On equations of KP-type
01 Jan 1998-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (Royal Society of Edinburgh Scotland Foundation)-Vol. 128, Iss: 4, pp 725-743
TL;DR: In this article, the local Cauchy problem for the generalized Kadomtsev-Petviashvili equation is discussed in both the periodic and non-periodic settings.
Abstract: We discuss the local Cauchy problem for the generalised Kadomtsev–Petviashvili equation, namely , in the periodic and nonperiodic settings.
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TL;DR: In this paper, the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces, is studied.
Abstract: The main results of this paper are concerned with the "bad" behavior of the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces. This leads to some kind of ill-posedness of the corresponding Cauchy problem: the flow map cannot be of class $C\sp 2$ in any Sobolev space.
127 citations
15 Mar 2001
TL;DR: In this article, Fourier series and periodic distributions are applied to ODEs. But they are not applied to linear equations, they are applied only to partial differential equations, and not to linear evolution equations.
Abstract: Part I. Fourier Series and Periodic Distributions: 1. Preliminaries 2. Fourier series: basic theory 3. Periodic distributions and Sobolev spaces Part II. Applications to Partial Differential Equations: 4. Linear equations 5. Nonlinear evolution equations 6. The Korteweg-de Vries Part III. Some Nonperiodic Problems: 7. Distributions, Fourier transforms and linear equations 8. KdV, BO and friends Appendix A. Tools from the theory of ODEs Appendix B. Commutator estimates Bibliography Index.
124 citations
TL;DR: In this article, the KP-I initial value problem was shown to be NP-hard, and it was shown that the solution of the problem is polynomial in the number of vertices.
Abstract: We prove that the KP-I initial-value problem
$$\begin{cases} \partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0 \,\text{ on }\,\mathbb{R}^2_{x,y}\times\mathbb{R}_t;\\ u(0)=\phi, \end{cases}$$
is globally well-posed in the energy space
$$\mathbf{E}^1(\mathbb{R}^2)=\big\{\phi:\mathbb{R}^2\to\mathbb{R}: \|\phi\|_{\mathbf{E}^1(\mathbb{R}^2)}\approx\|\phi\|_{L^2}+\|\partial_x\phi\|_{L^2}+\big\|\partial_x^{-1}\partial_y\phi\big\|_{L^2}<\infty\big\}.$$
119 citations
TL;DR: In this article, it was shown that the KP-I initial value problem is globally well-posed in the natural energy space of the equation, and that it can be solved efficiently.
Abstract: We prove that the KP-I initial value problem is globally well-posed in the natural energy space of the equation.
111 citations
TL;DR: In this paper, the global well-posedness of the KP-I equation was proved for data of arbitrary size in a suitable Sobolev class, using a compactness method to get local solutions.
Abstract: We prove the global well-posedness of the KP-I equation in ${\Bbb R}^2$
for data of arbitrary size in a suitable Sobolev class. We use a compactness method to get local solutions. We extend the solutions thanks to some conservation laws, an almost conserved quantity and Strichartz estimates.
89 citations
References
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11 Feb 1992
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
Abstract: 1 Generation and Representation.- 1.1 Uniformly Continuous Semigroups of Bounded Linear Operators.- 1.2 Strongly Continuous Semigroups of Bounded Linear Operators.- 1.3 The Hille-Yosida Theorem.- 1.4 The Lumer Phillips Theorem.- 1.5 The Characterization of the Infinitesimal Generators of C0 Semigroups.- 1.6 Groups of Bounded Operators.- 1.7 The Inversion of the Laplace Transform.- 1.8 Two Exponential Formulas.- 1.9 Pseudo Resolvents.- 1.10 The Dual Semigroup.- 2 Spectral Properties and Regularity.- 2.1 Weak Equals Strong.- 2.2 Spectral Mapping Theorems.- 2.3 Semigroups of Compact Operators.- 2.4 Differentiability.- 2.5 Analytic Semigroups.- 2.6 Fractional Powers of Closed Operators.- 3 Perturbations and Approximations.- 3.1 Perturbations by Bounded Linear Operators.- 3.2 Perturbations of Infinitesimal Generators of Analytic Semigroups.- 3.3 Perturbations of Infinitesimal Generators of Contraction Semigroups.- 3.4 The Trotter Approximation Theorem.- 3.5 A General Representation Theorem.- 3.6 Approximation by Discrete Semigroups.- 4 The Abstract Cauchy Problem.- 4.1 The Homogeneous Initial Value Problem.- 4.2 The Inhomogeneous Initial Value Problem.- 4.3 Regularity of Mild Solutions for Analytic Semigroups.- 4.4 Asymptotic Behavior of Solutions.- 4.5 Invariant and Admissible Subspaces.- 5 Evolution Equations.- 5.1 Evolution Systems.- 5.2 Stable Families of Generators.- 5.3 An Evolution System in the Hyperbolic Case.- 5.4 Regular Solutions in the Hyperbolic Case.- 5.5 The Inhomogeneous Equation in the Hyperbolic Case.- 5.6 An Evolution System for the Parabolic Initial Value Problem.- 5.7 The Inhomogeneous Equation in the Parabolic Case.- 5.8 Asymptotic Behavior of Solutions in the Parabolic Case.- 6 Some Nonlinear Evolution Equations.- 6.1 Lipschitz Perturbations of Linear Evolution Equations.- 6.2 Semilinear Equations with Compact Semigroups.- 6.3 Semilinear Equations with Analytic Semigroups.- 6.4 A Quasilinear Equation of Evolution.- 7 Applications to Partial Differential Equations-Linear Equations.- 7.1 Introduction.- 7.2 Parabolic Equations-L2 Theory.- 7.3 Parabolic Equations-Lp Theory.- 7.4 The Wave Equation.- 7.5 A Schrodinger Equation.- 7.6 A Parabolic Evolution Equation.- 8 Applications to Partial Differential Equations-Nonlinear Equations.- 8.1 A Nonlinear Schroinger Equation.- 8.2 A Nonlinear Heat Equation in R1.- 8.3 A Semilinear Evolution Equation in R3.- 8.4 A General Class of Semilinear Initial Value Problems.- 8.5 The Korteweg-de Vries Equation.- Bibliographical Notes and Remarks.
11,637 citations
03 Jul 1975-Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences
TL;DR: For the Korteweg-de-vries model equation, existence, uniqueness, regularity, and continuous dependence results are established for both the pure initial value problem and the continuous dependence model equation as mentioned in this paper.
Abstract: For the Korteweg-de Vries equation u t + u x + u u x + u x x x = 0 , existence, uniqueness, regularity and continuous dependence results are established for both the pure initial-value problem (posed on -∞ x x ⩽ l with periodic initial data). The results are sharper than those obtained previously in that the solutions provided have the same number of L 2 -derivatives as the initial data and these derivatives depend continuously on time, as elements of L 2 . The same equation with dissipative and forcing terms added is also examined. A by-product of the methods used is an exact relation between solutions of the Korteweg-de Vries equation and solutions of an alternative model equation recently studied by Benjamin, Bona & Mahony (1972). It is proven that in the long-wave limit under which these equations are generally derived, the solutions of the two models posed for the same initial data are the same. In the process of carrying out this programme, new results are obtained for the latter model equation.
715 citations
455 citations
TL;DR: In this article, the authors proved the existence, uniqueness, and continuous dependence on the initial data for the local (in time) solution of the Korteweg-de Vries equation on the real line.
Abstract: Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line, with the initial function ϕ in the Sobolev space of order s>3/2 and the solution u(t) staying in the same space, s=∞ being included For the proper KdV equation, existence of global solutions follows if s≥2. The proof is based on the theory of abstract quasilinear evolution equations developed elsewhere.
280 citations