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Soliton resolution for equivariant wave maps to the sphere
TLDR
In this article, the authors consider finite energy corotationnal wave maps with a target manifold and 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 linear scattering term (in global case), up to an error which tends to 0 in the energy space.Abstract:
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 linear scattering term (in the global case), up to an error which tends to 0 in the energy space.read more
Citations
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Two-bubble dynamics for threshold solutions to the wave maps equation
Jacek Jendrej,Andrew Lawrie +1 more
TL;DR: In this article, the authors consider the energy-critical wave maps equation and prove that the solution is defined for all time and either converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction.
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Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations
TL;DR: In this paper, a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution was shown to converge in some weak sense along a sequence of times and up to scaling and space translation to a sum of solitary waves.
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Solutions of the focusing nonradial critical wave equation with the compactness property
TL;DR: In this article, the authors proved that any solution which is bounded in the energy space converges, along a sequence of times and in some weak sense, to a solution with the compactness property, that is a solution whose trajectory stays in a compact subset of energy space up to space translation and scaling.
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Asymptotic decomposition for semilinear wave and equivariant wave map equations
Hao Jia,Carlos E. Kenig +1 more
TL;DR: In this paper, a unified proof to the soliton resolution conjecture along a sequence of times was given for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities.
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Profiles for the Radial Focusing 4 d Energy-Critical Wave Equation
TL;DR: In this article, a finite energy radial solution to the focusing energy critical semilinear wave equation in 1 + 4 dimensions is considered, and it is shown that along a sequence of times tending to the maximal forward time of existence, the solution decomposes into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space.
References
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Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case
Carlos E. Kenig,Frank Merle +1 more
TL;DR: In this paper, it was shown that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller compared to that of a standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5.
Journal ArticleDOI
High frequency approximation of solutions to critical nonlinear wave equations
Hajer Bahouri,Patrick Gérard +1 more
TL;DR: In this paper, the authors describe bounded energy sequences of solutions to the linear wave equation (1) in terms of their energy, up to remainder terms small in energy norm and in every Strichartz norm.
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Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation
Carlos E. Kenig,Frank Merle +1 more
TL;DR: In this article, it was shown that for Cauchy data (u, u_1) whose energy is smaller than that of (W, 0), where W is the well-known radial positive solution to the corresponding ellipyic equation, there is blow-up in finite time.
Journal ArticleDOI
Renormalization and blow up for charge one equivariant critical wave maps
TL;DR: In this paper, the authors prove the existence of equivariant finite-time blow-up solutions for the wave map problem from ℝ2+1→S petertodd 2 of the form $u(t,r)=Q(\lambda(t)r)+\mathcal{R}( t,r)$cffff where u is the polar angle on the sphere, $Q(r)=2\arctan r$cffff is the ground state harmonic map, λ(t)=t -1-ν, and $\mathcal {R} (t
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