On well-posedness for the Benjamin–Ono equation

By Nicolas Burq, F. Planchon. 2005

111 citations recorded by the source corpus.

We prove existence and uniqueness of solutions for the Benjamin–Ono equation with data in $$H^{s}({\mathbb{R}})$$ , s > 1/4. Moreover, the flow is hölder continuous in weaker topologies.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/55ee2c7a4b597b86ed8754d1d2f9b2de71fab4bb

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Citations in the corpus

Asymptotic stability of solitons for the Benjamin-Ono equation (2008, background): “The Cauchy problem for (1) is globally well-posed in Hs, 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 ).”
Global well-posedness of the Benjamin–Ono equation in low-regularity spaces (2005, background): “See also the very recent papers [3] and [15] (which became available after the submission of this work) for other well-posedness results on the Benjamin-Ono equation and the periodic Benjamin-Ono equation.”
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.”
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(R) for respectively s > 1 4 and s ≥ 0.”
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]).”
Invariance of the Gibbs measure for the Benjamin-Ono equation (2012, methodology): “For the Euclidean version we now also have well-posedness in L2, see Burq-Planchon [4] and Ionescu-Kenig [10].”
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”
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.”