Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation
J Ginibre,Giorgio Velo +1 more
TLDR
In this paper, it was shown that the generalized Benjamin-Ono equation exhibits smoothing properties similar to those of the generalized Korteweg-de Vries equation (GKdV), which is the special case μ = 1 of (∗), if V satisfies suitable estimates at zero and at infinity.About:
This article is published in Journal of Differential Equations.The article was published on 1991-09-01 and is currently open access. It has received 83 citations till now. The article focuses on the topics: Benjamin–Ono equation.read more
Citations
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MonographDOI
Nonlinear dispersive equations : local and global analysis
TL;DR: In this paper, the Korteweg de Vries equation was used for ground state construction in the context of semilinear dispersive equations and wave maps from harmonic analysis.
Journal ArticleDOI
Well-posedness of the initial value problem for the Korteweg-de Vries equation
TL;DR: In this paper, the authors studied the local and global well-posedness of the IVP (1.1) in classical Sobolev spaces Hs(R) and showed that the global wellposedness depends on the available local theory and on the conservation laws satisfied by solutions of (1).
Journal ArticleDOI
Smoothing properties and retarded estimates for some dispersive evolution equations
J. Ginibre,Giorgio Velo +1 more
TL;DR: In this article, a general formulation for dispersive evolution equations is presented, which makes the separation between the two types of ingredients as clear as possible, and illustrate it with the examples of the Schrodinger equation, of the wave equation, and of a class of 1+1 dimensional equations related to the Benjamin-Ono equation.
Journal ArticleDOI
Ill-Posedness Issues for the Benjamin--Ono and Related Equations
TL;DR: It is established that the Cauchy problem for the Benjamin--Ono equation and for a rather general class of nonlinear dispersive equations with dispersion slightly weaker than that of the Korteweg--de Vries equation cannot be solved by an iteration scheme based on the Duhamel formula, and the flow map fails to be smooth.
Journal ArticleDOI
Global well-posedness of the Benjamin–Ono equation in low-regularity spaces
TL;DR: This paper proved that the Benjamin-Ono initial value problem is globally well-posed in the Sobolev spaces. But they did not prove that the problem is solvable in the Euclidean space.
References
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Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Journal ArticleDOI
The Korteweg–deVries Equation: A Survey of Results
TL;DR: A survey of results for the Korteweg-deVries equation can be found in this paper, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method.
Journal ArticleDOI
Local smoothing properties of dispersive equations
TL;DR: In this article, the authors describe a general local smoothing effect for dispersive equations and systems, including the K-dV, Benjamin-Ono, intermediate long wave, various Boussinesq, and Schrodinger equations.