scholarly journals Large data local well-posedness for a class of KdV-type equations

2014 ◽  
Vol 367 (2) ◽  
pp. 755-773 ◽  
Author(s):  
Benjamin Harrop-Griffiths
Keyword(s):  
Large Data ◽  
Well Posedness ◽  
Kdv Type Equations

scholarly journals Sharp local well-posedness of KdV type equations with dissipative perturbations

2016 ◽  
Vol 74 (3) ◽  
pp. 571-594 ◽  
Author(s):  
Xavier Carvajal ◽  
Mahendra Panthee
Keyword(s):  
Well Posedness ◽  
Kdv Type Equations

scholarly journals Local well-posedness of the fifth-order KdV-type equations on the half-line

2019 ◽  
Vol 18 (5) ◽  
pp. 2607-2661 ◽  
Author(s):  
Márcio Cavalcante ◽  
◽  
Chulkwang Kwak ◽  
Keyword(s):  
Half Line ◽  
Well Posedness ◽  
Kdv Type Equations ◽  
Fifth Order

Global well-posedness and scattering for the defocusing cubic Schrödinger equation on waveguide ℝ2 × 𝕋2

2019 ◽  
Vol 16 (01) ◽  
pp. 73-129 ◽  
Author(s):  
Zehua Zhao
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Data ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Resonant System ◽  
Induction Method ◽  
Profile Decomposition ◽  
Cubic Schrödinger Equation ◽  
Cubic Nonlinear Schrödinger Equation

We consider the problem of large data scattering for the defocusing cubic nonlinear Schrödinger equation on [Formula: see text]. This equation is critical both at the level of energy and mass. The key ingredients are global-in-time Stricharz estimate, resonant system approximation, profile decomposition and energy induction method. Assuming the large data scattering for the 2d cubic resonant system, we prove the large data scattering for this problem. This problem is the cubic analogue of a problem studied by Hani and Pausader.

scholarly journals Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities

2014 ◽  
Vol 215 ◽  
pp. 67-149 ◽  
Author(s):  
Jerry L. bona ◽  
Jonathan Cohen ◽  
Gang Wang
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Coupled Systems ◽  
Well Posedness ◽  
Initial Value ◽  
Quadratic Polynomials ◽  
Partial Differentiation ◽  
Quadratic Nonlinearities ◽  
Kdv Type Equations

AbstractIn this paper, coupled systemsof Korteweg-de Vries type are considered, where u = u(x, t), v = v(x, t) are real-valued functions and where x, t∈R. Here, subscripts connote partial differentiation andare quadratic polynomials in the variables u and v. Attention is given to the pure initial-value problem in which u(x, t) and v(x, t) are both specified at t = 0, namely,for x ∈ ℝ. Under suitable conditions on P and Q, global well-posedness of this problem is established for initial data in the L2-based Sobolev spaces Hs(ℝ) × Hs(ℝ) for any s > ‒3/4.

scholarly journals BILINEAR ESTIMATES AND APPLICATIONS TO NONLINEAR WAVE EQUATIONS

2002 ◽  
Vol 04 (02) ◽  
pp. 223-295 ◽  
Author(s):  
SERGIU KLAINERMAN ◽  
SIGMUND SELBERG
Keyword(s):  
Systematic Review ◽  
Cauchy Problem ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Large Data ◽  
Nonlinear Wave Equations ◽  
Well Posedness ◽  
Bilinear Estimates ◽  
The Cauchy Problem

We undertake a systematic review of results proved in [26, 27, 30-32] concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we give a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates.

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