Two-bubble dynamics for threshold solutions to the wave maps equation

Abstract

We consider the energy-critical wave maps equation R 1+2 → S 2 in the equivariant case, with equivariance degree k ≥ 2. It is known that initial data of energy < 8πk and topological degree zero leads to global solutions that scatter in both time directions. We consider the threshold case of energy 8πk. We prove that the solution is defined for all time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps.The proof combines the classical concentration-compactness techniques of Kenig-Merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation.Theorem 1.2 (Sequential Decomposition). [8,25] Let ψ(t) ∈ H ℓπ be a smooth solution to (1.5) on [0, T + ). Then there exists a sequence of times t n → T + , an integer J ∈ N, a regular map ϕ ∈ H 0 , sequences of scales λ n,j and signs ι j ∈ {−1, 1} for j ∈ {1, . . . , J}, so thatIn the case of finite time blow-up at least one scale λ n,1 → 0 as n → ∞ and ϕ(t) → ϕ(1) is a finite energy map with E( ϕ(1)) = E( ψ) − JE( Q). In the case of a global solution, ϕ(t) can be taken to be a solution to the linear wave equation (2.2) and signs ι j are required to match up so that lim r→∞ ψ(0, r) = ℓπ = lim r→∞ J j=1 ι j Q λn,j (r).Remark 1.3. A decomposition into bubbles for a sequence of times for the full nonequivariant model was obtained by Grinis [21] up to an error that vanishes in a weaker Besov-type norm. Duyckaerts, Jia, Kenig and Merle [12] proved that for energies slightly above E( Q) a one-bubble decomposition holds for continuous time.The same authors obtained in [13] a sequential decomposition into bubbles in the case of the focusing energy critical power-type nonlinear wave equation (NLW).Theorem 1.2 raises two natural questions:• Are there any solutions to (1.5) with J ≥ 2 in (1.7), i.e., are there any solutions that form more than one bubble? • And, if so, does the decomposition (1.7) hold continuously in time, i.e., does soliton resolution hold for (1.5)?

Inelasticity of soliton collisions for the 5D energy critical wave equation

“…In [11,13], the above version of the soliton resolution conjecture was proved in the non-radial case for a sequence of times t n → +∞ in 3, 4 and 5D. We also refer to previous results of classification in [8,43,25,26] and to constructions of special solutions in [27,19,20,21]. We expect that, beyond its own interest, the full understanding of the collision problem will be a key to the proof of the soliton resolution conjecture for the whole sequence of time.…”

The threshold conjecture for the energy critical hyperbolic Yang–Mills equation

“…For as long as it exists it is gauge equivalent with A inside the cone, since they are gauge equivalent initially, see Theorem 2. 18. This shows that it cannot blow-up inside the cone.…”

“…Also, as far as the soliton bubbling off property is concerned, blow up solutions with this property are known to exist. The constructions in [25,41] give such examples 4 whose energies may be arbitrarily close to the threshold 2E GS , and the recent work [18] provides 5 a blow up solution at exactly the threshold energy 2E GS . These solutions concentrate at the blowup point following a rescaled soliton profile, where the soliton scale differs logarithmically from the self-similar scale.…”

“…Similarly, solutions where bubbling occurs at infinity also exist. For Yang-Mills in the topologically trivial class, this was achieved in [17]; interestingly, the nonscattering solution in [17] has exactly the threshold energy 2E GS (see also [18]). Such solutions have been obtained for related models such as the energy critical wave maps equation in the one-bubble case, see [8].…”

Construction of multi-bubble solutions for the energy-critical wave equation in dimension 5

“…For the energy-critical wave equation in dimension larger than 6, a global radial solution decomposing asymptotically as a concentrating bubble on the top of a standing soliton of same sign is constructed in [21]. Note that this behavior corresponds to a specific choice of sign and blow-up rate; see a nonexistence result in [19] and a classification result in a similar framework in [22]. In [29], a solution of (1.1) containing an arbitrary number K of bounded traveling solitons is constructed under some restrictions on the speeds ℓ k of the solitons.…”