Global well-posedness in L2 for the periodic Benjamin-Ono equation

By Luc Molinet. 2006

84 citations recorded by the source corpus.

We prove that the Benjamin-Ono equation is globally well-posed in $ H^s({\Bbb T}) $ for $ s\ge 0 $. Moreover we show that the associated flow-map is Lipschitz on every bounded set of $ H^s_0({\Bbb T}) $, $s\ge 0$, and even real-analytic in this space for small times. This result is sharp in the sense that the flow-map (if it can be defined and coincides with the standard flow-map on $ {H}_0^\infty({\Bbb T}) $) cannot be of class $ C^{1+\alpha} $, $\alpha>0 $, from $ H_0^s({\Bbb T}) $ into $ H_0^s({\Bbb T}) $ as soon as $ s< 0 $.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4093e1d800b7c6e5ceb4dec929e577f490e96344

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

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.”
A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations (2011, background): “For the global well-posedness of the Cauchy problem of the above equation, see [13], [14].”
On the Integrability of the Benjamin‐Ono Equation on the Torus (2019, result): “solution t∈ R→ u(t) ∈ L2 r,0 is almost periodic. Its orbit is relatively compact in L2 r,0. Remark 1.2. (i) The solutions of the BO equation of Theorem 2 coincide with the ones obtained by Molinet in [18] (cf. also [19]). Since the average is conserved by (1.1), the results of Theorem 2 easily exten”
The Cauchy problem for the Benjamin-Ono equation in $L^2$ revisited (2010, methodology): “This leads to a great simplification of the proof the global well-posedness in L2(T) proved in [Molinet 2008].”
On well-posedness for the Benjamin–Ono equation (2005, background): “Statements which are somewhat similar may be found in [19], where uniqueness is obtained in a class which includes the solution and its renormalization by itself.”
Remarks on the Gibbs measures for nonlinear dispersive equations (2014, background): “ment being however not explicited in [7]. In a second time we deal with the Benjamin-Ono equation on the circle S1 = R/(2πZ). This model arises in the study of one-dimensional internal long waves. In [31, 32] L. Molinet has shown that the equation is globally well-posed in L2(S1) and that this resul”
Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type (2012, background): “everal authors in the last years: see for example Molinet, Saut & Tzvetkov [31], Colliander, Kenig & Staﬃlani [14], Tao [37], Kenig & Ionescu [20], Burq & Planchon [13], Molinet [29], [30], and the references therein. On the contrary, to the best of our knowledge, there are few works”