Global stability of some totally geodesic wave maps

Stability and instability of traveling wave solutions to nonlinear wave equations

“…In fact commuting with r 2 ∂ v or with an appropriate modification of the conformal Morawetz vector field K 0 = (t 2 + r 2 )∂ t + 2tr∂ r should introduce only √ t weights even though they have quadratic weights. 2 The fact that the boost in the x direction introduces better weights was already used in [1]. These observations could also be useful establishing better decay estimates.…”

“…Note that we will usually work with the quantities E 1 , E 2 which are the square roots of the energies E 2 1 , E 2 2 . This is purely a notational convenience.…”

“…In particular, for the relativistic membrane equation (also known as timelike or Lorentzian minimal/maximal surface equation or Lorentzian vanishing mean curvature flow ), [1] show stability of compactly supported planar waves in dimension 3 and higher while [29] show it in dimension 2 and higher (they also consider a more general class of traveling wave solution than the class considered in this paper which is schematically of the form (a + by)f (t − x), see Section 13 for further discussion). For the wave map equation, [2] prove stability for certain planar waves.…”

Hidden structure and sharp asymptotics for the Dirac--Klein-Gordon system in two space dimensions

“…To conclude the introduction, we remind the reader of numerous recent works concerning global existence and decay for coupled wave-Klein-Gordon equations in 1 + 3 dimensions. These include wave-Klein-Gordon equations derived from mathematical physics, such as the Dirac-Klein-Gordon model, the Dirac-Proca and U (1)-electroweak model [14,21,39,40], the Einstein-Klein-Gordon equations [20,25,27,41], the Klein-Gordon-Zakharov equations [15,33,38], the Maxwell-Klein-Gordon equations [24] and certain geometric problems derived from wave maps [1].…”

“…There has also been work by Stingo [37] and the first author [10] which does not require compactly supported data. Other work has also looked at the Klein-Gordon-Zakharov model in 1 + 2 dimensions [11,16,32] and the wave map model derived in [1] has been studied in the critical case of 1 + 2 dimensions in the recent works [13,16].…”