相关论文: Dynamics for the corotational energy-critical wave…

We consider the energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$. For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic…

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…

We exhibit a stable finite time blow up regime for the 1-corotational energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$ which reduces to the semilinear parabolic problem $$\partial_t u…

We study the blow-up dynamics for the energy-critical 1-corotational wave maps problem with 2-sphere target. In arXiv:0911.0692, Rapha\"el and Rodnianski exhibited a stable finite time blow-up dynamics arising from smooth initial data. In…

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…

We construct a one parameter family of finite time blow ups to the co-rotational wave maps problem from $S^2\times \RR$ to $S^2,$ parameterized by $\nu\in(1/2,1].$ The longitudinal function $u(t,\alpha)$ which is the main object of study…

We consider the half-wave equation $iu_t=Du-|u|u$ in two dimensions. For the initial data $u_0(x)\in H^{s}(\mathbb{R}^2)$, $s\in\left(\frac{3}{4},1\right)$, we obtain the non-radial ground state mass blow-up solutions with the blow-up speed…

We consider the half-wave equation $i u_t=Du-|u|^{\frac{2}{3}}u$ in three dimension and in the mass critical. For initial data $u(t_0,x)=u_0(x)\in H^{1/2}_{rad}(\mathbb{R}^3)$ with radial symmetry, we construct a new class of minimal mass…

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 that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ and restricted to the co-rotational setting with co-rotation index $k = 2$ admits finite time blow up solutions of finite energy on $(0,…

In this work we consider a semi-linear energy critical wave equation in ${\mathbb R}^d$ ($3\leq d \leq 5$) \[ \partial_t^2 u - \Delta u = \pm \phi(x) |u|^{4/(d-2)} u, \qquad (x,t)\in {\mathbb R}^d \times {\mathbb R} \] with initial data…

We consider the energy-supercritical harmonic map heat flow from $\mathbb{R}^d$ into $\mathbb{S}^d$, under an additional assumption of 1-corotational symmetry. We are interested by the 7 dimensional case which is the borderline between the…

We consider the wave maps problem with domain $\mathbb{R}^{2+1}$ and target $\mathbb{S}^{2}$ in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from $\mathbb{R}^{2}$ to…

We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb{R}$. We show an upper bound for any blow-up…

We consider the energy-critical half-wave maps equation $$\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling…

We study the dynamics of corotational wave maps from $\mathbb R^{1+2} \rightarrow \mathbb S^2$ at threshold energy. It is known that topologically trivial wave maps with energy $< 8\pi$ are global and scatter to a constant map. In this…

We consider the energy-critical wave maps equation $\mathbb R^{1+2} \to \mathbb S^2$ in the equivariant case, with equivariance degree $k \geq 2$. It is known that initial data of energy $ < 8k\pi$ and topological degree zero leads to…

We consider the energy-critical (corotational) 1-equivariant wave maps into the two-sphere. By the seminal work [53] of Rapha\"el and Rodnianski, there is an open set of initial data whose forward-in-time development blows up in finite time…

We consider the energy supercritical harmonic heat flow 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…

We analyze the finite-time blow-up of solutions of the heat flow for $k$-corotational maps $\mathbb R^d\to S^d$. For each dimension $d>2+k(2+2\sqrt{2})$ we construct a countable family of blow-up solutions via a method of matched…