We establish probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for scaling super-critical random initial data. The proof relies on an induction on frequency procedure and a mod ...
In this paperwe obtain a global characterization of the dynamics of even solutions to the one-dimensional nonlinear Klein–Gordon (NLKG) equation on the line with focusing nonlinearity |u|p−1u, p > 5, provided their energy exceeds that of the ground state o ...
We introduce a suitable concept of weak evolution in the context of the radial quintic focussing semilinear wave equation on R^{3+1}, that is adapted to continuation past type II singularities. We show that the weak extension leads to type I singularity fo ...
We establish basic local existence as well as a stability result concerning small perturbations of the Catenoid minimal surface in R-3 under hyperbolic vanishing mean curvature flow. ...
For the critical focusing wave equation □u=u5 on R3+1 in the radial case, we establish the role of the “center stable” manifold Σ constructed in [18] near the ground state (W,0) as a threshold between blowup and scattering ...
This monograph contains a study of the global Cauchy problem for the Yang-Mills equations on (6+1) and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space. (H) over dot(A)((n-4)/2). Regular ...
We consider finite-energy equivariant solutions for the wave map problem from ℝ2+1 to S2 which are close to the soliton family. We prove asymptotic orbital stability for a codimension-two class of initial data which is small with respect to a stronger topo ...
We consider the focusing L2-critical half-wave equation in one space dimension i∂tu=Du−∣u∣2u, where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold M∗>0 such that all ...
We study the Cauchy problem for the one-dimensional wave equation \[ \partial_t^2 u(t,x)-\partial_x^2 u(t,x)+V(x)u(t,x)=0. \] The potential V is assumed to be smooth with asymptotic behavior \[ V(x)\sim -\tfrac14 |x|^{-2}\mbox{ as } |x|\to \infty. \] ...
We exhibit non-equivariant perturbations of the blowup solutions constructed in [18] for energy critical wave maps into S2. Our admissible class of perturbations is an open set in some sufficiently smooth topology and vanishes near the light co ...
We consider the hyperbolic Yang-Mills equation on the Minkowski space \reels4+1. Our main result asserts that this problem is globally well-posed for all initial data whose energy is sufficiently small. This solves a longstanding open problem. ...
We study global behavior of radial solutions for the nonlinear wave equation with the focusing energy critical nonlinearity in three and five space dimensions. Assuming that the solution has energy at most slightly more than the ground states and gets away ...
In this paper we prove the global in time well-posedness of the following non-local diffusion equation with αε[0,2/3): ∂tu={(−Δ)−1u}Δu+αu2,u(t=0)=u0. The initial condition u0 is positive, rad ...
We establish probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for scaling super-critical random initial data. The proof relies on an induction on frequency procedure and a mod ...
For the critical focusing wave equation □u=u5 on R3+1 in the radial case, we prove the existence of type II blow up solutions with scaling parameter λ(t)=t−1ν for all ν>0. This extends the previous work by the autho ...
The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [25]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathem ...
In this paper we consider the equation for equivariant wave maps from R3+1 to S-3 and we prove global in forward time existence of certain C-infinity-smooth solutions which have infinite critical Sobolev norm (H) overdot(3/4) (R-3) x (H) overdot(1/2) (R-3) ...
We prove that the critical Wave Maps equation with target S2 and origin R2+1 admits energy class blow up solutions of the form \[ u(t, r) = Q(\lambda(t)r) + \eps(t, r) \] where Q:R2→S2 is the ground state harmonic map and $\lambd ...
We investigate the stability and stabilization of the cubic focusing Klein–Gordon equation around static solutions on the closed ball of radius L in R3. First we show that the system is linearly unstable near the static solution u ≡ 1 for any dissipative b ...
We formulate the half-wave maps problem with target S2 and prove global regularity in sufficiently high spatial dimensions for a class of small critical data in Besov spaces. ...
