / ∈ A (1 + | k | + | j | ) exp { σ ( k, j ) t } Z t exp {− σ ( k, j ) s } ds ! . The first factor on the right-hand side can simply be bounded by k h k B ρ . Abound on the second factor (i.e., the double supremum) can be found bydirectly computing the integral, which gives: k I + h k B ρ ≤ k h k B ρ " sup t ∈ [0 , ∞ ) sup ( k,j ) / ∈ A (1 + | k | + | j | )(1 − exp { σ ( k, j ) t }− σ ( k, j ) . The negative term in the numerator can be neglected. Therefore, we have k I + h k B ρ ≤ K k h k B ρ , where K = sup ( k,j ) / ∈ A | k | + | j |− σ ( k, j ) . We now turn to proving estimates on the semigroup. We show that e ( − ∆ − ∆) t maps B ρ into B ρ boundedly. In fact, we first observe that k e ( − ∆ − ∆) t w k B ρ ≤ X ( k,j ) / ∈ A e ρ ( | k | + | j | ) (1 + | k | + | j | ) sup t ∈ [0 , ∞ ) exp { σ ( k, j ) t } | ( w ) k,j | . The supremum is achieved at t = 0 for every ( k, j ) / ∈ A. (Recall that w issupported in Fourier space only on wavenumbers in the complement of theset A. ) We therefore have(23) k e ( − ∆ − ∆) t w k B ρ ≤ X ( k,j ) / ∈ A e ρ ( | k | + | j | ) (1 + | k | + | j | ) | ( w ) k,j | = k w k B ρ . Inductive argument and convergence
We are now ready to complete the proof of Theorem 1. First, in Propo-sition 3 we obtain uniform bounds on the iterates by induction, using thebounds already established. Then in Theorem 2, we state a precise versionof our main result, existence of a global mild solution φ , which follows bypassing to the limit n → ∞ and using compactness arguments. Proposition 3.
Fix ρ > . Let ε = max { ε , ε } , where ε i , i = 1 , , is givenin (6) . Assume the initial data a , (0) , a , (0) , a , (0) , a , (0) , and w (0) satisfy ( a , (0)) + ( a , (0)) ≤ M , ε , ( a , (0)) + ( a , (0)) ≤ M , ε , | a , (0) | ≤ M , ε , | a , (0) | ≤ M , ε , k w (0) k B ρ ≤ M ε / . HE 2D KURAMOTO-SIVASHINSKY EQUATION 17
Then there exists ε ∗ such that for all i ∈ { , } , for all ε i ∈ (0 , ε ∗ ) , and forall n, the following bounds are satisfied: (24) sup t ∈ [0 , ∞ ) | a n , | ≤ M / , ε / , sup t ∈ [0 , ∞ ) | a n , | ≤ M / , ε / , (25) sup t ∈ [0 , ∞ ) | a n , | ≤ M , ε , sup t ∈ [0 , ∞ ) | a n , | ≤ M , ε , (26) k w n k B ρ ≤ M ε / . (27) sup t ∈ [0 , ∞ ) | F n , ,x | ≤ Kε , sup t ∈ [0 , ∞ ) | F n , ,y | ≤ Kε , (28) sup t ∈ [0 , ∞ ) | F n , ,x | ≤ Kε , sup t ∈ [0 , ∞ ) | F n , ,y | ≤ Kε . (29) sup t ∈ [0 , ∞ ) | F n , ,x | ≤ Kε , sup t ∈ [0 , ∞ ) | F n , ,y | ≤ Kε , (30) sup t ∈ [0 , ∞ ) | F n , ,x | ≤ Kε , sup t ∈ [0 , ∞ ) | F n , ,y | ≤ Kε . Proof.
We initialize our iterative scheme with a , = a , = 0 , a , = a , (0) = 0 , and w = 0 . The bounds (24), (25), (26), (27), (28), (29),and (30) are trivially satisfied by a , , a , , a , , a , , and w . We assume(24), (25), (26), (27), (28), (29), and (30), as our inductive hypothesis. Wenow prove the analogues of these for the next iterate. We recall the definitionof φ n from (10).An appeal to Proposition 1 with i = 1 immediately proves the desiredbound on a n +11 , , and another appeal to Proposition 1 with i = 2 immediatelyproves the desired bound on a n +10 , . Then appealing twice to Proposition 2again immediately proves the desired bounds on a n +12 , and a n +10 , . Next, we write the mild fomulation of the equation for w n +1 from (11).Since P is a projection, we may write P = P . Using the definition of I + introduced in Section 4, we have w n +1 = e ( − ∆ − ∆) t w + I + ( P (( φ nx ) + ( φ ny ) )) . Using the bounds developed in Section 4, we can then estimate w n +1 asfollows:(31) k w n +1 k B ρ ≤ k w k B ρ + K (cid:13)(cid:13) P (( φ nx ) ) (cid:13)(cid:13) B ρ + K (cid:13)(cid:13) P (( φ ny ) ) (cid:13)(cid:13) B ρ . In order to close the induction argument, we need to express φ nx and φ ny interms of the quantities we are estimating. To this end, we will use a different decomposition for ( φ nx ) and ( φ ny ) than the one used in Section 1.1. For thepartial derivative with respect to x, we write φ nx = − πa n , L sin (cid:18) πxL (cid:19) − πa n , L sin (cid:18) πxL (cid:19) + w nx , and decompose ( φ nx ) in the following way:(32) ( φ nx ) = Φ + Φ + Φ w nx + ( w nx ) , where Φ , Φ , and Φ are given byΦ = 4 π ( a n , ) L sin (cid:18) πxL (cid:19) , Φ = 16 π a n , a n , L sin (cid:18) πxL (cid:19) sin (cid:18) πxL (cid:19) + 16 π ( a n , ) L sin (cid:18) πxL (cid:19) , Φ = − πa n , L sin (cid:18) πxL (cid:19) − πa n , L sin (cid:18) πxL (cid:19) . One reason for this decomposition is that P Φ = 0 . Another reason is thatthe term Φ is larger than the remaining terms; Φ , Φ w nx , and ( w nx ) areall of order ε / or smaller, while the same is not true for Φ . We now estimate (cid:13)(cid:13) P (( φ nx ) ) (cid:13)(cid:13) B ρ , using the fact that P Φ = 0 and thefact P is bounded:(33) (cid:13)(cid:13) P (( φ nx ) ) (cid:13)(cid:13) B ρ ≤ k Φ n k B ρ + k Φ n k B ρ k w n k B ρ + k w n k B ρ . Above, we have also used the algebra property for B ρ and that k w nx k B ρ ≤k w n k B ρ , which is a direct consequence of the definition.We then can make some straightforward estimates of Φ and Φ . For Φ we have k Φ k B ρ ≤ M / , M , K ε / + M , K ε . Of course, to get this bound, we have employed the inductive hypothesis andthe fact that ε ≤ ε. We recall that M ≤ K (2 M / , M , K ) to concludethat k Φ k B ρ ≤ M K ε / + M , K ε . We take ε small enough so that M , K ε ≤ M K ε / . We thus have(34) k Φ k B ρ ≤ M K ε / . We next turn to bounding Φ . By using again the inductive hypothesisand the definition of the constant K , it is straightforward that k Φ k B ρ ≤ M / , K ε / + M , K ε. HE 2D KURAMOTO-SIVASHINSKY EQUATION 19
Another application of the inductive hypothesis gives that k Φ k B ρ k w n k B ρ ≤ M / , M K ε + M , M K ε / . We take ε small enough so that2 M / , M K ε + M , M K ε / ≤ M K ε / . We then have(35) k Φ k B ρ k w n k B ρ ≤ M K ε / . We proceed in a similar fashion to bound the quadratic term: k w n k B ρ ≤ M ε . We take ε small enough so that(36) M ε ≤ M K ε / . We then have(37) k w n k B ρ ≤ M K ε / . Having concluded our treatment of ( φ nx ) , we now consider ( φ ny ) . We maytreat ( φ ny ) analogously to the way we treated ( φ nx ) in (32), leading to thedecomposition ( φ ny ) = Φ + Φ + Φ w ny + ( w ny ) , with the formulas Φ = 4 π ( a n , ) L sin (cid:18) πyL (cid:19) , Φ = 16 π a n , a n , L sin (cid:18) πyL (cid:19) sin (cid:18) πyL (cid:19) + 16 π ( a n , ) L sin (cid:18) πyL (cid:19) , Φ = − πa n , L sin (cid:18) πyL (cid:19) − πa n , L sin (cid:18) πyL (cid:19) . Similarly to the case for Φ , we have P Φ = 0 . We may then bound ( φ ny ) as(38) k P ( φ ny ) k B ρ ≤ k Φ k B ρ + k Φ k B ρ k w n k B ρ + k w n k B ρ . We then proceed to estimating the remaining terms in a manner analogousto the previous case, omitting details:(39) k Φ k B ρ ≤ M K ε / , (40) k Φ k B ρ k w n k B ρ ≤ M K ε / , (41) k w n k B ρ ≤ M K ε / , where ε is chosen sufficiently small.We recall the condition on the initial data for w, namely,(42) k w k B ρ ≤ M ε / . We then combine (31), (33), (34), (35), (37), (38), (39), (40), (41), and (42)to conclude(43) k w n +1 k B ρ ≤ M ε / . It remains to demonstrate the estimates for the forcing terms; we will showthe argument for F n +11 , ,x and F n +12 , ,x in detail, and omit the details for F n +11 , ,y ,F n +12 , ,y , F n +10 , ,x , F n +10 , ,y , F n +10 , ,x , and F n +10 , ,y , as they are completely analogous.We begin with F n +11 , ,x . We introduce another decomposition of ( φ n +1 x ) , analogous to that in Section 1.1, namely( φ n +1 x ) = X i =1 Ψ n +1 i , with the terms Ψ n +1 i defined asΨ n +11 = 4 π ( a n +11 , ) L sin (cid:18) πxL (cid:19) , Ψ n +12 = 16 π ( a n +12 , ) L sin (cid:18) πxL (cid:19) , Ψ n +13 = ( w n +1 x ) , Ψ n +14 = 16 π a n +11 , a n +12 , L sin (cid:18) πxL (cid:19) sin (cid:18) πxL (cid:19) , Ψ n +15 = − πa n +11 , L w n +1 x sin (cid:18) πxL (cid:19) , Ψ n +16 = − πa n +12 , L w n +1 x sin (cid:18) πxL (cid:19) . Notice that P , Ψ n +11 = P , Ψ n +12 = 0 . Also there is no contribution to F n +11 , ,x from Ψ n +14 since this term is explicitly accounted for in the a n +11 , evolutionequation. We thus have F n +11 , ,x cos (cid:18) πxL (cid:19) = P , (cid:2) Ψ n +13 + Ψ n +15 + Ψ n +16 (cid:3) . We treat this term now as we treated the previous F -terms, namely,(44) | F n +11 , ,x | ≤ k Ψ n +13 k B ρ + k Ψ n +15 k B ρ + k Ψ n +16 k B ρ (cid:13)(cid:13)(cid:13) cos (cid:16) πxL (cid:17)(cid:13)(cid:13)(cid:13) B ρ . HE 2D KURAMOTO-SIVASHINSKY EQUATION 21
For Ψ n +13 we have the estimate(45) k Ψ n +13 k B ρ ≤ k w n +1 x k B ρ ≤ k w n +1 k B ρ ≤ M ε . For Ψ n +15 we again use (14), as well as the inductive hypothesis, finding(46) k Ψ n +15 k B ρ ≤ M / , M K ε . The next term, Ψ n +16 , is similar, and we bound it as(47) k Ψ n +16 k B ρ ≤ M , M K ε / . From the definition of K we have the bound K ≤ M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πxL (cid:17)(cid:13)(cid:13)(cid:13) B ρ . By exploiting this bound, combining (44), (45), (46), and (47), and taking ε sufficiently small, we have | F n +11 , ,x | ≤ Kε . We next prove the corresponding bound for F n +12 , ,x . There is no contri-bution from Ψ n +11 to F n +12 , ,x , because this contribution has already been ac-counted for in the a n +12 , evolution equation. There is no contribution fromΨ n +12 or Ψ n +14 to F n +12 , ,x , because P , Ψ n +12 = P , Ψ n +14 = 0 . We then canestimate, as before, | F n +12 , ,x | ≤ k Ψ n +13 k B ρ + k Ψ n +15 k B ρ + k Ψ n +16 k B ρ (cid:13)(cid:13)(cid:13) cos (cid:16) πxL (cid:17)(cid:13)(cid:13)(cid:13) B ρ . We have already estimated Ψ n +13 , Ψ n +15 , and Ψ n +16 , and we reuse thesebounds. Now we bound K differently, namely, K ≤ M / , M K (cid:13)(cid:13)(cid:13) cos (cid:16) πxL (cid:17)(cid:13)(cid:13)(cid:13) B ρ , which follows again from its definition. As before, taking ε sufficiently small,we have | F n +12 , ,x | ≤ Kε . We omit the details of the bounds for F n +11 , ,y , F n +12 , ,y , F n +10 , ,x , F n +10 , ,y , F n +10 , ,x , and F n +10 , ,y , as already stated. Therefore, this completes the proof. (cid:3) We now state our main theorem.
Theorem 2.
Let ¯ ψ ∈ R be given. Fix < T < ∞ . Let ε i = − (cid:16) πL i (cid:17) + (cid:16) πL i (cid:17) , and set ε = max { ε , ε } . There exists ε ∗ > such that if ε ∈ (0 , ε ∗ ) ,and if ( a , (0)) + ( a , (0)) ≤ M , ε / , | a , (0) | ≤ M , ε / , ( a , (0)) + ( a , (0)) ≤ M , ε / , | a , (0) | ≤ M , ε / , k w (0) k B ρ ≤ M ε / / , then the Kuramoto-Sivashinsky equation (1) on the torus T = [0 , L ] × [0 , L ] with initial data ψ ( x, y,
0) = ¯ ψ + a , (0) cos (cid:18) πxL (cid:19) + a , (0) cos (cid:18) πxL (cid:19) + a , (0) cos (cid:18) πyL (cid:19) + a , (0) cos (cid:18) πyL (cid:19) + w ( x, y ) has a mild solution that is analytic in space on [0 , T ] .Proof. By Proposition 3, the family { φ n } n ∈ N , where φ n is given in (10), isa uniformly bounded family of functions analytic in space, with radius ofanalyticity ρ independent of t , and continuous and bounded in t ∈ [0 , ∞ ).Upon passing to a subsequence if necessary, not relabeled, we may thus finda limit as n → ∞ that is analytic in space by Montel’s Theorem, continuousand bounded in time. This is enough regularity to pass to the limit in themild formulation of the evolution equations. Thus the limit of the iterates, φ, exists and solves (2) on [0 , ∞ ).We next turn to show the existence of ψ solving the 2D KSE (1) on[0 , T ] for an arbitrary 0 < T < ∞ . We may write ψ = φ + ¯ ψ , where ¯ ψ solves (3) with initial condition ¯ ψ . As noted in the introduction, the initialvalue problem for ¯ ψ can be solved on the time interval [0 , T ] as long as φ ∈ L ([0 , T ]; ˙ H ), which is the case given the regularity established aboveon φ . (cid:3) Notice that we state this theorem on the interval [0 , T ] for arbitrary T, rather than using the interval [0 , ∞ ) . This is because we do not know thatthe mean, ¯ ψ, has a well-defined limit as t → ∞ since the mild formulation for φ does not imply that φ ∈ L ([0 , ∞ ); ˙ H ). Nevertheless we have achievedthe stated goal, showing that the two-dimensional Kuramoto-Sivashinskyequation has small solutions for all time in the presence of two linearlygrowing Fourier modes, one in each direction. Remark 1.
We note that we have not always used ε as opposed to theindividual parameters ε and ε ; this is because bounding a , and a , with HE 2D KURAMOTO-SIVASHINSKY EQUATION 23 ε and bounding a , and a , with ε is a slightly more detailed descriptionof the dynamics, although bounding them just in terms of ε = max { ε , ε } could have instead provided a more streamlined exposition. Remark 2.
As we have stated, our prior work [7] proved global existencewhen L and L are each in the interval (0 , π ) , while here we have shownglobal existence when they are each slightly larger than π. The present re-sults do also extend to the case when only one of these lengths is slightlylarger than π, and the other is smaller. We also expect that argumentssimilar to those in this work and in [7] will give the existence of a mild so-lution with initial data in L under hypotheses akin to those in Theorem 2.For brevity and clarity, we chose not to pursue this result here. Conclusion
We have shown the existence of small solutions of the Kuramoto-Sivashinskyequation in two space dimensions for all time, when the size of the domainadmits a linearly growing mode in each direction. To our knowledge, thisis the first result of this kind. The method of proof is new, in combininga dynamical systems approach for a finite number of modes with functionspace estimates for the remaining infinitely many modes. This approachraises the possibility that significant further progress could be possible, ex-tending beyond the present case of a pair of slightly growing modes, bydesigning different Lyapunov functions or making different choices of func-tion spaces. Another possible area of extension is extracting more detailedinformation about the solutions; while we showed that w ∼ ε / , a finerdescription of amplitudes could be made for the modes encompassed by w. The method of the present work could also be extended to other systems,including more fundamental systems in flame propagation. That is, theKuramoto-Sivashinksy equation is a weakly nonlinear model, and can beproved to be a valid approximation for coordinate-free models; global exis-tence of solutions for these coordinate-free and other models is of interest[1], [6], [14].
References [1] B.F. Akers and D.M. Ambrose. Efficient computation of coordinate-free models offlame fronts. 2020. Preprint.[2] D.M. Ambrose. Small strong solutions for time-dependent mean field games with localcoupling.
C. R. Math. Acad. Sci. Paris , 354(6):589–594, 2016.[3] D.M. Ambrose. Strong solutions for time-dependent mean field games with non-separable Hamiltonians.
J. Math. Pures Appl. (9) , 113:141–154, 2018.[4] D.M. Ambrose. The radius of analyticity for solutions to a problem in epitaxial growthon the torus.
Bull. Lond. Math. Soc. , 51(5):877–886, 2019.[5] D.M. Ambrose, J.L. Bona, and T. Milgrom. Global solutions and ill-posedness for theKaup system and related Boussinesq systems.
Indiana Univ. Math. J. , 68(4):1173–1198, 2019.[6] D.M. Ambrose, F. Hadadifard, and J.D. Wright. Well-posedness and asymptotics ofa coordinate-free model of flame fronts. 2020. Preprint. arXiv:2010.00737. [7] D.M. Ambrose and A.L. Mazzucato. Global existence and analyticity for the 2DKuramoto-Sivashinsky equation.
J. Dynam. Differential Equations , 31(3):1525–1547,2019.[8] H. Bellout, S. Benachour, and E.S. Titi. Finite-time singularity versus global regular-ity for hyper-viscous Hamilton-Jacobi-like equations.
Nonlinearity , 16(6):1967–1989,2003.[9] S. Benachour, I. Kukavica, W. Rusin, and M. Ziane. Anisotropic estimates for thetwo-dimensional Kuramoto-Sivashinsky equation.
J. Dynam. Differential Equations ,26(3):461–476, 2014.[10] J.C. Bronski and T.N. Gambill. Uncertainty estimates and L bounds for theKuramoto-Sivashinsky equation. Nonlinearity , 19(9):2023–2039, 2006.[11] J. Campos, O. Duque, and G. Rodr´ıguez-Blanco. The Cauchy problem associatedwith a periodic two-dimensional Kuramoto-Sivashinsky type equation.
Rev. Colom-biana Mat. , 45(1):1–17, 2011.[12] J. Duchon and R. Robert. Global vortex sheet solutions of Euler equations in theplane.
J. Differential Equations , 73(2):215–224, 1988.[13] Y. Feng and A.L. Mazzucato. Global existence for the two-dimensional Kuramoto-Sivashinsky equation with advection. 2020. Preprint. arXiv:2009.04029.[14] M.L. Frankel and G.I. Sivashinsky. On the nonlinear thermal diffusive theory of curvedflames.
J. Physique , 48:25–28, 1987.[15] L. Giacomelli and F. Otto. New bounds for the Kuramoto-Sivashinsky equation.
Comm. Pure Appl. Math. , 58(3):297–318, 2005.[16] M. Goldman, M. Josien, and F. Otto. New bounds for the inhomogenous Burg-ers and the Kuramoto-Sivashinsky equations.
Comm. Partial Differential Equations ,40(12):2237–2265, 2015.[17] J. Goodman. Stability of the Kuramoto-Sivashinsky and related systems.
Comm.Pure Appl. Math. , 47(3):293–306, 1994.[18] A. Kalogirou, E.E. Keaveny, and D.T. Papageorgiou. An in-depth numerical study ofthe two-dimensional Kuramoto-Sivashinsky equation.
Proc. A. , 471(2179):20140932,20, 2015.[19] Y. Kuramoto and T. Tsuzuki. Persistent propagation of concentration waves in dis-sipative media far from thermal equilibrium.
Progress of Theoretical Physics , 55:356–369, 1976.[20] A. Larios and K. Yamazaki. On the well-posedness of an anisotropically-reduced two-dimensional Kuramoto-Sivashinsky equation.
Phys. D , 411:132560, 14, 2020.[21] T. Milgrom and D.M. Ambrose. Temporal boundary value problems in interfacialfluid dynamics.
Appl. Anal. , 92(5):922–948, 2013.[22] L. Molinet. A bounded global absorbing set for the Burgers-Sivashinsky equation inspace dimension two.
C. R. Acad. Sci. Paris S´er. I Math. , 330(7):635–640, 2000.[23] L. Molinet. Local dissipativity in L for the Kuramoto-Sivashinsky equation in spatialdimension 2. J. Dynam. Differential Equations , 12(3):533–556, 2000.[24] B. Nicolaenko, B. Scheurer, and R. Temam. Some global dynamical properties ofthe Kuramoto-Sivashinsky equations: Nonlinear stability and attractors.
Phys. D ,16(2):155–183, 1985.[25] F. Otto. Optimal bounds on the Kuramoto-Sivashinsky equation.
J. Funct. Anal. ,257(7):2188–2245, 2009.[26] F.C. Pinto. Nonlinear stability and dynamical properties for a Kuramoto-Sivashinskyequation in space dimension two.
Discrete Contin. Dynam. Systems , 5(1):117–136,1999.[27] G.R. Sell and M. Taboada. Local dissipativity and attractors for the Kuramoto-Sivashinsky equation in thin 2D domains.
Nonlinear Anal. , 18(7):671–687, 1992.[28] G.I. Sivashinsky. Nonlinear analysis of hydrodynamic instability in laminar flames. I.Derivation of basic equations.
Acta Astronaut. , 4(11-12):1177–1206, 1977.
HE 2D KURAMOTO-SIVASHINSKY EQUATION 25 [29] E. Tadmor. The well-posedness of the Kuramoto-Sivashinsky equation.
SIAM J.Math. Anal. , 17(4):884–893, 1986.[30] R.J. Tomlin, A. Kalogirou, and D.T. Papageorgiou. Nonlinear dynamics of a disper-sive anisotropic Kuramoto-Sivashinsky equation in two space dimensions.
Proc. A. ,474(2211):20170687, 19, 2018.
Department of Mathematics, Drexel University, Philadelphia, PA 19104,USA
Email address : [email protected] Department of Mathematics, Penn State University, University Park, PA16802, USA
Email address ::