Existence and generalized Gevrey regularity of solutions to the Kuramoto–Sivashinsky equation in Rn

Abstract

Motivated by the work of Foias and Temam [C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal. 87 (1989) 359-369], we prove the existence and Gevrey regularity of local solutions to the Kuramoto-Sivashinsky equation in R n with initial data in the space of distributions. The control on the Gevrey norm provides an explicit estimate of the analyticity radius in terms of the initial data. In the particular case when n = 1, our analysis allows for initial data that are less smooth than that considered by Grujić and Kukavica [Z. Grujić, I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in L p , J. Funct. Anal. 152 (1998) 447-466].

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

“…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.…”

“…Lemma 4.2. Suppose p ∈ 2N+2, s ∈ {0, 1}, u (1) , u (2) , • • • , u (p) , v, w ∈ D(A e τ A 1/2 ), τ > 0, where A = −∆. Then, if Condition 1 is satisfied, the following estimate holds:…”

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

“…The Cahn-Hilliard equation (3) has been extensively studied in the existing literature, at both the theoretical and numerical levels. In particular, the Gevrey regularity solution has been proven by [35] for the Cahn-Hilliard equation with dimensions d = 1 to d = 5; a more recent work [40] gives a further analysis with a rough initial data.…”

“…While there have been extensive numerical works for the given model [9,11,12,16,17,24,43], a theoretical justification of the smoothness and analyticity for the PDE solution has been limited. To obtain a PDE solution with real analytic regularity, the Gevrey norm has been a widely-used tool for the analysis for many time-dependent nonlinear PDEs; see the related works for 2-D and 3-D incompressible Navier-Stokes equation [4,19], Kuramoto-Sivashinsky equation [3], nonlinear parabolic equation [8,18], 3-D Navier-Stokes-Voigt equation [26], porous media flow [34]. Other than the Gevrey regularity solutions, a more general class of analytic solutions for different models of incompressible fluid have been discussed in [5,23,27,28,29,30,31,32], etc.…”

“…Theorem 4.1. Assume that φ 0 ∈ H 3 per . Then the FCH equation 6has a unique, global-in-time regular solution, such that τ → e τ A 1/2 φ(τ ) is analytic on (0, T ), for any T > 0.…”