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

Abstract

The existence and uniqueness of Gevrey regularity solutions for the functionalized Cahn-Hilliard (FCH) and Cahn-Hilliard-Willmore (CHW) equations are established. The energy dissipation law yields a uniform-in-time H 2 bound of the solution, and the polynomial patterns of the nonlinear terms enable one to derive a local-in-time solution with Gevrey regularity. A careful calculation reveals that the existence time interval length depends on the H 3 norm of the initial data. A further detailed estimate for the original PDE system indicates a uniform-in-time H 3 bound. Consequently, a global-in-time solution becomes available with Gevrey regularity.

Well-posedness of a Hydrodynamic Phase-field Model for Functionalized Membrane-Fluid Interaction

“…We note that the boundary condition (1.6) does not imply ∂ n ∆ 2 φ = 0 on ∂Ω for the solution to problem (1.1)-(1.7) whenever the trace of the normal derivative of ∆ 2 φ makes sense. Thus, we cannot apply the argument in the periodic setting (see [9,57]) to derive a globalin-time estimate on ∇∆φ by testing the phase-field equation (1.3) with −∆ 3 φ (or equivalently, testing the equation (1.4) by ∂ t ∆φ). Actually, we have…”

“…A lot of work has been done to understand morphological changes of membranes. We refer to [2,3,7,8,18-22, 33, 39] for mathematical modeling and numerical simulations, see also [9,11,13,17,23,37,57] for rigorous analysis.…”

“…When the fluid coupling is neglected (i.e., setting u = 0 in (1.3)), for the functionalized Cahn-Hilliard equation subject to periodic boundary conditions, the authors of [14] proved the existence of global weak solutions in the case of a regular potential and a degenerate mobility. For η ∈ R, m = 1, and a regular potential, existence and uniqueness of global solutions in the Gevrey class were established in [9], again in the periodic setting. In the recent work [48], the authors considered the case η ∈ R, m = 1 with a physically relevant logarithmic potential 1).…”

On a Class of Sixth-order Cahn-Hilliard Type Equations with Logarithmic Potential

“…For the functionalized Cahn-Hilliard equation (i.e., η < 0) subject to periodic boundary conditions, in [14], the authors proved existence of global weak solutions in the case of regular potential and degenerate mobility. Besides, for η ∈ R, M = 1 and a regular potential, existence and uniqueness of global solutions in the Gevrey class were established in [11], again in the periodic setting. We note that in those contributions [11,14,20,31] mentioned above, the potential F is always assumed to be a regular one.…”

“…Besides, for η ∈ R, M = 1 and a regular potential, existence and uniqueness of global solutions in the Gevrey class were established in [11], again in the periodic setting. We note that in those contributions [11,14,20,31] mentioned above, the potential F is always assumed to be a regular one. Indeed, this choice plays a crucial role in the mathematical analysis therein.…”