TY - JOUR
T1 - On the well-posedness of an anisotropically-reduced two-dimensional Kuramoto–Sivashinsky equation
AU - Larios, Adam
AU - Yamazaki, Kazuo
N1 - Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/10
Y1 - 2020/10
N2 - The Kuramoto–Sivashinsky equations (KSE) arise in many diverse scientific areas, and are of much mathematical interest due in part to their chaotic behavior, and their similarity to the Navier–Stokes equations. However, very little is known about their global well-posedness in the 2D case. Moreover, regularizations of the system (e.g., adding large diffusion, etc.) do not seem to help, due to the lack of any control over the L2 norm. In this work, we propose a new “reduced” 2D model that modifies only the linear part of (the vector form of) the 2D KSE in only one component. This new model shares much in common with the 2D KSE: it is 4th-order in space, it has an identical nonlinearity which does not vanish in energy estimates, it has low-mode instability, and it lacks a maximum principle. However, we prove that our reduced model is globally well-posed. We also examine its dynamics computationally. Moreover, while its solutions do not appear to be close approximations of solutions to the KSE, the solutions do seem to hold many qualitative similarities with those of the KSE. We examine these properties via computational simulations comparing solutions of the new model to solutions of the 2D KSE.
AB - The Kuramoto–Sivashinsky equations (KSE) arise in many diverse scientific areas, and are of much mathematical interest due in part to their chaotic behavior, and their similarity to the Navier–Stokes equations. However, very little is known about their global well-posedness in the 2D case. Moreover, regularizations of the system (e.g., adding large diffusion, etc.) do not seem to help, due to the lack of any control over the L2 norm. In this work, we propose a new “reduced” 2D model that modifies only the linear part of (the vector form of) the 2D KSE in only one component. This new model shares much in common with the 2D KSE: it is 4th-order in space, it has an identical nonlinearity which does not vanish in energy estimates, it has low-mode instability, and it lacks a maximum principle. However, we prove that our reduced model is globally well-posed. We also examine its dynamics computationally. Moreover, while its solutions do not appear to be close approximations of solutions to the KSE, the solutions do seem to hold many qualitative similarities with those of the KSE. We examine these properties via computational simulations comparing solutions of the new model to solutions of the 2D KSE.
KW - Anisotropic
KW - Global well-posedness
KW - Kuramoto–Sivashinsky
KW - Regularization
KW - Two-dimensional
UR - http://www.scopus.com/inward/record.url?scp=85085564392&partnerID=8YFLogxK
U2 - 10.1016/j.physd.2020.132560
DO - 10.1016/j.physd.2020.132560
M3 - Article
AN - SCOPUS:85085564392
SN - 0167-2789
VL - 411
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
M1 - 132560
ER -