On the Cauchy Problem for the Generalized Korteweg-de Vries Equation

By Luc Molinet, Francis Ribaud. 2003

681 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/56a589986330ec60fd7d8502829388ac4b289289

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Existence of blow-up solutions in the energy space for the critical generalized KdV equation (2001, background): “Local existence in time of solutions of (2) in the energy space is now well understood; see Kato [10], Ginibre and Tsutsumi [8] for the Hs theory (s > 32 ), Kenig, Ponce and Vega [11] for the L2 theory in the case of equation (1) and sharp Hs theory for (2), and Bourgain [3] and [4] for the periodic”
Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations (2010, neutral): “By direct computations, we have the following generalization of the well-known Kato identity ([16]) d dt Mφ(t) = μ− 1 2 ρt ∫ u(2)φA(x̃)dx+ ∫ utuφA(x̃)dx (51)”
A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane (2001, background): “T. Kato, On the Korteweg-de Vries equation, Manuscripta Math.”
Unique continuation and decay for the Korteweg-de Vries equation with localized damping (2005, methodology): “We then apply the classical smoothing properties as in [8, 10], showing that the solution is smooth.”
Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton (2014, background): “(Kato localization identities [9]) Let g be any C(3) function and v(t, x) be a solution of (1.”
On the Cauchy problem for higher-order nonlinear dispersive equations (2007, methodology): “The use of Besov spaces is inspired by the works of Molinet and Ribaud on the Korteweg-de Vries equation [13] and on the Benjamin-Ono equation [14], and of Planchon on the nonlinear Schrödinger equation [16].”
Blow up in finite time and dynamics of blow up solutions for the L^2-critical generalized KdV equation (2002, result): “We refer to Kato [4] and Ginibre and Tsutsumi [3] for previous results on the well-posedness of the Cauchy problem for (4) and to Bourgain [2] for the periodic case.”
An Initial Boundary-Value Problem in a Half-Strip for the Korteweg–De Vries Equation in Fractional-Order Sobolev Spaces (2005, result): “…(1969), Tsutsumi and Mukasa (1971), Bona and Smith (1975), Saut and Temam (1976), Bona and Scott (1976), Kato (1979), Kruzhkov and Faminskii (1983), Kato (1983), Faminskii (1988b), Ginibre and Tsutsumi (1989), Ginibre et al. (1990), Kenig et al. (1989, 1991, 1993a,b, 1996), Bourgain (1993),…”