An official website of the United States government
Abstract We study the global well-posedness of the supercritical dissipative surface quasi-geostrophic (SQG) equation, a key model in geophysical fluid dynamics. While local well-posedness is known, achieving global well-posedness for large initial data remains open. Motivated by enhanced decay in radial solutions, we aim to establish global well-posedness for small perturbations of potentially large radial data. Our main result shows that for small perturbations of radial data, the SQG equation admits a unique global solution.
more »
« less
Well-Posedness for the Dispersive Hunter–Saxton Equation
Abstract This article represents a 1st step toward understanding the well-posedness of the dispersive Hunter–Saxton equation, which arises in the study of nematic liquid crystals. Although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character. Further, the lack of spatial decay obstructs access to dispersive tools, including local smoothing estimates. Here, we give the 1st proof of local and global well-posedness for the Cauchy problem. Secondly, we improve our well-posedness results with respect to the low regularity of the initial data. The key techniques we use include constructing modified energies to realize a normal form analysis in our quasilinear setting, and frequency envelopes to prove continuous dependence with respect to the initial data.
more »
« less
- Award ID(s):
- 1928930
- PAR ID:
- 10364954
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2023
- Issue:
- 9
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- p. 7883-7924
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter–Shu–Zhang (2021Pure Appl. Anal.3403–72) established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation’s nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim–Tataru.more » « less
-
Abstract We derive a PDE that models the behavior of a boundary layer solution to the incompressible porous media (IPM) equation posed on the 2D periodic half-plane. This 1D IPM model is a transport equation with a non-local velocity similar to the well-known Córdoba–Córdoba–Fontelos (CCF) equation. We discuss how this modification of the CCF equation can be regarded as a reasonable model for solutions to the IPM equation. Working in the class of bounded smooth periodic data, we then show local well-posedness for the 1D IPM model as well as finite time blow-up for a class of initial data.more » « less
-
Abstract The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${\mathbb {R}}^{d+2}$$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions$$d \geqq 4$$ . In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\geqq 2$$ . This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.more » « less
-
Abstract The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${{\mathbb {R}}}^{d+2}$$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\ge 4$$ .more » « less
