GLOBAL WELL-POSEDNESS OF THE BENJAMIN–ONO EQUATION IN H1(R)

By Terence Tao. 2003

202 citations recorded by the source corpus.

We show that the Benjamin–Ono equation is globally well-posed in Hs(R) for s≥1. This is despite the presence of the derivative in the nonlinearity, which causes the solution map to not be uniformly continuous in Hs for any s [18]. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0453725c499ec21c4f9ebeafed79f7acb5c8ec52

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Citations in the corpus

Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory (2004, background): “This has been improved by Kenig and Koenig [31] to s> 9/8 and more recently by Tao [ 45 ] to H 1(R).”
Global well-posedness of the Benjamin–Ono equation in low-regularity spaces (2005, methodology): “0) = 0. (2.4) We now let P +higheu = e−iU0w +; P−higheu = eiU0w−; P loweu = w 0, (2.5) 4 ALEXANDRU D. IONESCU AND CARLOS E. KENIG where U 0 is a suitable gauge that depends only on u 0. As in [18], we deﬁne ﬁrst U(0,t) on the time axis x= 0 by the formula ∂tU 0(0,t)+ 1 2 H∂xu 0(0,t)+ 1 4 u2 0 (0,”
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 ).”
The Cauchy problem for the Benjamin-Ono equation in $L^2$ revisited (2010, methodology): “Once again, these works combined Tao’s gauge transformation with estimates in Bourgain’s spaces.”
Invariance of the Gibbs measure for the Benjamin-Ono equation (2012, background): “In [18], Tao introduces a gauge transform to prove the well-posedness result in H(1) for the Euclidean counterpart of (1.”
Global well-posedness in L2 for the periodic Benjamin-Ono equation (2006, methodology): “Ribaud and the author [19] have improved the global well-posedness result to H1(T) by using the gauge transform introduced by T. Tao [ 23 ] combining”
Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces (2008, methodology): “[23] T. Tao, Global well-posedness of the Benjamin–Ono equation in H1(R), J. Hyperbolic Differ.”
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”
Nonlinear wave interactions for the Benjamin-Ono equation. (2004, background): “Even more is true: For initial data in H1(R) there exists a unique global solution u with ∂xu ∈ L(1)loc(R;L(R)) see [17].”
A priori bounds and weak solutions for the nonlinear Schr (2006, unknown): “the solution ucan obtained by analyzing Φϕ(t,u) for weights ϕwhich are more general than hξi2s, and are speciﬁcally adapted to the initial datum u0 (cf. the “frequency envelopes” used for instance in [12]). We have actually proved the following statement more general than that announced earlier. NON”
On well-posedness for the Benjamin–Ono equation (2005, methodology): “• We use (a variant of) onormal spa es Xs,b; in fa t, as remarked by Tao [18℄, should one silently drop the low-high intera tion in the original equation mentioned above, the resulting equation an be proved to be well-posed in X0 +, 1 2 + .”
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.”