A Hopf bifurcation in the planar Navier-Stokes equations
aa r X i v : . [ m a t h . A P ] S e p A Hopf bifurcation in the planar Navier-Stokes equations
Gianni Arioli , and Hans Koch Abstract.
We consider the Navier-Stokes equation for an incompressible viscous fluid on asquare, satisfying Navier boundary conditions and being subjected to a time-independent force.As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo aHopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof isconstructive and uses computer-assisted estimates.
1. Introduction and main result
We consider the Navier-Stokes equations ∂ t u − ν ∆ u + ( u · ∇ ) u + ∇ p = f , ∇ · u = 0 on Ω , (1 . u = u ( t, x, y ) of an incompressible fluid on a planar domain Ω, satisfyingsuitable boundary conditions for ( x, y ) ∈ ∂ Ω and initial conditions at t = 0. Here, p denotes the pressure, and f = f ( x, y ) is a fixed time-independent external force.Our focus is on solution curves and bifurcations as the kinematic velocity ν is beingvaried. In order to reduce the complexity of the problem, the domain Ω is chosen to beas simple as possible, namely the square Ω = (0 , π ) . Following [21], we impose Navierboundary conditions on ∂ Ω, which are given by u = ∂ x u = 0 on { , π } × (0 , π ) ,u = ∂ y u = 0 on (0 , π ) × { , π } . (1 . f that satisfies( ∂ x f − ∂ y f )( x, y ) = 5 sin( x ) sin(2 y ) −
13 sin(3 x ) sin(2 y ) . (1 . Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano. Supported in part by the PRIN project “Equazioni alle derivate parziali e disuguaglianze analitico-geometriche associate”. Department of Mathematics, The University of Texas at Austin, Austin, TX 78712. ARIOLI & KOCH was given in [6] for the rotating B´enard problem. A proof exists also for the Couette-Taylor problem [7,9]. Sufficient conditions for the existence of a Hopf bifurcation in aNavier-Stokes setting are presented in [13].Before giving a precise statement of our result, let us replace the vector field u in theequation (1.1) by ν − u . The equation for the rescaled function u is α∂ t u − ∆ u + γ ( u · ∇ ) u + ∇ p = f , ∇ · u = 0 on Ω , (1 . γ = ν − . The value of α that corresponds to (1.1) is ν − , but this can be changedto any positive value by rescaling time.Numerically, it is possible to find stationary solutions of (1.4) for a wide range ofvalues of the parameter γ . At a value γ ≈ . . . . we observe a Hopf bifurcationthat leads to a branch of periodic solutions for γ > γ .For a fixed value of α , the time period τ of the solution varies with γ . Instead oflooking for τ -periodic solution of (1.4) for fixed α , we look for 2 π -periodic solutions, where α = 2 π/τ has to be determined. To simplify notation, a 2 π -periodic function will beidentified with a function on the circle T = R / (2 π Z ). Our main theorem is the following. Theorem 1.1.
There exists a real number γ = 83 . . . . , an open interval I including γ , and a real analytic function ( γ, x, y ) u γ ( x, y ) from I × Ω to R , such that u γ is astationary solution of (1.4) and (1.2) for each γ ∈ I . In addition, there exists a real number α = 4 . . . . , an open interval J centered at the origin, two real analytic functions γ and α on J that satisfy γ (0) = γ and α (0) = α , respectively, as well as two realanalytic functions ( s, t, x, y ) u s, e ( t, x, y ) and ( s, t, x, y ) u s, o ( t, x, y ) from J × T × Ω to R , such that the following holds. For any given β ∈ C satisfying β ∈ J , the vector field u = u s, e + βu s, o with s = β is a solution of (1.4) and (1.2) with γ = γ ( s ) and α = α ( s ) .Furthermore, u , e ( t, . , . ) = u γ and ∂ t u , o ( t, . , . ) = 0 . To our knowledge, this is the first result establishing the existence of a Hopf bifurcationfor the Navier-Stokes equation in a stationary environment.Our proof of this theorem is computer-assisted. The solutions are obtained by rewrit-ing (1.4) and (1.2) as a suitable fixed point equation for scalar vorticity of u . Here wetake advantage of the fact that the domain is two-dimensional. We isolate the periodicbranch from the stationary branch by using a scaling that admits two distinct limits at thebifurcation point. This approach is also known as the blow-up method, which is a commontool in the study of singularities and bifurcations [8].Computer-assisted methods have been applied successfully to many different problemsin analysis, mostly in the areas of dynamical systems and partial differential equations.Here we will just mention work that concerns the Navier-Stokes equation or Hopf bifur-cations. For the Navier-Stokes equation, the existence of symmetry-breaking bifurcationsamong stationary solutions has been established in [10,21]. Periodic solutions for theNavier-Stokes flow in a stationary environment have been obtained in [20]. In the case ofperiodic forcing, the problem of existence and stability of periodic orbits has been inves-tigated in [14]. Concerning the existence of Hopf bifurcations, a computer-assisted proofwas given recently in [22] for a finite-dimensional dynamical system; and an extension of opf bifurcation in Navier-Stokes t = 0 and t = π of a solution u : T × Ω → R of theequations (1.4) with boundary conditions (1.2) and forcing (1.3), obtained numerically forthe parameter value γ ≈ . . . . . Figure 1.
Snapshots at two distinct times of a time-periodic solution for γ ≈ . . . . As mentioned earlier, a system similar to the one considered here is known to exhibita symmetry-breaking bifurcation within the class of stationary solutions [21]. The brokensymmetry is y π/ − y . Based on a numerical computation of eigenvalues, we expectan analogous bifurcation to occur here at γ ≈ γ . We have not tried to prove theexistence of a symmetry-breaking bifurcation for the forcing (1.3), since such an analysiswould duplicate the work in [21] and go beyond the scope of the present paper.The remaining part of this paper is organized as follows. In Section 2, we first rewrite(1.4) as an equation for the function Φ = ∂ y u − ∂ x u , which is the scalar vorticity of − u . After a suitable scaling Φ = U β φ , the problem of constructing the solution branchesdescribed in Theorem 1.1 is reduced to three fixed point problems for the function φ . Thesefixed point equations are solved in Section 3, based on estimates described in Lemmas 3.3,3.4, and 3.6. Section 4 is devoted to the proof of these estimates, which involves reducingthem to a large number of trivial bounds that can be (and have been) verified with theaid of a computer [24].
2. Fixed point equations
The goal here is to rewrite the equation (1.4) with boundary conditions (1.2) as a fixedpoint problem. Applying the operator ∂ : ( u , u ) ∂ u − ∂ u on both sides of theequation (1.4), we obtain α∂ t Φ − ∆Φ + γu · ∇ Φ = ∂ f , Φ = ∂ u . (2 . ∂ ( u · ∇ ) u = u · ∇ Φ. Using the divergence-free condition ∇ · u = 0,one also finds that ∆ u = J ∇ Φ , J = h − i . (2 . ARIOLI & KOCH
If Φ vanishes on the boundary of ∂ Ω, then the equation (2.2) can be inverted to yield u = ∂ − Φ def = J ∇ ∆ − Φ , (2 . t, x, y ) = X j,k ∈ N Φ j,k ( t ) sin( jx ) sin( ky ) , (2 . T . Here, and inwhat follows, N denotes the set of all positive integers. If Φ admits such an expansion,then the equation (2.3) yields u ( t, x, y ) = X j,k ∈ N − kj + k Φ j,k ( t ) sin( jx ) cos( ky ) ,u ( t, x, y ) = X j,k ∈ N jj + k Φ j,k ( t ) cos( jx ) sin( ky ) . (2 . u = ( u , u ) satisfies theNavier boundary conditions (1.2). So a solution u of (1.4) and (1.2) can be obtained via(2.5) from a solution Φ of the equation (2.1). For convenience, we write (2.1) as( α∂ t − ∆)Φ + γ L (Φ)Φ = ∂ f , (2 . L is the symmetric bilinear form defined by L ( φ ) ψ = ( ∇ φ ) · 6 ∂ − ψ + ( ∇ ψ ) · 6 ∂ − φ . (2 . j,k in the series (2.4) are 2 π -periodic functions and thus admit an expan-sion Φ j,k = X n ∈ Z Φ n,j,k cosi n , cosi n ( t ) = (cid:26) cos( nt ) if n ≥ − nt ) if n <
0. (2 . N the set of all nonnegative integers. For any subset N ⊂ N we define E N Φ = X n ∈ Z | n |∈ N X j,k ∈ N Φ n,j,k cosi n × sin j × sin k , (2 . m ( z ) = sin( mz ). In particular, the even frequency part Φ e (odd frequency partΦ o ) of Φ is defined to be the function E N Φ, where N is the set of all even (odd) nonnegativeintegers. This leads to the decomposition Φ = Φ e + Φ o that will be used below.To simplify the discussion, consider first non-stationary periodic solutions. For γ nearthe bifurcation point γ , we expect Φ to be nearly time-independent. So in particular,Φ o is close to zero. Consider the function φ = φ e + φ o obtained by setting φ e = Φ e and opf bifurcation in Navier-Stokes φ o = β − Φ o . The scaling factor β = 0 will be chosen below, in such a way that φ e and φ o are of comparable size. SubstitutingΦ = U β φ def = φ e + βφ o (2 . α∂ t − ∆) φ + γ L s ( φ ) φ = ∂ f , (2 . s = β and L s ( φ ) ψ = L ( φ e ) ψ e + L ( φ e ) ψ o + L ( φ o ) ψ e + s L ( φ o ) ψ o . (2 . α∂ t − ∆ toboth sides. Setting g = ( − ∆) − ∂ f , the resulting equation is ˜ φ = φ , where˜ φ = g − γ | ∆ | / ( α∂ t − ∆) − ˆ φ , ˆ φ def = | ∆ | − / L s ( φ ) φ . (2 . φ , , = 0. Inaddition, we choose β = θ − Φ − , , , where θ is some fixed constant that will be specifiedlater. This leads to the normalization conditions Aφ def = φ , , = 0 , Bφ def = φ − , , = θ . (2 . β enters our main equation ˜ φ = φ only via its square s = β . It is convenient toregard s to be the independent parameter and express γ as a function of s . The functions γ = γ ( s ) and α = α ( s ) are determined by the condition that ˜ φ satisfies the normalizationconditions (2.14). Applying the functionals A and B to both sides of (2.11), using theidentities A ∆ = − A , A∂ t = B , B ∆ = − B , B∂ t = − A , and imposing the conditions A ˜ φ = 0 and B ˜ φ = θ , we find that γ = − / θB ˆ φ , α = 2 A ˆ φB ˆ φ . (2 . s , define F s ( φ ) = ˜ φ , where ˜ φ is given by (2.13), with γ = γ ( s, φ )and α = α ( s, φ ) determined by (2.15). The fixed point equation for F s is used to findnon-stationary time-periodic solutions of (2.11). Remark 1.
The choice (2.15) guarantees that A ˜ φ = 0 and B ˜ φ = θ , even if φ does notsatisfy the normalization conditions (2.14). Thus, the domain of the map F s can includenon-normalized function φ . (The same is true for the map F γ described below.) But afixed point of this map will be normalized by construction.In order to determine the bifurcation point γ and the corresponding frequency α ,we consider the map F : φ ˜ φ given by (2.13) with s = 0. The values of γ and α are ARIOLI & KOCH again given by (2.15), so that A ˜ φ = 0 and B ˜ φ = θ . We will show that this map F has afixed point φ with the property that φ n,j,k = 0 whenever | n | >
1. The values of γ and α for this fixed point define γ and α .A similar map F γ : φ ˜ φ , given by (2.13) with s = 0, is used to find stationarysolutions of the equation (2.6). In this case, the value of γ is being fixed, and φ o is takento be zero. The goal is to show that this map F γ has a fixed point φ γ that is independentof time t . Then Φ = φ γ is a stationary solution of (2.6).We finish this section by computing the derivative of the map F s described after (2.15).The resulting expressions will be needed later. Like some of the above, the following ispurely formal. A proper formulation will be given in the next section. For simplicity,assume that φ depends on a parameter. The derivative of a quantity q with respect to thisparameter will be denoted by ˙ q . Define L α = | ∆ | / ( α∂ t − ∆) − , L ′ α = ∂ t ( α∂ t − ∆) − . (2 . F s ( φ ) = g − γ L α ˆ φ with ˆ φ = | ∆ | − / L s ( φ ) φ , the parameter-derivative of F s ( φ )is given by D F s ( φ ) ˙ φ = − L α h(cid:0) ˙ γ − γ ˙ α L ′ α (cid:1) ˆ φ + γ ˙ˆ φ i , ˙ˆ φ = 2 | ∆ | − / L s ( φ ) ˙ φ , (2 . γ = 2 − / γ θ B ˙ˆ φ , ˙ α = 2 − / αγθ B ˙ˆ φ − − / γθ A ˙ˆ φ . (2 . γ and ˙ α are obtained by differentiating (2.15).
3. The associated contractions
In this section, we formulate the fixed point problems for the maps F , F γ , and F s in asuitable functional setting. The goal is to reduce the problems to a point where we caninvoke the contraction mapping theorem. After describing the necessary estimates, we givea proof of Theorem 1.1 based on these estimates.We start by defining suitable function spaces. Given a real number ρ >
1, denote by A the space of all functions h ∈ L ( T ) that have a finite norm k h k , where k h k = | h | + X n ∈ N p | h n | + | h − n | ρ n , h = X n ∈ Z h n cosi n . (3 . n are the trigonometric function defined in (2.8). It is straightforward to checkthat A is a Banach algebra under the pointwise product of functions. That is, k gh k ≤k g kk h k for any two functions g, h ∈ A . We also identify functions on T with 2 π -periodicfunctions on R . In this sense, a function in A extends analytically to the strip T ( ρ ) = { z ∈ C : | Im z | < log ρ } . opf bifurcation in Navier-Stokes ̺ >
1, we denote by B the space of all function Φ : T → A thatadmit a representation (2.4) and have a finite norm k Φ k = X j,k ∈ N k Φ j,k k ̺ j + k . (3 . x, y ) ( t Φ( t, x, y )) in this space will also be identified with a function( t, x, y ) Φ( t, x, y ) on T , or with a function on R that is 2 π -periodic in each argument.In this sense, every function in B extends analytically to T ( ρ ) × T ( ̺ ) .We consider A and B to be Banach spaces over F ∈ { R , C } . In the case F = R , thefunctions in these spaces are assumed to take real values for real arguments.Clearly, a function Φ ∈ B admits an expansion (2.9) with N = N . The sequence ofFourier coefficients Φ n,k,j converges to zero exponentially as | n | + j + k tends to infinity. Ifall but finitely many of these coefficients vanish, then Φ is called a Fourier polynomial. Theequation (2.9) with N ⊂ N non-empty defines a continuous projection E N on B whoseoperator norm is 1. Using Fourier series, it is straightforward to see that the equation(2.16) defines two bounded linear operators L α and L ′ α on B , for every α ∈ C . Theoperator L α is in fact compact. Specific estimates will be given in Section 4. The followingwill be proved in Section 4 as well. Proposition 3.1. If Φ and φ belong to B , then so does | ∆ | − / L (Φ) φ , and (cid:13)(cid:13) | ∆ | − / L (Φ) φ (cid:13)(cid:13) ≤ (cid:13)(cid:13) | ∆ | − / Φ (cid:13)(cid:13) k φ k + k Φ k (cid:13)(cid:13) | ∆ | − / φ (cid:13)(cid:13) . (3 . φ ˜ φ , given by (2.13) for fixedvalues of s , γ and α , is well-defined and compact as a map from B to B .As is common in computer-assisted proofs, we reformulate the fixed point equationfor the map φ ˜ φ as a fixed point problem for an associated quasi-Newton map. Sincewe need three distinct versions of this map, let us first describe a more general setting.Let F : D → B be a C map defined on an open domain D in a Banach space B .Let h ϕ + Lh be a continuous affine map on B . We define quasi-Newton map N for( D , F , ϕ, L ) by setting N ( h ) = F ( ϕ + Lh ) − ϕ + (I − L ) h . (3 . N is defined to be the set of of all h ∈ B with the property that ϕ + Lh ∈ D .Notice that, if h is a fixed point of N , then ϕ + Lh is a fixed point of F . In our applications, ϕ is an approximate fixed point of F and L is an approximate inverse of I − D F ( ϕ ).The following is an immediate consequence of the contraction mapping theorem. Proposition 3.2.
Let F : D → B be a C map defined on an open domain in a Banachspace B . Let h ϕ + Lh be a continuous affine map on B . Assume that the quasi-Newtonmap (3.4) includes a non-empty ball B δ = { h ∈ B : k h k < δ } in its domain, and that kN (0) k < ε , k D N ( h ) k < K , h ∈ B δ , (3 . ARIOLI & KOCH where ε, K are positive real numbers that satisfy ε + Kδ < δ . Then F has a fixed point in ϕ + LB δ . If L is invertible, then this fixed point is unique in ϕ + LB δ . In our applications below, B is always a subspace of B . The domain parameter ρ andthe constant θ that appears in the normalization condition (2.14) are chosen to have thefixed values ρ = 2 , θ = 2 − . (3 . ̺ is defined implicitly in our proofs. That is, the lemmas belowhold for ̺ > γ and the associatedfrequency α . Let B = E { , } B over R . For every δ > B δ = { h ∈ B : k h k < δ } .Let s = 0, and denote by D the set of all functions φ ∈ B with the property that B ˆ φ = 0.Define F : D → B to be the map φ ˜ φ given by (2.13), with γ = γ ( φ ) and α = α ( φ )defined by the equation (2.15). Clearly, F is not only C but real analytic on D . Lemma 3.3.
With F as described above, there exists an affine isomorphism h ϕ + L h of B and real numbers ε, δ, K > satisfying ε + Kδ < δ , such that the following holds. Thequasi-Newton map N associated with ( B , F , ϕ, L ) includes the ball B δ in its domain andsatisfies the bounds (3.5). The domain of F includes the ball in B of radius r = δ k L k ,centered at ϕ . For every function φ in this ball, γ ( φ ) = 83 . . . . and α ( φ ) =4 . . . . . Our proof of this lemma is computer-assisted and will be described in Section 4.By Proposition 3.2, the map F has a unique fixed point φ ∗ ∈ ϕ + L B δ . We define γ = γ ( φ ∗ ) and α = α ( φ ∗ ).Our next goal is to construct a branch of periodic solutions for the equation (2.6).Consider B = B over F ∈ { R , C } . By continuity, there exists an open ball J ⊂ F centeredat the origin, and an open neighborhood D of φ ∗ in B , such that B ˆ φ = B | ∆ | − / L s ( φ ) φ isnonzero for all s ∈ J and all φ ∈ D . For every s ∈ J , define F s : D → B to be the map φ ˜ φ given by (2.13), with γ = γ ( s, φ ) and α = α ( s, φ ) defined by the equation (2.15). Lemma 3.4.
Let F = R . There exists a isomorphism L of B such that the followingholds. If N denotes the the quasi-Newton map associated with ( D , F , φ ∗ , L ) , then thederivative D N (0) of N at the origin is a contraction. Our proof of this lemma is computer-assisted and will be described in Section 4. Asa consequence we have the following.
Corollary 3.5.
Consider F = C . There exists an open disk J ⊂ C , centered at the origin,and an analytic curve s φ s on J with values in D , such that F s ( φ s ) = φ s for all s ∈ J .If s belongs to the real interval J ∩ R , then φ s is real. Furthermore, φ = φ ∗ . Proof.
Consider still F = C . For s ∈ I , the derivative of N s on its domain is given by D N s ( h ) = D F s ( φ ∗ + Lh ) L + I − L . (3 . ψ ∈ B satisfies D F ( φ ∗ ) ψ = ψ . We may assume that ψ takesreal values for real arguments. A straightforward computation shows that D N (0) L − ψ = opf bifurcation in Navier-Stokes L − ψ . Since D N (0) is a contraction in the real setting, by Lemma 3.4, this implies that ψ = 0. So the operator D F ( φ ∗ ) does not have an eigenvalue 1. This operator is compact,since it is the composition of a bounded linear operator with the compact operator L α .Thus, D F ( φ ∗ ) has no spectrum at 1. By the implicit function theorem, there exists acomplex open ball J , centered at the origin, such that the fixed point equation F s ( φ ) = φ has a solution φ = φ s for all s ∈ J . Furthermore, the curve s φ s is analytic, passesthrough φ ∗ at s = 0, and there is a unique curve with this property. By uniqueness, wealso have φ ¯ s = φ s for all s ∈ J , so φ s is real for real values of s ∈ J . QED
A branch of stationary periodic solutions for (2.6) is obtained similarly. Consider B = E { } B over F ∈ { R , C } . For every γ ∈ F , define F γ : B → B to be the map φ ˜ φ given by (2.13), with s = α = 0. Notice that φ ∗ e is a fixed point of F γ . Lemma 3.6.
Let F = R . There exists an isomorphism L of B such that the followingholds. If N γ denotes the the quasi-Newton map associated with ( B , F γ , φ ∗ e , L ) , then thederivative D N γ (0) of N γ at the origin is a contraction. Our proof of this lemma is computer-assisted and will be described in Section 4. Asa consequence we have the following.
Corollary 3.7.
Consider F = C . There exists an open disk I ⊂ C , centered at γ , andan analytic curve γ φ γ on I with values in B , such that F γ ( φ γ ) = φ γ for all γ ∈ I . If γ belongs to the real interval I ∩ R , then φ γ is real. Furthermore, φ γ = φ ∗ e . The proof of this corollary is analogous to the proof of Corollary 3.5.We note that the disk
I ∋ γ is disjoint from the disk J ∋ γ φ γ and s φ s for the curve ofstationary and periodic solutions, respectively, of the equation (2.11),Based on the results stated in this section, we can now give a Proof of Theorem 1.1.
As described in the preceding sections, the curve γ φ γ for γ ∈ I yields a curve γ u γ of stationary solutions of the equation (1.4), where u γ = ∂ − φ γ . By our choice of function spaces, the function ( γ, x, y ) u γ ( x, y ) is realanalytic on I × T , where I = I ∩ R .Similarly, the curve s φ s for s ∈ J defines a family of of non-stationary periodicsolutions for (1.4), with γ = γ s and α = α s determined via the equation (2.15). To be moreprecise, the even frequency part φ s, e of φ s determines a vector field u s, e = ∂ − φ s, e , and theodd frequency part φ s, o determines a vector field u s, o = ∂ − φ s, o . If β is a complex numbersuch that s = β ∈ J , then u = u s, e + βu s, o is a periodic solution of (1.4), with γ = γ s and α = α s . Here, we have used the decomposition (2.10). By our choice of function spaces,the functions ( s, t, x, y ) u s, e ( t, x, y ) and ( s, t, x, y ) u s, o ( t, x, y ) are real analytic on J × T , where J = J ∩ R . Clearly, ∂ t u , o ( t, . , . ) = 0, due to the normalization condition φ − , , = θ imposed in (2.14). And by construction, we have u = u γ for s = 0. QED ARIOLI & KOCH
4. Remaining estimates
What remains to be proved are Lemmas 3.3, 3.4, and 3.6. Our method used in the proofof Lemma 3.3 can be considered perturbation theory about the approximate fixed point ϕ of F . The function ϕ is a Fourier polynomial with over 20000 nonzero coefficients, so alarge number of estimates are involved.We start by describing bounds on the bilinear function L and on the linear operators L α and L ′ α . These are the basic building blocks for our transformations F , F s , and F γ .The “mechanical” part of these estimates will be described in Subsection 4.4. L and a proof of Proposition 3.1 Consider the bilinear form L defined by (2.7). Using the identity (2.3), we have L (Φ) φ = ( ∇ Φ) · J ∇ ∆ − φ + ( ∇ φ ) · J ∇ ∆ − Φ= (cid:2) ( ∂ x Φ)∆ − ∂ y φ − ( ∂ y Φ)∆ − ∂ x φ (cid:3) − (cid:2) (∆ − ∂ x Φ) ∂ y φ − (∆ − ∂ y Φ) ∂ x φ (cid:3) . (4 . L (Φ) φ in terms of the Fourier coefficients of Φ and φ . Given that L is bilinear, and that theidentity (4.1) holds pointwise in t , it suffices to compute L (Φ) φ for the time-independentmonomials Φ = sin J × sin K , φ = sin j × sin k , (4 . J, K, j, k >
0. A straightforward computation shows that L (Φ) φ = Θ( J k + jK ) (cid:2) sin J + j × sin K − k − sin J − j × sin K + k (cid:3) + Θ( J k − jK ) (cid:2) sin J + j × sin K + k − sin J − j × sin K − k (cid:3) , (4 . | ∆ | − / L (Φ) φ = X σ,τ = ± N σ,τ sin σJ + j × sin τK + k , (4 . N σ,τ = Θ σJ k − τ Kj p ( σJ + j ) + ( τ K + k ) , Θ = 14 (cid:18) J + K − j + k (cid:19) . (4 . Proof of Proposition 3.1.
Using the Cauchy-Schwarz inequality in R , we find that | N σ,τ | = | Θ | | ( σJ + j ) k − ( τ K + k ) j | p ( σJ + j ) + ( τ K + k ) ≤ | Θ | p j + k . (4 . N σ,τ is invariant under an exchange of ( j, k ) and ( J, K ), thisimplies that | N σ,τ | ≤ / p j + k ∨ / √ J + K , (4 . opf bifurcation in Navier-Stokes a ∨ b = max( a, b ) for a, b ∈ R . As a result, we obtain the bound (cid:13)(cid:13) | ∆ | − / L (Φ) φ (cid:13)(cid:13) ≤ (cid:13)(cid:13) | ∆ | − / Φ (cid:13)(cid:13) ̺,ǫ k φ k + k Φ k (cid:13)(cid:13) | ∆ | − / φ (cid:13)(cid:13) . (4 . A is a Banach algebra for thepointwise product of functions, this bound extends by bilinearity to arbitrary functionsΦ , φ ∈ B . QED
We note that the bound (4.8) exploits the cancellations that lead to the expression(4.3). A more straightforward estimate loses a factor of 2 with respect to (4.8). But it isnot just this factor of 2 that counts for us. The expressions (4.5) for the coefficients N σ,τ and the bounds (4.7) are used in our computations and error estimates. The expressionon the right hand side of (4.7) is a decreasing function of the wavenumbers j, k, J, K , so itcan be used to estimate L (Φ) φ when Φ and/or φ are “tails” of Fourier series. L α and L ′ α Consider the linear operators L α and L ′ α defined in (2.16), with α real. A straightforwardcomputation shows that ψ n,j,k = p j + k ( j + k ) φ n,j,k − αnφ − n,j,k ( j + k ) + α n , ψ = L α φ . (4 . R , this yields the estimate q | ψ n,j,k | + | ψ − n,j,k | ≤ C n,j,k q | φ n,j,k | + | φ − n,j,k | , (4 . C n,j,k = s j + k ( j + k ) + α n ≤ p | αn | ∧ p j + k (4 . n = 0, where a ∧ b = min( a, b ) for a, b ∈ R . The last bound in (4.11) is a decreasingfunction of | n | , j, k and can be used to estimate L α φ when φ is the tail of a Fourier series.For the operator L ′ α we have ψ n,j,k = n ( j + k ) φ − n,j,k + αnφ n,j,k ( j + k ) + α n , ψ = L ′ α φ . (4 . ψ = L ′ α φ , with C n,j,k = s n ( j + k ) + α n . (4 . n = ±
1, since these are the onlynonzero frequencies of the function ˆ φ = | ∆ | − / L ( φ ) φ with φ ∈ E { , } B . And for fixed n ,the right hand side of (4.13) is decreasing in j and k .2 ARIOLI & KOCH
Recall that a function φ ∈ B admits a Fourier expansion φ = X n ∈ Z X j,k ∈ N φ n,j,k θ n,j,k , θ n,j,k def = cosi n × sin j × sin k , (4 . φ is given by k φ k = X j,k ∈ N (cid:20) | φ ,j,k | + X n ∈ N q | φ n,j,k | + | φ − n,j,k | ρ n (cid:21) ̺ j + k . (4 . n ≥
0. A linear combination c + θ n,j,k + c − θ − n,j,k will be referred to as a modewith frequency n and wavenumbers ( j, k ) or as a mode of type ( n, j, k ). We assume ofcourse that c − = 0 when n = 0. Since (4.15) is a weighted ℓ norm, except for the ℓ normused for modes, we have a simple expression for the operator norm of a continuous linearoperator L : B → B , namely kLk = sup j,k ∈ N sup n ∈ N sup u kL u k / k u k , (4 . u of type ( n, j, k ).Let now n, j, k ≥ L θ ± n,j,k is known explicitly, weuse the following estimate. Denote by L n,j,k the restriction of L to the subspace spannedby the two functions θ ± n,j,k . For q ≥ kL n,j,k k q = sup ≤ p 3, then ˙ˆ φ = 2 | ∆ | − / L ( φ ) ˙ φ belongs to E N B with N = { n − , n, n + 1 } . Thus, we have ˙ γ = ˙ α = 0, and D F ( φ ) u n = − γ L α | ∆ | − / L ( φ ) u n . (4 . L α in this equation, if u n = c + θ n,j,k + c − θ − n,j,k with ( j, k ) and c ± fixed,then the ratios k D F ( φ ) u n k / k u n k (4 . n for n ≥ 3. And the limit as n → ∞ of this ratio is zero. opf bifurcation in Navier-Stokes L = D F ( φ ), the supremum over n ∈ N in (4.16) reduces toa maximum over finitely many terms. The same holds for the operator L = D N (0) = D F ( φ ∗ ) L + I − L that is described in Lemma 3.4. This is a consequence of the followingchoice. Remark 2. The operator L chosen in Lemma 3.4 is a “matrix perturbation” of theidentity, in the sense that Lθ n,j,k = θ n,j,k for all but finitely many indices ( n, j, k ). Thesame is true for the operators L and L chosen in Lemma 3.3 and Lemma 3.6, respectively. Lemmas 3.3, 3.6, and 3.4 assert the existence of certain objects that satisfy a set of strictinequalities. The goal here is to construct these objects, and to verify the necessaryinequalities by combining the estimates that have been described so far.The above-mentioned “objects” are real numbers, real Fourier polynomials, and linearoperators that are finite-rank perturbations of the identity. They are obtained via purelynumerical computations. Verifying the necessary inequalities is largely an organizationaltask, once everything else has been set up properly. Roughly speaking, the procedurefollows that of a well-designed numerical program, but instead of truncation Fourier seriesand ignoring rounding errors, we determine rigorous enclosures at every step along thecomputation. This part of the proof is written in the programming language Ada [25]. Thefollowing is meant to be a rough guide for the reader who wishes to check the correctnessof our programs. The complete details can be found in [24].An enclosure for a function φ ∈ B is a set in B that includes φ and is defined in termsof (bounds on) a Fourier polynomial and finitely many error terms. We define such setshierarchically, by first defining enclosures for elements in simpler spaces. In this context,a “bound” on a map f : X → Y is a function F that assigns to a set X ⊂ X of a giventype ( Xtype ) a set Y ⊂ Y of a given type ( Ytype ), in such a way that y = f ( x ) belongs to Y for all x ∈ X . In Ada, such a bound F can be implemented by defining a procedureF(X: in Xtype; Y: out Ytype) .Our most basic enclosures are specified by pairs S=(S.C,S.R) , where S.C is a repre-sentable real number ( Rep ) and S.R a nonnegative representable real number ( Radius ).Given a Banach algebra X with unit , such a pair S defines a ball in X which we denoteby h S , X i = { x ∈ X : k x − ( S . C ) k ≤ S . R } .When X = R , then the data type described above is called Ball . Bounds on somestandard functions involving the type Ball are defined in the package Flts Std Balls .Other basic functions are covered in the packages Vectors and Matrices . Bounds of thistype have been used in many computer-assisted proofs; so we focus here on the moreproblem-specific aspects of our programs.Consider now the space A for a fixed domain radius ̺ > Radius . Asmentioned before Remark 2, we only need to consider Fourier polynomials in A . Ourenclosures for such polynomials are defined by an array(-I c .. I c ) of Ball . This datatype is named NSPoly , and the enclosure associated with data P of this type is h P , Ai def = I c X i = − I c (cid:10) P ( i ) , R (cid:11) cosi ν ( i ) , (4 . ARIOLI & KOCH where ν is an increasing index function with the property that ν ( − i ) = − ν ( i ). Thetype NSPoly is defined in the package NSP , which also implements bounds on some basicoperations for Fourier polynomials in A . Among the arguments to NSP is a nonnegativeinteger n (named NN ). Our proof of Lemma 3.6 and Lemma 3.3 uses I c = n = 0 and I c = n = 1, respectively, and ν ( i ) = i . Values n ≥ L u for the operator L = D N (0), with u a mode of frequency n . In this case, ν takesvalues in {− n, n } or {− n − , − n, − n + 1 , , n − , n, n + 1 } , depending on whether n isodd or even. (The value ν = 0 is being used only for n = 2.) The package NSP also definesa data type NSErr as an array(0 .. I c ) of Radius . This type will be used below.Given in addition a positive number ̺ ≥ Radius , our enclosures for functionsin B are defined by pairs (F.C,F.E) , where F.C is an array(1 .. J c ,1 .. K c ) of NSPoly and F.E is an array(1 .. J e ,1 .. K e ) of NSErr ; all for a fixed value of the parameter NN .This data type is named Fourier3 , and the enclosure associated with F=(F.C,F.E) is h F , B i def = J c X j =1 K c X k =1 (cid:10) F . C ( j , k ) , A (cid:11) × sin j × sin k + J e X J =1 K e X K =1 H J,K ( F . E ( J , K )) . (4 . H J,K ( E ) denotes the set of all functions φ = P I c i =0 φ i with k φ i k ≤ E ( i ), where φ i canbe any function in B whose coefficients φ in,j,k vanish unless j ≥ J , k ≥ K , and | n | = ν ( i ).The type Fourier3 and bounds on some standard functions involving this type aredefined in the child package NSP.Fouriers . This package is a modified version of thepackage Fouriers2 that was used earlier in [11,15,21]. The procedure Prod is now abound on the bilinear map | ∆ | − / L . The error estimates used in Prod are based onthe inequality (4.7). The package NSP.Fouriers also includes bounds InvLinear and DtInvLinear on the linear operators L α and L ′ α , respectively. These bounds use theestimates described in Subsection 4.3.As far as the proof of Lemma 3.3 is concerned, it suffices now to compose existingbounds to obtain a bound on the map F and its derivative D F . This is done by theprocedures GMap and DGMap in Hopf.Fix . Here we use enclosures of type NN=1 .The type of quasi-Newton map N defined by (3.4) has been used in several computer-assisted proof before. So the process of constructing a bound on N from a bound on F hasbeen automated in the generic packages Linear and Linear.Contr . (Changes comparedto earlier versions are mentioned in the program text.) This includes the computation of anapproximate inverse L for the operator I − D F ( ϕ ). A bound on N is defined (in essence)by the procedure Linear.Contr.Contr , instantiated with Map => GMap . And a boundon D N is defined by Linear.Contr.Contr , with DMap => DGMap . Bounds on operatornorms are obtained via Linear.OpNorm . Another problem-dependent ingredient in theseprocedures, besides Map and DMap , are data of type Modes . These data are constructed bythe procedure Make in the package Hopf . They define a splitting of the given space B intoa finite direct sum. For details on how such a splitting is defined and used we refer to [16].If the parameter NN has the value 0, then the procedures GMap and DGMap define boundson the map F γ and its derivative, respectively. The operator L used in Lemma 3.6 hasthe property that M = L − I satisfies M = P M P , where P = E { } P m for somepositive integer m . Here, and in what follows, P m denotes the canonical projection in B opf bifurcation in Navier-Stokes with the property that P m φ is obtained from φ by restricting the second sum in (4.14) towavenumbers j, k ≤ m .If NN has a value n ≥ 2, then the procedure DGMap defines a bound on the map( φ, ψ ) D F ( φ ) ψ , restricted to the subspace E { , } B × E { n } B . The linear operator L that is used in Lemma 3.4 admits a decomposition L = I + M + M + . . . + M N ofthe following type. After choosing a suitable sequence n m n of positive integers, weset M n = P n ( L − I) P n , where P = E { , } P m and P n = E { n } P m n for n = 2 , , . . . , N .This structure of L simplifies the use of (4.16) for estimating the norm of L = D N (0).Furthermore, to check that L is invertible, it suffices to verify that I + M n is invertible onthe finite-dimensional subspace P n B , for each positive n ≤ N .The linear operator L that is used in Lemma 3.3 is of the form L = I + M with M as described above.All the steps required in the proofs of Lemmas 3.3, 3.6, and 3.4 are organized in themain program Check . As n ranges from 0 to N = 305, this program defines the parametersthat are used in the proof for NN = n , instantiates the necessary packages, computes theappropriate matrix M n , verifies that I + M n is invertible, reads ϕ from the file BP.approx ,and then calls the procedure ContrFix from the (instantiated version of the) package Hopf.Fix to verify the necessary inequalities.The representable numbers ( Rep ) used in our programs are standard [27] extendedfloating-point numbers (type LLFloat ). High precision [28] floating-point numbers (type MPFloat ) are used as well, but not in any essential way. Both types support controlledrounding. Radius is always a subtype of LLFloat . Our programs were run successfullyon a 20-core workstation, using a public version of the gcc/gnat compiler [26]. For furtherdetails, including instruction on how to compile and run our programs, we refer to [24]. References [1] E. Hopf, Abzweigung einer periodischen L¨osung von einer station¨aren L¨osung eines Differ-entialsystems , Ber. Math.-Phys. Kl. Siichs. Akad. Wiss. Leipzig, , 3–22 (1942).[2] J. Serrin, A Note on the Existence of Periodic Solutions of the Navier-Stokes Equations ,Arch. Rational Mech. Anal. The Hopf bifurcation theorem in infinite dimensions ,Arch. Rational Mech. Anal. , 53–72 (1977).[4] J. Marsden, M. McCracken, The Hopf bifurcation and its applications , Springer AppliedMathematical Sciences Lecture Notes Series, Vol. 19, 1976.[5] D. Ruelle, F. Takens, On the Nature of Turbulence , Commun. Math. Phys. 20, 167–192(1971)[6] P. Kloeden, R. Wells, An explicit example of Hopf bifurcation in fluid mechanics , Proc. Roy.Soc. London Ser. A , 293–320 (1983).[7] P. Chossat, G. Iooss, Primary and secondary bifurcations in the Couette-Taylor problem ,Japan J. Appl. Math. , 37–68 (1985).[8] F. Dumortier, Techniques in the theory of local bifurcations: blow-up, normal forms, nilpo-tent bifurcations, singular perturbations ; in: Bifurcations and periodic orbits of vector fields ,(D. Schlomiuk, ed., Kluwer Acad. Pub.) NATO ASI Ser. C Math. Phys. Sci. , 10–73(1993). ARIOLI & KOCH [9] P. Chossat, G. Iooss, The Couette-Taylor problem , Applied Mathematical Sciences, 102.Springer-Verlag, New York, 1994[10] M.T. Nakao, Y. Watanabe, N. Yamamoto, T. Nishida, M.-N. Kim, Computer assisted proofsof bifurcating solutions for nonlinear heat convection problems , J. Sci. Comput. , 388–401(2010).[11] G. Arioli, H. Koch, Non-symmetric low-index solutions for a symmetric boundary valueproblem , J. Differ. Equations , 448–458 (2012).[12] G. Arioli, H. Koch, Some symmetric boundary value problems and non-symmetric solutions ,J. Differ. Equations , 796–816 (2015).[13] G.P. Galdi, On bifurcating time-periodic flow of a Navier-Stokes liquid past a cylinder , Arch.Rational Mech. Anal. , 285–315 (2016). Digital Object Identifier (DOI) 10.1007/s00205-016-1001-3[14] C.-H. Hsia, C.-Y. Jung, T.B. Nguyen, and M.-C. Shiu, On time periodic solutions, asymptoticstability and bifurcations of Navier-Stokes equations , Numer. Math. , 607–638 (2017).[15] G. Arioli, H. Koch, Spectral stability for the wave equation with periodic forcing , J. Differ.Equations , 2470–2501 (2018).[16] G. Arioli, H. Koch, Non-radial solutions for some semilinear elliptic equations on the disk ,Nonlinear Analysis , 294308 (2019).[17] M.T. Nakao, M. Plum, Y. Watanabe, Numerical verification methods and computer-assistedproofs for partial differential equations , Springer Series in Computational Mathematics, Vol.53, Springer Singapore, 2019[18] J. G´omez-Serrano, Computer-assisted proofs in PDE: a survey , SeMA , 459–484 (2019).[19] D. Wilczak, P. Zgliczy´nski, A geometric method for infinite-dimensional chaos: Symbolicdynamics for the Kuramoto-Sivashinsky PDE on the line , J. Differ. Equations , 8509–8548 (2020).[20] J. B. van den Berg, M. Breden, J.-P. Lessard, L. van Veen, Spontaneous periodic orbits inthe Navier-Stokes flow , Preprint 2019,[21] G. Arioli, F. Gazzola, H. Koch, Uniqueness and bifurcation branches for planar steady Navier-Stokes equations under Navier boundary conditions , Preprint 2020.[22] J. B. van den Berg, J.-P. Lessard, E. Queirolo, Rigorous verification of Hopf bifurcations viadesingularization and continuation , Preprint 2020.[23] J. B. van den Berg, E. Queirolo, Validating Hopf bifurcation in the Kuramoto-SivashinkyPDE , in preparation.[24] G. Arioli, H. Koch, Programs and data files for the proof of Lemmas 3.3, 3.6, 3.4, and , https://web.ma.utexas.edu/users/koch/papers/nshopf/ [25] Ada Reference Manual, ISO/IEC 8652:2012(E), available e.g. at [26] A free-software compiler for the Ada programming language, which is part of the GNUCompiler Collection; see gnu.org/software/gnat/ [27] The Institute of Electrical and Electronics Engineers, Inc., IEEE Standard for Binary Float-ing–Point Arithmetic , ANSI/IEEE Std 754–2008.[28] The MPFR library for multiple-precision floating-point computations with correct rounding;see, ANSI/IEEE Std 754–2008.[28] The MPFR library for multiple-precision floating-point computations with correct rounding;see