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26
Concentration compactness for critical wave maps
, 2009
"... Abstract. By means of the concentrated compactness method of BahouriGerard [1] and KenigMerle [13], we prove global existence and regularity for wave maps with smooth data and large energy from R 2+1 → H 2. The argument yields an apriori bound of the Coulomb gauged derivative components of our wav ..."
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Cited by 23 (2 self)
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Abstract. By means of the concentrated compactness method of BahouriGerard [1] and KenigMerle [13], we prove global existence and regularity for wave maps with smooth data and large energy from R 2+1 → H 2. The argument yields an apriori bound of the Coulomb gauged derivative components of our wave map relative to a suitable norm ‖ · ‖S (which holds the solution) in terms of the energy alone. As a byproduct of our argument, we obtain a phasespace decomposition of the gauged derivative components analogous to the one of BahouriGerard.
Global regularity of wave maps VII. Control of delocalised or dispersed solutions
, 2009
"... This is the final paper in the series [18], [19], [20], [21] that establishes global regularity for twodimensional wave maps into hyperbolic targets. In this paper we establish the remaining claims required for this statement, namely a divisible perturbation theory, and a means of synthesising sol ..."
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Cited by 14 (4 self)
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This is the final paper in the series [18], [19], [20], [21] that establishes global regularity for twodimensional wave maps into hyperbolic targets. In this paper we establish the remaining claims required for this statement, namely a divisible perturbation theory, and a means of synthesising solutions for frequencydelocalised, spatiallydispersed, or spatiallydelocalised data out of solutions of strictly smaller energy. As a consequence of the perturbation theory here and the results obtained earlier in the series, we also establish spacetime bounds and scattering properties of wave maps into hyperbolic space.
ON STABLE SELF–SIMILAR BLOW UP FOR EQUIVARIANT WAVE MAPS: THE LINEARIZED PROBLEM
, 2010
"... We consider co–rotational wave maps from (3 + 1) Minkowski space into the three– sphere. 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 supp ..."
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Cited by 12 (10 self)
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We consider co–rotational wave maps from (3 + 1) Minkowski space into the three– sphere. 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. In this paper we develop a rigorous linear perturbation theory around f0. This is an indispensable prerequisite for the study of nonlinear stability of the self–similar blow up which is conducted in the companion paper [11]. In particular, we prove that f0 is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than 1 2. The remaining compact region is well–studied numerically and all available results strongly suggest the nonexistence of unstable modes.
Global regularity of wave maps VI. Abstract theory of minimalenergy blowup solutions
, 2009
"... Abstract. In [16], [17], [18], the global regularity conjecture for wave maps from twodimensional Minkowski space R 1+2 to hyperbolic space H m was reduced to the problem of constructing a minimalenergy blowup solution which is almost periodic modulo symmetries in the event that the conjecture fai ..."
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Cited by 12 (4 self)
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Abstract. In [16], [17], [18], the global regularity conjecture for wave maps from twodimensional Minkowski space R 1+2 to hyperbolic space H m was reduced to the problem of constructing a minimalenergy blowup solution which is almost periodic modulo symmetries in the event that the conjecture fails. In this paper, we show that this problem can be reduced further, to that of showing that solutions at the critical energy which are either frequencydelocalised, spatiallydispersed, or spatiallydelocalised have bounded “entropy”. These latter facts will be demonstrated in the final paper [19] in this series. 1.
Characterization of large energy solutions of the equivariant wave map problem
"... Abstract. We consider 1equivariant wave maps from R1+2 → S2. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic projection. We deduce this result via the concentration compactness/rigidity ..."
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Cited by 11 (8 self)
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Abstract. We consider 1equivariant wave maps from R1+2 → S2. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic projection. We deduce this result via the concentration compactness/rigidity method developed by the second author and Merle. In particular, we establish a classification of equivariant wave maps with trajectories that are precompact in the energy space up to the scaling symmetry of the equation. Indeed, a wave map of this type can only be either 0 or Q up to a rescaling. This gives a proof in the equivariant case of a refined version of the threshold conjecture adapted to the degree zero theory where the true threshold is 2E(Q), not E(Q). The aforementioned global existence and scattering statement can also be deduced by considering the work of Sterbenz and Tataru in the equivariant setting. For wave maps of topological degree one, we establish a classification of solutions blowing up in finite time with energies less than three times the energy of Q. Under this restriction on the energy, we show that a blowup solution of degree one is essentially the sum of a rescaled Q plus a remainder term of topological degree zero of energy less than twice the energy of Q. This result reveals the universal character of the known blowup constructions for degree one, 1equivariant wave maps of Krieger, the fourth author, and Tataru as well as Raphaël and Rodnianski.
Global wellposedness of hedgehog solutions for the (3+1) Skyrme model, preprint
 12 DANANDREI GEBA, KENJI NAKANISHI, AND SARADA G. RAJEEV
, 2011
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Profiles for bounded solutions of dispersive equations, with applications to energycritical wave and Schrödinger equations
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GLOBAL WELLPOSEDNESS FOR THE MAXWELLKLEIN GORDON EQUATION IN 4 + 1 DIMENSIONS. SMALL ENERGY.
"... Abstract. We prove that the critical MaxwellKlein Gordon equation on R 4+1 is globally wellposed for smooth initial data which are small in the energy. This reduces the problem of global regularity for large, smooth initial data to precluding concentration of energy. 1. ..."
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Cited by 2 (0 self)
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Abstract. We prove that the critical MaxwellKlein Gordon equation on R 4+1 is globally wellposed for smooth initial data which are small in the energy. This reduces the problem of global regularity for large, smooth initial data to precluding concentration of energy. 1.
The Cauchy Problem for Wave Maps on a Curved Background
 Calculus of Variations and Partial Differential Equations 45
, 2012
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