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Diffusive stability and self-similar decay for the harmonic map heat flow
- Publication Year :
- 2023
-
Abstract
- In this paper we study the harmonic map heat flow on the euclidean space $\mathbb{R}^d$ and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space $\dot{B}^{\frac{d}{p}}_{p,\infty}(\mathbb{R}^d)$ where $d<p<\infty$. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form $\|\nabla u(t) \|_{L^\infty(\mathbb{R}^d)}\leq Ct^{-\frac12}$. Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity.<br />Comment: 20 pages
- Subjects :
- Mathematics - Differential Geometry
Mathematics - Analysis of PDEs
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2301.02067
- Document Type :
- Working Paper