Global Well-Posedness of the Kuramoto--Sivashinsky Equation in One and Two Dimensions under a Codimension-One Constraint
Abstract
We study the Kuramoto--Sivashinsky (KS) equation in one, two, and three spatial dimensions subject to a codimension-one constraint. The constraint preserves the L2-norm of the solution and consequently provides a uniform a priori bound that plays a central role in the higher-order energy estimates. In one spatial dimension, we establish global well-posedness by means of a Fourier--Galerkin approximation and compactness arguments. In two and three dimensions, we first prove the existence and uniqueness of local strong solutions using analytic semigroup theory and the Banach fixed-point theorem. In two dimensions, the preserved L2-norm, combined with the Agmon inequality and suitable higher-order energy estimates, allows us to control the nonlinear terms and obtain global-in-time a priori bounds. Consequently, we establish the existence and uniqueness of global strong solutions for arbitrary sufficiently regular initial data, without any smallness assumption. This provides a global well-posedness result for the constrained two-dimensional KS equation, in contrast with the corresponding unconstrained problem, for which global well-posedness of strong solutions remains open in general. In three dimensions, although the constraint yields a uniform L2-bound, the available interpolation and Agmon estimates are insufficient to close the higher-order energy estimate. This identifies a genuine dimension-dependent obstruction to extending the global regularity argument to three dimensions.
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BibTeX
@misc{tiwari2026global,
title = {{Global Well-Posedness of the Kuramoto--Sivashinsky Equation in One and Two Dimensions under a Codimension-One Constraint}},
author = {Harsh Tiwari and Manil T. Mohan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00248},
eprint = {2610.00248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00248},
}
RIS
TY - UNPB
TI - Global Well-Posedness of the Kuramoto--Sivashinsky Equation in One and Two Dimensions under a Codimension-One Constraint
AU - Harsh Tiwari
AU - Manil T. Mohan
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00248
UR - https://arxiv.org/abs/2610.00248
AB - We study the Kuramoto--Sivashinsky (KS) equation in one, two, and three spatial dimensions subject to a codimension-one constraint. The constraint preserves the L^2-norm of the solution and consequently provides a uniform a priori bound that plays a central role in the higher-order energy estimates. In one spatial dimension, we establish global well-posedness by means of a Fourier--Galerkin approximation and compactness arguments. In two and three dimensions, we first prove the existence and uniqueness of local strong solutions using analytic semigroup theory and the Banach fixed-point theorem. In two dimensions, the preserved L^2-norm, combined with the Agmon inequality and suitable higher-order energy estimates, allows us to control the nonlinear terms and obtain global-in-time a priori bounds. Consequently, we establish the existence and uniqueness of global strong solutions for arbitrary sufficiently regular initial data, without any smallness assumption. This provides a global well-posedness result for the constrained two-dimensional KS equation, in contrast with the corresponding unconstrained problem, for which global well-posedness of strong solutions remains open in general. In three dimensions, although the constraint yields a uniform L^2-bound, the available interpolation and Agmon estimates are insufficient to close the higher-order energy estimate. This identifies a genuine dimension-dependent obstruction to extending the global regularity argument to three dimensions.
ER -