ترغب بنشر مسار تعليمي؟ اضغط هنا

On the $H^1(ds)$-gradient flow for the length functional

150   0   0.0 ( 0 )
 نشر من قبل Glen Wheeler
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

In this article we consider the length functional defined on the space of immersed planar curves. The $L^2(ds)$ Riemannian metric gives rise to the curve shortening flow as the gradient flow of the length functional. Motivated by the triviality of the metric topology in this space, we consider the gradient flow of the length functional with respect to the $H^1(ds)$-metric. Circles with radius $r_0$ shrink with $r(t) = sqrt{W(e^{c-2t})}$ under the flow, where $W$ is the Lambert $W$ function and $c = r_0^2 + log r_0^2$. We conduct a thorough study of this flow, giving existence of eternal solutions and convergence for general initial data, preservation of regularity in various spaces, qualitative properties of the flow after an appropriate rescaling, and numerical simulations.



قيم البحث

اقرأ أيضاً

147 - Li Ma , Anqiang Zhu 2008
In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C-infinity topology as t goes to infinity.
217 - Hui-Ling Gu , Xi-Ping Zhu 2007
In this paper we prove the existence of Type II singularities for the Ricci flow on $S^{n+1}$ for all $ngeq 2$.
244 - Fei He , Changliang Wang 2019
In this note, we establish certain regularity estimates for the spinor flow introduced and initially studied in cite{AWW2016}. Consequently, we obtain that the norm of the second order covariant derivative of the spinor field becoming unbounded is th e only obstruction for long-time existence of the spinor flow. This generalizes the blow up criteria obtained in cite{Sc2018} for surfaces to general dimensions. As another application of the estimates, we also obtain a lower bound for the existence time in terms of the initial data. Our estimates are based on an observation that, up to pulling back by a one-parameter family of diffeomorphisms, the metric part of the spinor flow is equivalent to a modified Ricci flow.
184 - Jeff Streets 2010
We investigate the low-energy behavior of the gradient flow of the $L^2$ norm of the Riemannian curvature on four-manifolds. Specifically, we show long time existence and exponential convergence to a metric of constant sectional curvature when the in itial metric has positive Yamabe constant and small initial energy.
In this paper we consider the steepest descent $L^2$-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (e ither immersed locally convex of class $C^2$ or embedded of class $W^{2,2}$ bounding a convex domain), the flow converges smoothly to a round expanding multiply-covered circle.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا