Anisotropic Estimates for the Two-Dimensional Kuramoto–Sivashinsky Equation

Abstract

Abstract. We address the global solvability of the Kuramoto-Sivashinsky equation in a rect-We give sufficient conditions on the width L 2 of the domain, depending on the length L 1 , so that the obtained solutions are global. Our proofs are based on anisotropic estimates.

Global existence for the two-dimensional Kuramoto–Sivashinsky equation with a shear flow

“…Local well-posedness holds in L p spaces [7,27]. Global existence is known only under fairly restrictive assumptions, such as for thin domains and for the anisotropically reduced KSE [6,30,32,36], without growing modes [1,17], or with only one growing mode in each direction [2], for small data. The attractor and determining modes were studied in [34], under a uniform bound on higher Sobolev norms that yields global existence of solutions.…”

“…for some ε 0 > 0, independent of ν, and for every t ≥ 0. For AKSE, we use (6) to show that solutions are global, as stated in the next theorem.…”

Global Solutions for the Kuramoto-Sivashinsky Equation Posed on Unbounded 3D Grooves

“…In [5][6][7][8], mathematical results on initial and initial boundary value problems for one-dimensional (1) are presented, see references there for more information. Multi-dimensional problems for various types of (1) can be found in [9][10][11][12][13][14][15] with some results on the existence, regularity and, nonlinear stability of solutions. In [10][11][12][13], stability of global solutions for the K-S equation posed on a rectangle with the width L 2 and the length L 1 , where L 2 is a decreasing function of L 1 and initial data depend on L 1 , was studied.…”

“…Multi-dimensional problems for various types of (1) can be found in [9][10][11][12][13][14][15] with some results on the existence, regularity and, nonlinear stability of solutions. In [10][11][12][13], stability of global solutions for the K-S equation posed on a rectangle with the width L 2 and the length L 1 , where L 2 is a decreasing function of L 1 and initial data depend on L 1 , was studied. This did not allow to study initial boundary value problems posed on strips.…”

Nonlinear dynamics of a dispersive anisotropic Kuramoto–Sivashinsky equation in two space dimensions

“…In two spatial dimensions, this equation has been derived to describe the propagation of a planar flame front [4], and has been suggested (with the addition of stochastic noise) as an empirical model for the evolution of surfaces eroded by ion bombardment [10,48,49]. A number of authors [50][51][52][53] have considered (1.11) analytically, proving global existence of solutions on sufficiently thin domains for restricted classes of initial conditions. Kalogirou et al [54] provided a comprehensive numerical study to complement this analytical work.…”