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18
Global wellposedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in R 1+4
, 2006
"... We obtain global wellposedness, scattering, uniform regularity, and global L6 t,x spacetime bounds for energyspace solutions to the defocusing energycritical nonlinear Schrödinger equation in R×R 4. Our arguments closely follow those in [11], though our derivation of the frequencylocalized inte ..."
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Cited by 70 (15 self)
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We obtain global wellposedness, scattering, uniform regularity, and global L6 t,x spacetime bounds for energyspace solutions to the defocusing energycritical nonlinear Schrödinger equation in R×R 4. Our arguments closely follow those in [11], though our derivation of the frequencylocalized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the L6 t,xnorm
Sharp Strichartz estimates on nontrapping asymptotically conic manifolds
, 2004
"... We obtain the Strichartz inequalities âu â q Lt Lr x ([0,1]ÃM) â¤ Câu(0)âL2 (M) for any smooth ndimensional Riemannian manifold M which is asymptotically conic at infinity (with either shortrange or longrange metric perturbation) and nontrapping, where u is a solution to the SchrÃ¶d ..."
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Cited by 39 (6 self)
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We obtain the Strichartz inequalities âu â q Lt Lr x ([0,1]ÃM) â¤ Câu(0)âL2 (M) for any smooth ndimensional Riemannian manifold M which is asymptotically conic at infinity (with either shortrange or longrange metric perturbation) and nontrapping, where u is a solution to the SchrÃ¶dinger equation iut + 1 2âMu = 0, and 2 < q, r â¤ â are admissible Strichartz exponents ( 2 q + n n =). This corresponds with the estimates available for Euclidean space r 2 (except for the endpoint (q, r) = (2, 2n) when n> 2). These estimates imply nâ2 existence theorems for semilinear SchrÃ¶dinger equations on M, by adapting arguments from Cazenave and Weissler [4] and Kato [14]. This result improves on our previous result in [10], which was an L4 t,x Strichartz estimate in three dimensions. It is closely related to the results in [22], [1], [26], [19], which consider the case of asymptotically flat manifolds.
Phase space analysis on some black hole manifolds
 J. Funct. Anal
"... The Schwarzschild and ReissnerNordstrøm solutions to Einstein’s equations describe space times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space time of this type. We show that for solutions with initial ..."
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Cited by 33 (13 self)
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The Schwarzschild and ReissnerNordstrøm solutions to Einstein’s equations describe space times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L 6 norm in space decays like t − 1 3. This
Strichartz estimates and local smoothing estimates for asympototically flat Schrödinger equations
 J. Funct. Anal
"... Abstract. In this article we study globalintime Strichartz estimates for the Schrödinger evolution corresponding to longrange perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [28] of the third author, where it is proved that local smoothing estimates im ..."
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Cited by 28 (12 self)
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Abstract. In this article we study globalintime Strichartz estimates for the Schrödinger evolution corresponding to longrange perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [28] of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. By [28] the local smoothing estimates are known to hold for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without nontrapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with nontrapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum. 1.
Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space
"... We study the long time behavior of radial solutions to nonlinear Schrödinger equations on hyperbolic space. We show that the usual distinction between short range and long range nonlinearity is modified: the geometry of the hyperbolic space makes every powerlike nonlinearity short range. The proofs ..."
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Cited by 18 (7 self)
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We study the long time behavior of radial solutions to nonlinear Schrödinger equations on hyperbolic space. We show that the usual distinction between short range and long range nonlinearity is modified: the geometry of the hyperbolic space makes every powerlike nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates for a broader family of admissible pairs, and on Morawetz type inequalities. The latter are established without symmetry assumptions.
A New Frequencyuniform Coercive Boundary Integral Equation for Acoustic Scattering
, 2011
"... A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2.� / (where � is the surface of the scatte ..."
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Cited by 14 (2 self)
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A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2.� / (where � is the surface of the scatterer) for all Lipschitz starshaped domains. Moreover, the coercivity is uniform in the wavenumber k D!=c,where! is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “starcombined ” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors’ knowledge, it is the only secondkind integral operator for which convergence of the Galerkin method in L2.� / is proved without smoothness assumptions on � except that it is Lipschitz. The coercivity of the starcombined operator implies frequencyexplicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the highfrequency case. The proof of coercivity of the starcombined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains.
DECAY FOR THE WAVE AND SCHRÖDINGER EVOLUTIONS ON MANIFOLDS WITH CONICAL ENDS, PART II
, 801
"... Abstract. Let Ω ⊂ RN be a compact imbedded Riemannian manifold of dimension d ≥ 1 and define the (d+1)dimensional Riemannian manifold M: = {(x, r(x)ω) : x ∈ R, ω ∈ Ω} with r> 0 and smooth, and the natural metric ds2 = (1+r ′(x) 2)dx2 +r2(x)ds2 Ω. We require that M has conical ends: r(x) = x+O ..."
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Cited by 14 (7 self)
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Abstract. Let Ω ⊂ RN be a compact imbedded Riemannian manifold of dimension d ≥ 1 and define the (d+1)dimensional Riemannian manifold M: = {(x, r(x)ω) : x ∈ R, ω ∈ Ω} with r> 0 and smooth, and the natural metric ds2 = (1+r ′(x) 2)dx2 +r2(x)ds2 Ω. We require that M has conical ends: r(x) = x+O(x −1) as x → ±∞. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrödinger evolution e it ∆ M and the wave evolution e it √ − ∆ M are obtained for data of the form f(x, ω) = Yn(ω)u(x) where Yn are eigenfunctions of −∆Ω with eigenvalues µ 2 n. In this paper we discuss all cases d + n> 1. If n ̸ = 0 there is the following accelerated local decay estimate: with q 0 < σ = 2µ 2 n + (d − 1)2 d − 1
On scattering for NLS: from Euclidean to hyperbolic space, preprint
, 2008
"... Abstract. We prove asymptotic completeness in the energy space for the nonlinear Schrödinger equation posed on hyperbolic space H n in the radial case, for n � 4, and any energysubcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estim ..."
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Cited by 13 (4 self)
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Abstract. We prove asymptotic completeness in the energy space for the nonlinear Schrödinger equation posed on hyperbolic space H n in the radial case, for n � 4, and any energysubcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which kind of interpolate between Euclidean space and hyperbolic space, showing that the family of short range nonlinearities becomes larger and larger as the space approaches the hyperbolic space. Finally, we describe the large time behavior of radial solutions to the free dynamics. 1.
Weighted Strichartz estimates for radial Schrödinger equation on noncompact manifolds, preprint
, 2007
"... Abstract. We prove global weighted Strichartz estimates for radial solutions of linear Schrödinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and DamekRicci spaces. This yields classical Strichartz estimates with a larger class o ..."
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Cited by 12 (4 self)
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Abstract. We prove global weighted Strichartz estimates for radial solutions of linear Schrödinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and DamekRicci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose volume element grows polynomially or exponentially at infinity, are characterized essentially by negativity conditions on the curvature, which shows in particular that the rich algebraic structure of the Hyperbolic and DamekRicci spaces is not the cause of the improved dispersive properties of the equation. The proofs are based on known dispersive results for the equation with potential on the Euclidean space, and on a new one, valid for C 1 potentials decaying like 1/r 2 at infinity. Contents