Results 1  10
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14
A refined global wellposedness result for Schrödinger equations with derivative
 SIAM J. Math. Anal
, 2002
"... Abstract. In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally wellposed in H s, for s> 1 2 for data small in L2. To understand the strength of this result one should recall that for s < 1 the Cauchy problem is illposed, in the 2 sense tha ..."
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Cited by 59 (18 self)
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Abstract. In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally wellposed in H s, for s> 1 2 for data small in L2. To understand the strength of this result one should recall that for s < 1 the Cauchy problem is illposed, in the 2 sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the “Imethod ” used by the same authors. The same argument can be used to prove that any to obtain global wellposedness for s> 2 3 quintic nonlinear defocusing Schrödinger equation on the line is globally wellposed for large data in H s, for s> 1 2. 1.
Multilinear estimates for periodic KdV equations, and applications
"... Abstract We prove an endpoint multilinear estimate for the X s;b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local wellposedness results for periodic generalized KdV equations, as well as some global wellposedness results below the energy norm. r ..."
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Cited by 56 (14 self)
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Abstract We prove an endpoint multilinear estimate for the X s;b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local wellposedness results for periodic generalized KdV equations, as well as some global wellposedness results below the energy norm. r
Global Wellposedness for Schrödinger equations with derivative
 SIAM J. Math. Anal
"... Abstract. We prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally wellposed in H s, for s> 2/3 for small L 2 data. The result follows from an application of the “Imethod”. This method allows to define a modification of the energy norm H 1 that is “almost cons ..."
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Cited by 52 (25 self)
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Abstract. We prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally wellposed in H s, for s> 2/3 for small L 2 data. The result follows from an application of the “Imethod”. This method allows to define a modification of the energy norm H 1 that is “almost conserved ” and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally wellposed for large data in H s, for s> 2/3.
Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin
 Dyn. Syst
"... Abstract. We study the longtime behaviour of the focusing cubic NLS on R in the Sobolev norms H s for 0 < s < 1. We obtain polynomial growthtype upper bounds on the H s norms, and also limit any orbital H s instability of the ground state to polynomial growth at worst; this is a partial anal ..."
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Cited by 14 (3 self)
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Abstract. We study the longtime behaviour of the focusing cubic NLS on R in the Sobolev norms H s for 0 < s < 1. We obtain polynomial growthtype upper bounds on the H s norms, and also limit any orbital H s instability of the ground state to polynomial growth at worst; this is a partial analogue of the H 1 orbital stability result of Weinstein [27], [26]. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the “Imethod ” from earlier papers [9][15] which pushes down from the energy norm, as well as an “upsidedown Imethod ” which pushes up from the L 2 norm. 1.
Local wellposedness of the YangMills equation in the temporal gauge below the energy norm, preprint
"... Abstract. We show that the YangMills equation in three dimensions in the Temporal gauge is locally wellposed in H s for s> 3/4 if the H s norm is sufficiently small. The temporal gauge is slightly less convenient technically than the more popular Coulomb gauge, but has the advantage of uniquene ..."
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Cited by 12 (0 self)
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Abstract. We show that the YangMills equation in three dimensions in the Temporal gauge is locally wellposed in H s for s> 3/4 if the H s norm is sufficiently small. The temporal gauge is slightly less convenient technically than the more popular Coulomb gauge, but has the advantage of uniqueness even for large initial data, and does not require solving a nonlinear elliptic problem. To handle the temporal gauge correctly we project the connection into curlfree and divergencefree components, and develop some new bilinear estimates of X s,b type which can handle integration in the time direction. 1.
Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm
 Commun. Pure Appl. Anal
"... Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing nonlinear Schrödinger equations in the H s (R n) norm for 1 − ε < s < 1, for small ε. In the L 2subcritical case we obtain a polynomial bound for the time required to move ..."
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Cited by 9 (3 self)
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Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing nonlinear Schrödinger equations in the H s (R n) norm for 1 − ε < s < 1, for small ε. In the L 2subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the H 1subcritical case then we cannot show this, but for defocussing equations we obtain global wellposedness and polynomial growth of H s norms for s sufficiently close to 1. 1.
FACHBEREICH MATHEMATIK UND NATURWISSENSCHAFTEN FACHGRUPPE PHYSIK
, 2008
"... Measurements and searches with top quarks Measurement of the top quark pair production cross section, the ratio of branching fractions and searches for new physics in the top quark sector with the DØ experiment ..."
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Cited by 1 (0 self)
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Measurements and searches with top quarks Measurement of the top quark pair production cross section, the ratio of branching fractions and searches for new physics in the top quark sector with the DØ experiment
ORBITAL INSTABILITY BOUNDS FOR THE NONLINEAR SCHRÖDINGER EQUATION BELOW THE ENERGY NORM
, 2005
"... Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focusing nonlinear Schrödinger equations in the H s norm for 1 − ε < s < 1. In the L 2subcritical case we obtain a polynomial bound for the time required to move away from the ground ..."
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Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focusing nonlinear Schrödinger equations in the H s norm for 1 − ε < s < 1. In the L 2subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the H 1subcritical case then we cannot show this, but we can obtain global wellposedness and polynomial growth of H s norms for s sufficiently close to 1. 1.
Contents
, 2006
"... Abstract. We prove an endpoint multilinear estimate for the X s,b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local wellposedness results for periodic generalized KdV equations, as well as some global wellposedness results below the energy norm. ..."
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Abstract. We prove an endpoint multilinear estimate for the X s,b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local wellposedness results for periodic generalized KdV equations, as well as some global wellposedness results below the energy norm.
Multilinear weighted convolution of L² functions, and applications to nonlinear dispersive equations
 AMER. J. MATH
, 2000
"... The X S B Spaces as Used Beals, . . . KlainermanMachedon and others, are fundamental tools to study the lowregularity behaviour of nonlinear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel’s theorem and duality, t ..."
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The X S B Spaces as Used Beals, . . . KlainermanMachedon and others, are fundamental tools to study the lowregularity behaviour of nonlinear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel’s theorem and duality, these estimates reduce to estimating a weighted convolution integral in terms of the L² norms of the component functions. In this paper we systematically study weighted convolution estimates on L². As a consequence we obtain sharp bilinear estimates for the KdV, wave, and Schrödinger X s,b spaces.