Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle

Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis

doi:10.1002/cpa.3160460405

Cited by 1,146 publications

(1,228 citation statements)

References 73 publications

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“…, u(x, 0) = u 0 (x), (1.8) we showed in [KPV1] that for k ≥ 4 (1.8) is locally well posed in H s (R), s ≥ s(k) = (k − 4)/2k, as the scaling argument suggests, and in [BKPSV] that these results are sharp. Also in [KPV1] we established similar local existence results for k = 2, 3 and s(k) = 1/4, 1/12 respectively. In [KPV3], for the KdV equation, k = 1, we obtain local well posedness for s > −3/4 (see also [B]).…”

Section: Introductionmentioning

confidence: 87%

Quadratic forms for the 1-D semilinear Schrödinger equation

Kenig

Ponce

Vega

1996

Trans. Amer. Math. Soc.

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127

107

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Abstract. This paper is concerned with 1-D quadratic semilinear Schrödinger equations. We study local well posedness in classical Sobolev space H s of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result. We prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered.

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Section: Introductionmentioning

confidence: 87%

Quadratic forms for the 1-D semilinear Schrödinger equation

Kenig

Ponce

Vega

1996

Trans. Amer. Math. Soc.

Self Cite

127

107

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“…(1) The existence and uniqueness of the solutions of (3.7) is classical: see [24] for the local well-posedness of the KdV equation, and [12] for the global well-posedness. One obtains the estimate as usual: as we multiply equation (3.7) by Λ 2k u i (with 3/2 < k ≤ s + 2) and integrate with respect to the space variable, one obtains…”

Section: Rigorous Demonstrationmentioning

confidence: 99%

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Duchêne¹

2011

ESAIM: M2AN

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Abstract. We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Choi and R. Camassa, J. Fluid Mech. 313 (1996) 83-103]. We study the wellposedness of such systems, and the asymptotic convergence of their solutions towards solutions of the full Euler system. Then, we provide a rigorous justification of the so-called KdV approximation, stating that any bounded solution of the full Euler system can be decomposed into four propagating waves, each of them being well approximated by the solutions of uncoupled Korteweg-de Vries equations. Our method also applies for models with the rigid lid assumption, using the Boussinesq/Boussinesq models introduced in [J.L. Bona, D. Lannes and J.-C. Saut, J. Math. Pures Appl. 89 (2008) 538-566]. Our explicit and simultaneous decomposition allows to study in details the behavior of the flow depending on the depth and density ratios, for both the rigid lid and free surface configurations. In particular, we consider the influence of the rigid lid assumption on the evolution of the interface, and specify its domain of validity. Finally, solutions of the Boussinesq/Boussinesq systems and the KdV approximation are numerically computed, using a Crank-Nicholson scheme with a predictive step inspired from [C.

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“…For the KdV equation (1.1), the previous result of local well-posedness for s > 3=4 obtained in [12] by exploiting the Kato on fjxj g 1g) which produce low-frequency components (i.e., components in fjxj a 1g). We call such interactions high-high-low, and also define the interactions of high-low-high or high-high-high type in the same manner.…”

Section: Introductionmentioning

confidence: 79%

Counterexamples to Bilinear Estimates for the Korteweg-de Vries Equation in the Besov-Type Bourgain Space

Kishimoto

2010

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Abstract.Recently it was shown by Z. Guo and the author independently that the Cauchy problem for the nonperiodic Korteweg-de Vries equation is locally well-posed in the Sobolev space of the critical regularity. Their proofs were based on the iteration argument in the Besov endpoint of the Bourgain space with some additional modification in low frequency, and it was conjectured that the Bourgain space with the Besov-type modification only does not restore the bilinear estimate which is essential to the iteration. In the present article we give the answer to this conjecture by constructing an exact counterexample to the bilinear estimate.

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Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle

Cited by 1,146 publications

References 73 publications

Quadratic forms for the 1-D semilinear Schrödinger equation

Quadratic forms for the 1-D semilinear Schrödinger equation

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Counterexamples to Bilinear Estimates for the Korteweg-de Vries Equation in the Besov-Type Bourgain Space

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