Global-in-time Gevrey regularity solution for a class of bistable gradient flows

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.

Efficient unconditionally stable numerical schemes for a modified phase field crystal model with a strong nonlinear vacancy potential

“…Theorem 3. 4 The numerical scheme ( 47)-( 50) is unconditionally energy stable, that is, there holds…”

“…FIGURE 3 Time evolution of the free energy (9) and the mass with various time steps using the first-order scheme ( 17)-( 20) when h vac = 5000, S = 100. The last column presents the evolution of the free energy (9) when h vac = 5000, S = 0 (c) (b) (a) FIGURE 4 Time evolution of the free energy ( 9) and the mass with various time steps using the second-order backward differentiation formulas scheme ( 34)-( 37) when h vac = 5000, S = 100. The last column presents the evolution of the free energy (9) when h vac = 5000, S = 0 (c) (b) (a) FIGURE 5 Time evolution of the free energy (9) and the mass with various time steps using the CN scheme ( 47)-( 50) when h vac = 5000, S = 100.…”

“…There are also many other strategies to design energy stable numerical schemes for the PFC model [6,18,20,24,28,34,43,73]. In Reference [5], the authors proposed an energy stable second-order backward differentiation formulas (BDF2) Fourier pseudo-spectral numerical scheme for the SPFC model [4,15], using the preconditioned steepest descent (PSD) iteration [14] to solve the corresponding nonlinear system. The authors in Reference [35] developed a stabilized SAV scheme for a new Allen-Cahn type SPFC model.…”

Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation

“…Higher order H m estimate (beyond the norm given by the physical energy) is available for many gradient flows, due to the analytic property of the surface diffusion parabolic operator; see the related discussions in [4]. There have also been quite a few works of uniform in time H 2 estimate for certain energy stable numerical schemes for the Cahn-Hilliard equation [16,32,56], beyond the H 1 bound given by the energy estimate.…”

Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation

“…For the gradient flows with variational energy formulation, the Gevrey regularity solution has been proven for Cahn-Hilliard equation [35,40], and certain extensions to the Cahn-Hilliard-fluid models have been reported in [15,33]. In addition to these Cahn-Hilliard type problems, equations with p-Laplacian type nonlinearities has been analyzed in a more recent article [10], with an establishment of a global-in-time well-posedness.…”