Global well-posedness of the Benjamin–Ono equation in low-regularity spaces

By A. Ionescu, Carlos E. Kenig. 2005

249 citations recorded by the source corpus.

We prove that the Benjamin-Ono initial value problem is globally well-posed in the Sobolev spaces $H^\sigma_r$, $\sigma\geq 0$.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0380d887f7a44f6bd9ff71644db4a570c015c390

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In Applied Mathematics

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Citations in the corpus

Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation (2008, background): “Structures of this kind of spaces were introduced, for instance, in [11], [4] and [3] for the BO equation.”
Asymptotic stability of solitons for the Benjamin-Ono equation (2008, methodology): “The Cauchy problem for (1) is globally well-posed in H, for any s ≥ 0 (see Tao [25] for s ≥ 1 and Ionescu and Kenig [11] for the case s ≥ 0, see also Burq and Planchon [5] for the case s > 14 ).”
Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type (2012, background): “The local and global well-posedness in Sobolev class for (2.1) and many generalizations of it (other powers upux, other linear terms ∂x|Dx|αu, 1 α 2, etc) have been studied by several authors in the last years: see for example Molinet, Saut & Tzvetkov [31], Colliander, Kenig & Staffilani [14], T”
Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations (2010, unknown): “[(αn + 1)(2αn + 1)] −1 (∫ Q2[αn] )αn = j1,[αn](Q[αn]) ≤ j1,[αn](Q[2]) → j1,[2](Q[2]) = [15] (∫ Q(2)[2] )2 .”
The Cauchy problem for the Benjamin-Ono equation in $L^2$ revisited (2010, methodology): “Very recently, Burq and Planchon [7], and Ionescu and Kenig [12] were able to use Tao’s ideas in the context of Bourgain’s spaces to prove well-posedness for the Benjamin-Ono equation in H s ( R ) for respectively s > and s ≥ 0.”
Global well-posedness of Korteweg-de Vries equation in H^-3/4 (R) (2008, background): “Structures of this kind of spaces were introduced, for instance, in [20,8] and [7] for the BO equation.”
Global well-posedness in L2 for the periodic Benjamin-Ono equation (2006, background): “This approach has been improved very recently in [6] and [12] where respectively H s ( IR ), s > 0, and L 2 ( IR ) are reached.”
Improvement of the energy method for strongly non resonant dispersive equations and applications (2014, unknown): “This method was proved to be very efficient to obtain the lowest regularity index for solving canonical equations (see [28], [12], [6], [23] for the BO equation and [11] for dispersive generalized BO equation) but has the disadvantage to behave very bad with respect to perturbation of the equation.”
Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces (2008, background): “These l1-type X s,b structures were first introduced in [27] and used in [12,13,26,8].”
The IVP for the dispersion generalized Benjamin–Ono equation in weighted Sobolev spaces (2011, background): “In the case of the BO equation the same well-posedness problem has been considered in [41,1,24,40,31,27,43,35, 8], and [23] where global well-posedness was established for s 0 (for further discussion we refer to [34]).”
Invariant Measures and Long-Time Behavior for the Benjamin–Ono Equation (2014, result): “We also refer to [1, 3, 7, 9, 13, 14, 15, 17, 18] for previous results on the topic, as well as to [4] where the Cauchy problem has been studied in functional spaces larger than L2.”
Invariance of the Gibbs measure for the Benjamin-Ono equation (2012, methodology): “For the Euclidean version we now also have well-posedness in L(2), see Burq-Planchon [4] and Ionescu-Kenig [10].”
On well-posedness for the Benjamin–Ono equation (2005, methodology): “It would be of interest to investigate how the techniques of [10] could be combined with our argument to progress further toward uniqueness of weak L2 solutions (unconditional uniqueness in Lt (L 2 x) or uniqueness of the solutions from [7], which verify a local smoothing inequality).”
The IVP for the Benjamin-Ono equation in weighted Sobolev spaces (2010, unknown): “Let us recall them: in [31] s > 3 was proven, in [1] and [16] s > 3/2, in [30] s 3/2, in [23] s > 5/4, in [20] s > 9/8, in [34] s 1, in [4] s > 1/4, and finally in [15] s 0 was established.”
Global existence and uniqueness of Schrödinger maps in dimensions d... 4 (2007, background): “The resolution spaces constructed in [8] (see also [6] for the 1dimensional version of these spaces) are based on directional L e physical spaces, which are related to local smoothing; in particular, the nonlinear analysis is based on local smoothing and the simple inclusion L∞,2 e · L2,∞ e · L2,∞ e”
Low-regularity Schrödinger maps, II: global well-posedness in dimensions d ≥ 3 (2007, methodology): “These spaces are minor modifications of the resolution spaces already used by the authors in [5] (see also [ 4 ] for the one-dimensional analogues of these resolution spaces).”