https://neckpinch.com/authors/lixia-yuan/New arXiv papers by Lixia Yuan, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.08283arxiv:2610.08283Tue, 06 Oct 2026 00:00:00 GMTTianhong Pu, Xiaoqian Xin, Lixia Yuan — We study a positive radial graph evolving by V=H^α+b, with α>0 and b<0, in the unit cylinder and with uᵣ(1,t)=u(1,t). We separate finite-time continuation estimates from the time-uniform interior estimates needed for convergence of normalized translates. On existing classical intervals we prove height and gradient bounds and an explicit conversion of velocity bounds into estimates for H, uᵣ/r, and uᵣᵣ. We derive the correct boundary defect for comparisons with finite-slope translators. Under independent interior velocity bounds and two-sided slope trapping, we prove compactness and identify every translated limit with the cup translator. We identify the radial zero-number comparison still required to establish that trapping. Stationary caps and power-law Robin laws are included.Mean curvature flowSelf-shrinkers & translatorsSingularitiesLimit spaces & RCDBoundaries & non-compact https://arxiv.org/abs/2610.08270arxiv:2610.08270Tue, 06 Oct 2026 00:00:00 GMTTianhong Pu, Xiaoqian Xin, Lixia Yuan, Xiaoyu Zhang — We study the graphical forced mean curvature flow in a planar strip subject to Robin boundary conditions. The normal velocity is prescribed by V=H+b, where b>0 is constant and H denotes the curvature of the graph. We consider positive even solutions and reduce the problem to the half interval. The main results describe the degeneration produced by the Robin condition: the height, the instantaneous velocity, and the spatial gradient diverge, while the normalized slope uₓ/u converges to one away from the symmetry axis. The normalized velocity uₜ/u converges to the forcing parameter b. The curvature converges to zero in the interior in an integral sense, whereas its horizontal curvature measure concentrates at the symmetry axis.Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsBoundaries & non-compact https://arxiv.org/abs/2302.07831arxiv:2302.07831Fri, 03 Feb 2023 00:00:00 GMTXinfu Chen, Bendong Lou, Xiaoliu Wang, Lixia Yuan — We consider a graphical mean curvature flow in a cylinder with Robin boundary conditions, which arises as a geometric model for interface motion in the singular limit of the Allen–Cahn equation with nonlinear boundary conditions. It was shown in [26] that, in the planar case, every solution converges to a translating Grim Reaper with a fixed profile and finite speed. In this paper, we investigate the radially symmetric problem in higher dimensions and reveal a completely different asymptotic dynamics caused by the spatial dimension. In contrast to the planar case, there is no fixed translating profile governing the long-time behaviour. Instead, the solution propagates with an exponentially increasing speed, while both the gradient |Du| (away from the center) and the instantaneous speed uₜ diverge exponentially as t→∞. This reveals a fundamentally different asymptotic behaviour induced by the interaction between the Robin boundary condition and the spatial dimension, that is, the translating profile continuously degenerates and becomes asymptotically ray-like. Since the equation becomes asymptotically degenerate and no uniform-in-time C⁰, C¹, or C² estimates are available, our analysis relies on a new approach based on the zero number argument.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactLong-time existence & convergenceRotational symmetry & warped productsPhase fields & multiphase flows https://arxiv.org/abs/2205.14431arxiv:2205.14431Sat, 28 May 2022 00:00:00 GMTBendong Lou, Lixia Yuan — We study a generalized mean curvature flow involving a positive power of the mean curvature and a driving force. In this paper, we first construct all kinds of radially symmetric translating solutions, and then select one of them to satisfy a prescribed boundary angle in a cylinder. We then consider the flow starting at an initial hypersurface: showing the a priori estimates (especially the uniform-in-time bounds for the mean curvature which guarantee the uniform parabolicity of the corresponding fully nonlinear equation), giving the global existence for the solution of the initial boundary value problem, and proving its convergence to the corresponding translating solution. Our study provides a complete exposition on the influence of the dimension, the power of the mean curvature, the driving force and the boundary angles on the existence and stability of radially symmetric translating solutions.Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessLong-time existence & convergenceRotational symmetry & warped products