GLOBAL-IN-TIME GEVREY REGULARITY SOLUTION FOR A CLASS OF BISTABLE GRADIENT FLOWS

. In this paper, we prove the existence and uniqueness of a Gevrey regularity solution for a class of nonlinear bistable gradient ﬂows, where with the energy may be decomposed into purely convex and concave parts. Example equations include certain epitaxial thin ﬁlm growth models and phase ﬁeld crystal models. The energy dissipation law implies a bound in the leading Sobolev norm. The polynomial structure of the nonlinear terms in the chemical potential enables us to derive a local-in-time solution with Gevrey regularity, with the existence time interval length dependent on a certain H m norm of the initial data. A detailed Sobolev estimate for the gradient equations results in a uniform-in-time-bound of that H m norm, which in turn establishes the existence of a global-in-time solution with Gevrey regularity.

(Communicated by Shouhong Wang) Abstract. In this paper, we prove the existence and uniqueness of a Gevrey regularity solution for a class of nonlinear bistable gradient flows, where with the energy may be decomposed into purely convex and concave parts. Example equations include certain epitaxial thin film growth models and phase field crystal models. The energy dissipation law implies a bound in the leading Sobolev norm. The polynomial structure of the nonlinear terms in the chemical potential enables us to derive a local-in-time solution with Gevrey regularity, with the existence time interval length dependent on a certain H m norm of the initial data. A detailed Sobolev estimate for the gradient equations results in a uniform-in-time-bound of that H m norm, which in turn establishes the existence of a global-in-time solution with Gevrey regularity.
Our principal aim in this paper is to establish the Gevrey regularity of solutions for the following family of nonlinear gradient flow evolution equations: ∂ t φ + (−∆) s µ = 0, µ := δ φ E on Ω T := Ω × (0, T ), (1.4) where φ is Ω-periodic in space, and s = 0 or s = 1. Equation (1.4) is the L 2 gradient flow (for s = 0) and the H −1 gradient flow (for s = 1) with respect to E in (1.1). The rates of energy dissipation along the solution trajectories are 5) and the mass of the solution is a conserved quantity, meaning d t Ω φ(x, t)dx = 0, for all t ≥ 0. It is often useful to consider the model in the following, less compact form: The evolution equation is thus a nonlinear "parabolic" equation of order 2 in purely divergence form, and, considering the periodic boundary conditions, the mass conservation is assured.
There are a few special cases of great physical interest that we wish to point out. The first is the epitaxial thin film model with slope selection, also known as the regularized Cross-Newell equation [8,14]. This equation can be obtained setting s = 0, = 2, ℘ = 4, a 1 = −1: It has been used as a model for thin film roughening and coarsening [19,20,21,28,29,30,31,34,35]. Some numerical works for the equation can be found in more recent articles [6,7,39,42,44].
The second is the square phase field crystal (SPFC) model, which is obtained by setting s = 1, = 3, ℘ = 4, a 2 = −2δ 2 : (1.8) The SPFC equation is related to another crystal growth model known as the phase field crystal (PFC) equation [10,11,37,41], which is the gradient flow (1.9) The PFC model was proposed in [10] for simulating crystal dynamics at the atomic scale in space but on diffusive scales in time, with natural incorporation of elastic and plastic deformations, multiple crystal orientations and defects. The natural lattice for a crystal described by the PFC equation is hexagonal in 2D. The SPFC model, on the other hand, predicts a "square" symmetry crystal lattice in 2D rather than the usual hexagonal structure; see the related references [11,15,43]. While the standard PFC model (1.9) is not covered by the following analysis -because the form of the energy is different from and, in fact, somewhat simpler than what is considered in (1.1) -our results can be easily extended for (1.9). There have been many existing works to establish the existence of Gevrey regularity solutions for time-dependent nonlinear PDEs, such as [3,13] for 2-D and 3-D incompressible Navier-Stokes equation, [2] for Kuramoto-Sivashinsky equation, [5,12] for certain nonlinear parabolic equations, [18] for the 3-D Navier-Stokes-Voight equation, [33] for models porous media flow, to mention a few. For gradient flow-type models, Gevrey regularity solutions have been proven by [36] for the Cahn-Hilliard equation with dimension d = 1 to d = 5. A more recent work [40] gives a further analysis with potentially rough initial data. In addition, a few related works for the Cahn-Hilliard model combined with certain fluid motion equation have also been reported, such as [9] for the convective Cahn-Hilliard equation, and [32] for the Cahn-Hilliard-Hele-Shaw model. Other than the Gevrey regularity solutions, a more general class of analytic solutions for different models of incompressible fluid have been discussed in [4,16,22,23,24,25,26,27], etc.
A general framework to establish the existence of local-in-time Gevrey regularity solutions for nonlinear parabolic equations with periodic boundary conditions in R n , has been addressed in [5,12]. The analyses therein apply when the growth of F (r, s) := G(r, s) − r, in either the r or the s variable, is bounded by a polynomial, and F is assumed to be real analytic in both variables such that it possesses a majorant. In any case, it is clear that the analyses in [5,12] will not cover equation (1.1) considered in this article. The reason is that the p laplacian terms of the form ∇ · (|∇φ| p ∇φ), p ∈ 2N + 2 involve first and second order derivatives combined in a highly nonlinear way, and these terms cannot be recast in the form of F (φ, ∇φ).
While there has been some existing work considering Gevrey regularity of solutions for gradient flows with respect to the Cahn-Hilliard-type energy, no work has been undertaken to study gradient flows with respect to (1.1). For the nonlinear gradient flow considered here, (1.4), which covers a large class of models, the most current result to our knowledge is the proof of a smooth solution for the epitaxial thin film growth model (1.7), as reported by [30]: given any H m initial data (with m ≥ 2), there is a unique solution with a uniform-in-time H m estimate.
In this paper, we provide an analysis of a global-in-time Gevrey regularity solution for the general gradient flow given by (1.4) with respect to the energy (1.1). The paper is organized as follows. In Section 2 we go over some basic notation. In Section 3 we construct an approximate solution to the PDE using the standard Galerkin procedure and give the leading order energy estimate. In Section 4 we prove the existence and uniqueness of a local-in-time Gevrey regularity solution for (1.4), with the existence time interval length dependent upon (−∆) is presented in Section 5, so that a global-in-time Gevrey regularity solution may be established.
2. Notation and preliminaries. We use the standard symbols for Lebesgue and Sobolev spaces of complex-valued functions and their norms. To begin, for u, v ∈ L 2 (Ω, C) = L 2 (Ω), we set (u, v) : Let us also define the following function spaces: We denote the standard semi-norm and norm on W m,p (Ω) by | · | m,p,Ω = | · | m,p and · m,p,Ω = · m,p , respectively, dropping the subscript m whenever m = 0. Since the domain Ω = (0, 1) d is understood in our discussion, we usually also drop the subscript Ω in referencing the (semi-)norms. Define the operator A to be −∆ paired with Ω-periodic boundary conditions. We define the range of A as R(A) :=L 2 (Ω). The domain of A is simply D(A) = H 2 per (Ω), and A : D(A) → R(A) is a positive, self-adjoint linear operator that admits a compact inverse. The eigenfunctions of A may be chosen as Φ α ( x) = exp(2πi α · x) ∈C ∞ per (Ω), for all α ∈ Z d \ 0 =: Z d , in which case the eigenvalues are λ α = (2π) 2 | α| 2 > 0. SetB := Φ α α ∈ Z d ; this is an orthonormal basis for L 2 (Ω). We can increaseB so the resulting set is an orthonormal basis for all of L 2 (Ω); in particular, B :=B ∪ Φ 0 ≡ 1 serves this purpose.
Since A is symmetric and positive, we can define the following Hilbert spaces: for any s ≥ 0, set . It is possible to define the exponential operator exp(τ A s ) = e τ A s , for any τ, s ≥ 0. To do so we introduce the Hilbert space We introduce the Gevrey space G τ := D e τ A 1 /2 . This is a Hilbert space with the inner product and norm denoted by Observe that, for any u ∈ G τ , Since |u| τ is finite, it follows that every H k norm of u is also finite.
. The operator P M : L 2 (Ω) → G M is the canonical orthogonal projection: Of course, if u ∈L 2 (Ω), thenû 0 = 0. One can extend the domain of definition P M toH −r per (Ω), for any r ∈ (0, ∞), as follows: if u ∈H −r per (Ω), then which implies that Recall that H −r per (Ω), · H −r per is a Hilbert space using the standard operator norm. We have the following basic properties of the orthogonal projection that we state without proof [38]: , for any r, s ≥ 0. Then, for any u ∈ X, The results can be modified in a trivial way to accommodate functions that are not of mean zero.
We have the following interpolation inequalities [1]: (2.9) For integer values of the indices, we have where a constant of 1 suffices.

3.1.
Lower and upper bounds of the energy.
where C 1 , · · · , C 5 are positive constants that depend only on the model parameters.
Proof. First, we decompose the energy (1.1) into non-quadratic and quadratic parts: We begin with the lower bounds. The non-quadratic energy part obeys the following estimate A simple application of Hölder inequality, using |Ω| = 1, shows that ∇φ 2j ≤ ∇φ ℘ , for 2 ≤ j < ℘ /2. Then, with the help of Young's inequality, the following general estimate can be derived Consequently, where the constant C 6 := A 4 + A 6 + · · · + A ℘−2 > 0 only depends on the coefficients c 4 , c 6 , · · · , c ℘−2 . The quadratic part, Q(φ), is analyzed in two separate cases: s = 0 and s = 1. If s = 0, a direct observation gives Meanwhile, an application of the interpolation inequality (2.9), with r = , k = 1 and k < j < , shows that where Young's inequality was applied in the last step. We remark that the nonnegative constants M 2 , M 3 , · · · , M −1 only depend on |a 2 |, |a 3 |, · · · , |a −1 | and ε. Substitution of (3.10) into (3.9) yields . As before, a simple application of Hölder inequality, using |Ω| = 1, shows that ∇φ 2 ≤ ∇φ ℘ . The negative part in (3.11) can be controlled as with another application of Young's inequality. Again, note that C 8 > 0 only depends on ℘ and |a 2 |, |a 3 |, · · · , |a −1 | and ε. Consequently, we arrive at Finally, a combination of the non-quadratic part (3.8) and the quadratic part (3.13) results in Therefore, the energy estimate (3.1) with s = 0 is proven with The lower bound for the case with s = 1 can be analyzed in a similar manner. We omit the details for the sake of brevity. Likewise, the upper bounds are straightforward, in fact, easier than the lower bounds, and the details are omitted.
If Condition 1 holds, then the quadratic diffusion term has control over the plaplacian terms, and we have the following: where C 6 , C 7 > 0 depend only upon the model parameters.

3.2.
Approximate solutions and uniform energy estimates. We may write the gradient flow in operator form as denoting the nonlinear term as N (φ) := −A s ∇ · |∇φ| ℘−2 + c ℘−2 |∇φ| ℘−4 + · · · + c 4 |∇φ| 2 ∇φ , and the indefinite (unsigned) linear term as We refer to the term ε 2 A φ as the "surface diffusion" term, following the physics literature for solid thin film models. We seek the following Galerkin approximation of the original problem: for fixed M ∈ N, find . Note that we have assumed, for simplicity, that the initial data are mean zero: |Ω| −1 Ω φ 0 ( x) d x = 0. We will keep this convention for the remainder of the paper.
Proof. The approximation problem can be recast as a system of nonlinear ODE's; it has a unique solution up to some finite time T , such thatφ α, We define the test function (3.20) Observe µ M ∈ G M ∩L 2 (Ω). Testing this with the Equation (3.19) and integrating, we arrive (after a standard energy variation calculation) at the result Integrating this in time, we have, for any T ∈ [0, T ], As a consequence of Proposition 3.1, Lemma 3.5, and Corollary 3.4 the following result is valid. Corollary 3.6. Suppose that Condition 1 holds and φ 0 ∈H −s per (Ω). Then φ M and µ M , defined as in Lemma 3.5, exist for all time, and, moreover, for any T > 0 whatsoever,

22)
where C 9 depends on the initial data and the equation parameters, but is independent of M and T .
Proof. A combination of Proposition 3.1, Lemma 3.5, and Corollary 3.4 indicates that, for any 0 < t ≤ T ,

NAN CHEN, CHENG WANG AND STEVEN WISE
where Lemma 2.8 was employed in the last step. By regularity, there is a constant constant, C 10 such that for any ψ ∈H −s per (Ω). Therefore, estimate (3.22) is proven for T = T . But, since C 9 is independent of the final time, T , the Galerkin approximate solutions do not blow-up and can be extended up to any final time T > 0 [38]. where C 11 > 0 is independent of T . Passing to limits, one can prove that the pair (φ, µ) is a weak solution to the gradient equation (1.4). The details are standard and are skipped for brevity.

4.
A local-in-time solution with Gevrey regularity. In this section, we establish a crucial technical estimate that will be used in the Gevrey analysis of the solution for (1.4). In a standard way, we need to analyze the Galerkin approximate solution (3.19) and pass to the limit to obtain the desired results.

4.1.
A preliminary estimate of the nonlinear terms. We define the following nonlinear terms: for φ sufficiently regular and Ω-periodic, set Then, using the formula we find we find where A : B = d j,k=1 A j,k B j,k , for two-tensors (matrices) A and B. In the next lemma, we examine a single representative term of N s p , s = 0, 1.
The following theorem is the main result of this section.
Theorem 4.6. Suppose that Conditions 1 and 2 hold, and assume that φ 0 ∈ D(A /2 ). Then there exists T * that depends upon A /2 φ 0 2 , such that the weak solution is regular and unique on (0, T ), and t → e tA 1/2 φ(t) is analytic on (0, T ).
Using the stability of the L 2 projection, this result implies the uniform (in M ) estimate for 0 ≤ τ ≤ T 2 , where, We can now extract a further subsequence of φ M and pass to limits to obtain our estimates for the limit point φ, which is observed to be Gevrey regular on the time interval (0, T 2 ). The uniqueness analysis of the Gevrey regularity solution is straightforward, due to the high order regularity. The details are left to interested readers. The theorem is proven with T = T 2 .
5. Global-in-time existence of a Gevrey regularity solution. Note that the existence time interval length T in Theorem 4.6 for the Gevrey regularity solution depends on the initial data, specifically A 2 φ 0 2 . To obtain a global-in-time solution with Gevrey regularity, we have to establish a uniform-in-time bound for A 2 φ(t) 2 , so that the constructed solution can be extended to any time. For the case s = 0, this follows from Theorem 3.8 immediately, and we have the following: To establish a uniform-in-time bound for A 2 φ(t) 2 for the case s = 1, we will need another condition, namely Definition 5.2. We say that Condition 3 holds iff when s = 1, H per (Ω) , however large the final time T may be. Furthermore, we have the uniform-in-time bound Proof. For simplicity, we only focus on the case of odd . The case with an even can be handled in a similar way. Taking the inner product of (3 For the temporal derivative term, since is odd, we have Finally, a combination of (5.3), (5.4), (5.11) and (5.24) results in and making use of the elliptic regularity for every ψ ∈H 2 (Ω), we arrive at where C 24 > 0 is independent of t ≥ 0. Integrating in time yields (5.28) Therefore, a global-in-time, uniform in M bound of A 2 φ M (t) is available. We can now extract a further subsequence of the Galerkin approximation and pass to limits to establish the bound for the weak solution, φ. The proof is complete.
As a consequence of Theorem 4.6 and Theorem 5.3, we arrive at the following theorem, the main result of this paper. Remark 5.5. For the local-in-time solution, the mapping t → e tA 1/2 φ(t) is analytic within the time interval (0, T ). Meanwhile, let us denote by T the Gevrey regularity solution existence time interval length, determined by Theorem 4.6, with A /2 φ 0 2 ≤ C 19 , where C 19 > 0 is given in Theorem 5.3. After time T , we can only ensure that the norm of e T A 1/2 φ(t) is bounded; we cannot ensure, by the present theory, that the norm of e tA 1/2 φ(t) is bounded for large time.
Remark 5.6. Before we conclude, let us check that Conditions 1 -3 are not so stringent as to exclude all of the interesting PDE's introduced earlier.
• For the Slope Selection (SS) epitaxial thin film growth model (1.7), we have the parameters, s = 0, = 2 and ℘ = 4 in d = 2. Condition 1 is easily satisfied. For the calculation of exponents, we are in Case 4, and σ 2 = 1. Thus Condition 2 is satisfied. Condition 3 is not applicable. • One can envision an SS epitaxial growth model with s = 0, = 2 and ℘ = 6 in d = 2 [43]. Thus the highest nonlinear term is a 6-laplacian. Again, Conditions 1 is easily satisfied. For the calculation of exponents, we are again in Case 4, and σ 2 = 1. Conditions 1 and 2 are satisfied. Condition 3 is not applicable. • For the regularized Cross-Newell (RCN) equation, the parameters are the same as the SS equation (in fact the equation is the same), but the dimension d = 3 may be appropriate. In this instance, Conditions 1 is satisfied. For the RCN equation with d = 3, σ 2 = 3/2 and Condition 2 is also satisfied. Here exponents are calculated according to Case 2. Condition 3 is not applicable.
• We can imagine an RCN-type equation with the following parameters: s = 0, = 2, ℘ = 6, and d = 3. The exponents are covered by Case 1, and σ 2 = 2. Unfortunately, our analysis does not cover this model, since Condition 2 fails to hold. Condition 3 is not applicable.
Remark 5.7. For a gradient flow with the Allen-Cahn/Cahn-Hilliard type energy we see that the global-in-time Gevrey regularity solution could be derived in the same manner, based on the fact that the degree of nonlinearity associated with φ 4 4 is much lower than that of ∇φ 4 4 . On the other hand, the Gevrey regularity for the Cahn-Hilliard equation has already been proved in an existing work [36]. which includes a logarithmic term. The L 2 gradient flow with respect to this energy is For the NSS model (5.31), the existence of a global-in-time smooth solution has been established in [30]. However, the framework to establish the Gevrey regularity solution, as presented in this article, can not be directly applied to this problem. The primary difficulty derives from the fact that the preliminary estimate Lemma 4.2 is not available for this gradient flow, since the nonlinear term in (5.31) is not in a polynomial pattern; instead, the nonlinear denominator makes a Fourier-typeanalysis not feasible any more. The analysis of the analytic solution for the NSS model (5.31) will be explored in a future work. The techniques related to the analyticity radius for nonlinear parabolic equations in a bounded domain, as reported by [4,17,22,23,24], are expected to be useful for this work.