A Construction of Blow up Solutions for Co-rotational Wave Maps

Cârstea, Cătălin I.

doi:10.1007/s00220-010-1118-4

Cited by 9 publications

(33 citation statements)

References 8 publications

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“…A main objective in this respect was to prove small data global existence for various target manifolds and space dimensions, see e.g., [32], [15], [14], [41], [42], [37], [38], [16], [29], [43], [21], [22], [23], [24]. On the other hand, for large data and in the energy critical case, there are newer results on blow up, e.g., [34], [20], [26], [25], [4] and, very recently, also on global existence [19], [39], [40], [36]. We will comment below in more detail on some of those works which are most relevant for us.…”

Section: Introductionmentioning

confidence: 99%

On stable self-similar blowup for equivariant wave maps

Donninger

2011

Comm. Pure Appl. Math.

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Abstract. We consider co-rotational wave maps from (3 + 1) Minkowski space into the threesphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. We prove that the blow up via f0 is stable under the assumption that f0 does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blow up to a linear ODE spectral problem. Although we are unable, at the moment, to verify the mode stability of f0 rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f0.

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Section: Introductionmentioning

confidence: 99%

On stable self-similar blowup for equivariant wave maps

Donninger

2011

Comm. Pure Appl. Math.

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“…However, numerical evidences by Bizon, Chmaj and Tabor [6], Isenberg and Liebling [31] strongly suggest singularity development for certain positively curved targets. Later, the existence of finite time blowup solutions for equivariant wave maps from (2 + 1) Minkowski space to the S 2 -sphere has been constructively proved by Krieger, Schlag and Tataru [36], Cârstea [12], Rodnianski and Sterbenz [57], Raphaël and Rodnianski [53]. It is worth mentioning the work by Côte et al [18,19] where the authors establish a classification of blowup solutions of topological degree one 1 with energies less than 3E(Q, 0), where Q(r) = 2 arctan(r) is the unique (up to scaling) non trivial solution to the equation Q rr + 1 r Q r = sin(2Q) 2r 2 .…”

Section: Introductionmentioning

confidence: 99%

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

Ghoul

Ibrahim

Nguyen

2018

Journal of Differential Equations

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We consider the energy supercritical wave maps from R d into the d-sphere S d with d ≥ 7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation1 1-COROTATIONAL ENERGY SUPERCRITICAL WAVE MAPS 5 Remark 1.3. The proof of Theorem 1.1 involves a detailed description of the the set of initial data leading to the type II blowup with the quantization of the blowup rate (1.13). In particular, given ℓ ∈ N * , L ≫ 1 and s ∼ L, our initial data is of the formwhere Q b is a deformation of the ground state Q = (Q, 0), and b = (b 1 , · · · , b L ) correspond to possible unstable directions of the flow in theḢ s ×Ḣ s−1 topology in a suitable neighborhood of Q.We show that for all q 0 ∈ O ⊂ Ḣ σ ∩Ḣ s × Ḣ σ−1 ∩Ḣ s−1 , where the set O is built on the linearized operator (see Definition 3.1 for its precise description of O) and for all b 1 (0), b ℓ+1 (0), · · · , b L (0) small enough, there exists a choice of unstable directions b 2 (0), · · · , b ℓ (0) such that the solution of (1.3) with initial data (1.15) satisfies the conclusion of Theorem 1.1. The control of (ℓ − 1) unstable modes is done through a topological argument based on Brouwer's fixed point theorem. In some sense, the set of blowup solutions we construct lies on a (ℓ − 1) codimension manifold in the radial class whose proof would require some Lipschitz regularity of the set of initial data we consider and it would be addressed separately in detail.Remark 1.4. It is worth mentioning that our analysis relies only on the study of supercritical Sobolev norms built on the linearized operator, thus, the finiteness of the H 1 norm of the initial data is not requested. Roughly speaking, the initial data (u 0 , u 1 ) can be taken smooth and compactly supported, namely that if u = Q+ε, we take ε(r) ∼ −Q(r) for r ≫ 1. Since the energy is conserved, our constructed solution can be taken to be of finite energy or even compactly supported. As a matter of fact, the finite energy together with the constructed manifold mentioned in the previous remark ensures that the original solution Φ to the wave map equation (1.2) has the same the regularity as for the 1-corotational symmetric solution u described in Theorem 1.1.Remark 1.5. We note from (1.12) thatThis implies that our constructed solution is of Type II blowup in the sense of (1.5).Remark 1.6. Following the work by Côte et al. [18,19] where the question of the classification of the flow near the special class of stationary solution Q are considered in the energy critical setting, i.e. d = 2, we would address the same question for the energy supercritical case d ≥ 7. In Theorem 1.1, the constructed blowup solutions exhibit the decomposition of the form (1.12). Here we ask for a converse problem, namely that if blowup does occur for a solution u, in which energy regime and in what sense does such the decomposition (1.12) always hold? Remark 1.7. It is worth mentioning the work of Krieger-Schlag-Tataru [36], where the authors constructed for equation (1.3) in the...

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“…The construction of blow up solutions to the wave maps problem was initiated by Shatah [31] and Cazenave, Shatah, Tahvildar-Zadeh [4]. Following the bubbling results of Struwe [39] and [40] formation of singularities were further studied in the works of Krieger, Schlag, Tataru [23], Rodnianski, Sterbenz [30], Raphael, Rodnianski [29], and Carstea [3]. The interested reader can consult the historical surveys in [22], [23], [29], [32], and [34] for more information on the history of the problem.…”

Section: Introductionmentioning

confidence: 99%

“…To compute the second derivative of α we differentiate the Gauss-Bonnet equation (26) twice to see that α ′′ + β ′′ = 0. We can also differentiate (23) to write β ′′ in terms of α ′′ : = sin β(f 2 (r) cos 2 β − f 2 (r ′ ) cos 2 α) f 2 (r) cos 3 β(α ′ + β ′ ) (α ′ ) 3 .…”

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confidence: 99%

Stability of stationary wave maps from a curved background to a sphere

Shahshahani¹

2016

DCDS-A

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We study time and space equivariant wave maps from M × R → S 2 , where M is diffeomorphic to a two dimensional sphere and admits an action of SO(2) by isometries. We assume that metric on M can be written as dr 2 + f 2 (r)dθ 2 away from the two fixed points of the action, where the curvature is positive, and prove that stationary (time equivariant) rotationally symmetric (of any rotation number) smooth wave maps exist and are stable in the energy topology. The main new ingredient in the construction, compared with the case where M is isometric to the standard sphere (considered by ), is the the use of triangle comparison theorems to obtain pointwise bounds on the fundamental solution on a curved background.

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A Construction of Blow up Solutions for Co-rotational Wave Maps

Cited by 9 publications

References 8 publications

On stable self-similar blowup for equivariant wave maps

On stable self-similar blowup for equivariant wave maps

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

Stability of stationary wave maps from a curved background to a sphere

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