相关论文: Schrodinger Flow Near Harmonic Maps

We prove that the critical Wave Maps equation with target $S^2$ and origin $\mathbb{R}^{2+1}$ admits energy class blow up solutions of the form $$u(t,r)=Q(\lambda(t)r)+\epsilon(t,r)$$where $Q: \mathbb{R}^2 \to S^2$ is the ground state…

We show the existence of non-trivial self-expanding harmonic map flows starting from non-energy-minimizing 0-homogeneous maps to a regular ball or a closed hemisphere. In particular, given a non-minimizing but stationary 0-homogeneous…

We consider the Cauchy problem for the Schr\"odinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich…

We consider the initial-value problem for the equivariant Schr\"odinger maps near a family of harmonic maps. We provide some supplemental arguments for the proof of local well-posedness result by Gustafson, Kang and Tsai in [Duke Math. J.…

The existence of co-rotational finite time blow up solutions to the wave map problem from R^{2+1} into N, where N is a surface of revolution with metric d\rho^2+g(\rho)^2 d\theta^2, g an entire function, is proven. These are of the form…

In this paper, we study the blow-up phenomena on the $\alpha_k$-harmonic map sequences with bounded uniformly $\alpha_k$-energy, denoted by $\{u_{\alpha_k}: \alpha_k>1 \quad \mbox{and} \quad \alpha_k\searrow 1\}$, from a compact Riemann…

We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave…

In this paper, we investigate the blow-up phenomenon of the $H^2$ norm of solutions to the inhomogeneous biharmonic Schrodinger equation in two distinct scenarios. First, we consider the case of negative energy, analyzing separately the…

We consider the energy-critical wave maps equation from 1+2 dimensional Minkowski space into the 2-sphere, in the equivariant case. We prove that if a wave map decomposes, along a sequence of times, into a superposition of at most two…

We generalize the no-neck result of Qing-Tian \cite{QT} to show that there is no neck during blowing up for the $n$-harmonic flow as $t\to\infty$. As an application of the no-neck result, we settle a conjecture of Hungerb\"uhler \cite…

It has been known for a long time that the equivariant 2+1 wave map into the 2-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit…

We consider finite energy corotationnal wave maps with target manifold $\m S^2$. We prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blow case) or a…

We prove an estimate for the difference of two solutions of the Schr\"odinger map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes…

We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic…

In this paper, we show that for certain initial values, the (extrinsic) biharmonic map flow in dimension four must blow up in finite time.

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure $(M, g)$. This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex…

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel,…

In this paper, we study the solutions to the energy-critical quadratic nonlinear Schr\"odinger system in ${\dot H}^1\times{\dot H}^1$, where the sign of its potential energy can not be determined directly. If the initial data ${\rm u}_0$ is…

In this paper, we prove that the solution of the Landau-Lifshitz flow $u(t,x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map as $t\to\infty$. The essential observation is that although there exist infinite numbers of…

We consider energy-critical Schroedinger maps with target either the sphere S^2 or hyperbolic plane H^2 and establish that a unique solution may be continued so long as a certain space-time L^4 norm remains bounded. This reduces the large…