Global well-posedness of the KP-I initial-value problem in the energy space

Ionescu, Alexandru D.; Kenig, Carlos E.; Tataru, Daniel

doi:10.1007/s00222-008-0115-0

Cited by 133 publications

(248 citation statements)

References 30 publications

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“…Other examples are the bounds for the KP-I equation considered in [7]. A large class of bilinear and multilinear estimates have been systematically studied in [9]; however, this does not include the present setup.…”

Section: Setup and Main Resultsmentioning

confidence: 99%

A convolution estimate for two-dimensional hypersurfaces

Bejenaru¹,

Herr²,

Tataru³

2010

Rev. Mat. Iberoam.

111

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Abstract. Given three transversal and sufficiently regular hypersurfaces in R 3 it follows from work of Bennett-Carbery-Wright that the convolution of two L 2 functions supported of the first and second hypersurface, respectively, can be restricted to an L 2 function on the third hypersurface, which can be considered as a nonlinear version of the Loomis-Whitney inequality. We generalize this result to a class of C 1,β hypersurfaces in R 3 , under scaleable assumptions. The resulting uniform L 2 estimate has applications to nonlinear dispersive equations.

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Section: Setup and Main Resultsmentioning

confidence: 99%

A convolution estimate for two-dimensional hypersurfaces

Bejenaru¹,

Herr²,

Tataru³

2010

Rev. Mat. Iberoam.

111

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“…This method has been very effective in establishing a priori bounds on solutions in low regularities (yielding even uniqueness in some cases), in particular, where a solution map is known to fail to be locally uniformly continuous. See [13,24,28,21,29,22,25].…”

Section: 2mentioning

confidence: 99%

“…See Section 7. In the following, we use the formulation introduced by Ionescu-Kenig-Tataru [24]. In Section 3, we provide the precise definitions of the function spaces F s (T ), N s (T ), and E s (T ).…”

Section: 2mentioning

confidence: 99%

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Non-Existence of Solutions for the Periodic Cubic NLS below ${L}^{{2}}$

Guo

2016

Int Math Res Notices

137

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Abstract. We prove non-existence of solutions for the cubic nonlinear Schrödinger equation (NLS) on the circle if initial data belong to H s (T) \ L 2 (T) for some s ∈ (− 1 8, 0). The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short-time Fourier restriction norm method. where u is a complex-valued function. The Cauchy problem (1.1) has been studied extensively from both theoretical and applied points of view. It is known to be one of the simplest partial differential equations (PDEs) with complete integrability [1,2,19]. In the following, however, we only discuss analytical aspects of (1.1) without using the complete integrable structure of the equation. It is well known that (1.1) enjoys several symmetries. In the following, we concentrate on the scaling symmetry and the Galilean symmetry. The scaling symmetry states that if u(x, t) is a solution to (1.1) on R with initial condition u 0 , then u λ (x, t) = λ −1 u(λ −1 x, λ −2 t) is also a solution to (1.1) with the λ-scaled initial condition u λ 0 (x) = λ −1 u 0 (λ −1 x). Associated to the scaling symmetry, there is a scaling-critical Sobolev index s c such that the homogeneousḢ sc -norm is invariant under the dilation symmetry. It is commonly conjectured that a PDE is ill-posed in H s for s < s c . In the case of the one-dimensional cubic NLS, the scaling-critical Sobolev index is s c = − 1 While there is no dilation symmetry in the periodic setting, the heuristics provided by the scaling argument plays an important role even in the periodic setting. For example, a norm inflation result for (1.1) on T analogous to [12] also holds for s < s c = − x u 0 (x). This symmetry also holds on the circle for β ∈ 2Z. Note that the L 2 -norm is invariant under the Galilean symmetry.2 This induces another critical regularity s ∞ crit = 0. Indeed, there is a strong dichotomy in the behavior of solutions for s ≥ 0 and s < 0.Bourgain [3] introduced the so-called Fourier restriction norm method via the X s,bspaces (see (3.1) below) and proved local well-posedness of (1.1) in L 2 (T). Thanks to the L 2 -conservation, this result was immediately extended to global well-posedness in L 2 (T). On the other hand, (1.1) is known to be ill-posed below L 2 (T). Burq-Gérard-Tzvetkov [5] showed that the solution map Φ t : u 0 ∈ H s → u(t) ∈ H s is not locally uniformly continuous if s < 0. See also [10]. Moreover, Christ-Colliander-Tao [11] and Molinet [31] proved that the solution map is indeed discontinuous if s < 0. Our main result in this paper states an even stronger form of ill-posedness holds true for (1.1) in negative Sobolev spaces. Theorem 1.1 (Non-existence of solutions for the cubic NLS). Let s ∈ (− 1 8 , 0) andNote that the condition (ii) is a natural condition to impose since it would follow from the continuity of the solution map :, which is one of the essential components in the well-posedness theory of evolution equations. On the one hand, the previous ill-posedness results [5,10,11,12,31] ...

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“…Motivated by [35], firstly, by using the X s,b spaces introduced by [36][37][38][39][40] and developed in [8,41,42] and the Strichartz estimates established in [19,43], we prove that (1.3) with initial data…”

Section: Introductionmentioning

confidence: 99%