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We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocusing case, we prove that arbitrary initial data in the energy space lead to global solutions that…
We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is global-in-time and scatters to free waves in…
We study the equivariant harmonic map heat flow, Schr\"odinger maps equation, and generalized Landau-Lifshitz equation from $\mathbb{C}^n$ to $\mathbb{C}\mathbb{P}^n$. By means of a careful geometric analysis, we determine a new, highly…
We study the Yamabe equation in the Euclidean half-space. We prove that any sign-changing solution has at least twice the energy of a standard bubble. Moreover, a sharper energy lower bound of the sign-changing solution set is also…
We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two…
In this paper, we continue our study of equivariant \emph{wave maps on a wormhole} initiated in our companion paper. More precisely, we study finite energy $\ell$--equivariant wave maps from the (1+3)-dimensional spacetime $\mathbb R \times…
A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau type approximation.
We consider the non linear focusing wave equation $\partial_{tt}u-\Delta u-u|u|^{p-1}=0$ in large dimensions and for radially symmetric data, in the energy supercritical zone for p large enough. We construct finite time blow up solutions…
The coupled motion is investigated for a mechanical system consisting of water and a body freely floating in it. Water occupies either a half-space or a layer of constant depth into which an infinitely long surface-piercing cylinder is…
Previous work developed a K-matrix formalism applicable to positive energies for the scattering between two $s$-wave interacting particles with two internal states, isotropic spin-orbit coupling and vanishing center-of-mass momentum [H.…
In this paper, we consider the wave equation in space dimension 3 with an energy-supercritical, focusing nonlinearity. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined and…
We prove that the energy-critical half-wave maps equation \[ \partial_t \mathbf{S} =\mathbf{S} \times |\nabla| \mathbf{S}, \quad (t,x) \in \mathbb{R} \times \mathbb{T} \] arises as an effective equation in the continuum limit of completely…
By means of the concentrated compactness method of Bahouri-Gerard and Kenig-Merle, we prove global existence and regularity for wave maps with smooth data and large energy from 2+1 dimensions into the hyperbolic plane. The argument yields…
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…
We study equivariant wave maps from the $2+1$ dimensional wormhole to the 2-sphere. This model has explicit harmonic map solutions which, in suitable coordinates, have the form of the sine-Gordon kinks/anti-kinks. We conjecture that there…
The exact two-particle energy eigenstates in a generic asymmetric rectangular box with periodic boundary conditions in all three directions are studied. Their relation with the elastic scattering phases of the two particles in the continuum…
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…
In this paper we generalise our previous results [1] concerning scattering on the exterior of collapsing dust clouds to the charged case, including in particular the extremal case. We analyse the energy boundedness of solutions $\phi$ to…
A theory for multiple scattering of elastic waves is presented in a random medium bounded by two ideal free surfaces, whose horizontal size is infinite and whose transverse size is smaller than the mean free path of the waves. This geometry…
In this paper we consider finite energy, \ell-equivariant wave maps from 1+3-dimensional Minkowski space exterior to the unit ball at the origin into the 3-sphere. We impose a Dirichlet boundary condition at r=1, which in this context means…