Abstract
We consider the Kuramoto–Sivashinsky equation (KSE) on the two-dimensional torus in the presence of advection by a given background shear flow. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we prove global existence of solutions with data in L^2, using a bootstrap argument. The initial data can be taken arbitrarily large.
Highlights
In this article we consider the Kuramoto–Sivashinsky equation (KSE) in two-space dimension in the presence of advection by a given background shear flow
The KSE is a well-known model of large-scale instabilities, such as those arising in flame-front propagation
The KSE comes in a scalar, potential form, and a differentiated, vectorial form
Summary
In this article we consider the Kuramoto–Sivashinsky equation (KSE) in two-space dimension in the presence of advection by a given background shear flow. For the Laplace operator plus advection, decay rates akin to (8) were obtained in [10,39] for a shear with infinitely many critical points, using the pseudo-spectral approach, and for shear flows with finitely many critical points in [4], using hypocoercivity Such quantitative semigroup estimates are relevant in the investigation of enhanced diffusion for passive scalars [3,4,13,39], in the study of asymptotic stability of particular solutions to the two-dimensional Navier–Stokes equations [14,19,33,40], and have applications to several other nonlinear problems [5,25,28,29].
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