Related papers: Type II Singularities on complete non-compact Yama…
The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth…
We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing…
We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= \varphi \quad \text{on } \partial…
In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show…
We consider the energy critical four dimensional semi linear heat equation \partial tu-\Deltau-u3 = 0. We show the existence of type II finite time blow up solutions and give a sharp description of the corresponding singularity formation.…
On a compact strictly pseudoconvex CR manifold $(M,\th)$, we consider the CR Yamabe constant of its infinite conformal covering. By using the maximum principles, we then prove a uniqueness theorem for the CR Yamabe flow on a complete…
We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are…
In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial…
We consider the energy super critical nonlinear Schr\"odinger equation $$i\pa_tu+\Delta u+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical…
It is well-known that the two-dimensional Keller-Segel system admits finite time blowup solutions, which is the case if the initial density has a total mass greater than $8\pi$ and a finite second moment. Several constructive examples of…
In this paper, we rigorously analyze the scalar curvature of complete expanding gradient Yamabe solitons. We completely classify nontrivial complete expanding gradient Yamabe solitons in both cases: when the scalar curvature is greater than…
In this paper, we study the combinatorial Yamabe flow on infinite triangulated surfaces in Euclidean background geometry, aiming for solving discrete Yamabe problem on noncompact surfaces. Under suitable conditions, we establish the…
The only non-compact linearly stable singularity models for mean curvature flow are cylindrical by Colding-Minicozzi. The uniqueness of blowups at singularities modeled on the cylinders has been established by the same authors. In this…
We present a numerical study of the local stability of mean curvature flow of rotationally symmetric, complete noncompact hypersurfaces with Type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs…
We construct axisymmetric solutions to the three-dimensional parabolic-elliptic Keller-Segel system that blows up in finite time. In particular, the singularity is of type II, which admits locally a leading order profile of the rescaled…
Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out…
We show that any integral Brakke flow coming out of a rotationally symmetric double cone with entropy at most two must stay rotationally symmetric for all time, provided the flow is smooth for a short time. We also show the existence of a…
We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the $L^q$-setting. Moreover, we give a…
We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally…
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension…