The well-posedness of non-linear dispersive equations: some recent developments

2006 ◽  
pp. 53-61 ◽  
Author(s):  
Carlos Kenig
Keyword(s):  
Well Posedness ◽  
Dispersive Equations ◽  
Non Linear ◽  
Recent Developments

ISUM: Its Birth, Growth and Future

2016 ◽  
Author(s):  
Sherif Rashed
Keyword(s):  
Nonlinear Behavior ◽  
Impact Loads ◽  
Coarse Mesh ◽  
Structural Units ◽  
Large Size ◽  
Linear Behavior ◽  
Large Ship ◽  
Non Linear ◽  
Recent Developments ◽  
Future Potential

ISUM (The Idealized Structural Unit Method) was born in 1972 to efficiently and accurately analyze the behavior of large size structures up to and beyond their ultimate strength. In this method a structure is divided into large elements, basically its structural units (members). Geometric and material non-linear behavior inside the element is formulated and expressed at a limited number of nodal points at the element boundaries. In this way a large structure can be modeled using a coarse mesh while still being able to consider the nonlinear behavior until the collapse of the structure. Several ISUM elements have been formulated and used to analyze the non-linear behavior of large ship structures. In further developments, more elements with more accurate formulations have been developed and more types of structures have been analyzed using this method. The same ISUM concept has been applied to the analysis of welding deformation of large welded structures and to failure analysis of structural and mechanical components subjected to impact loads. In this paper, the basic ISUM concept is outlined, and several elements are presented. Examples of applications to ships and marine structures are presented demonstrating the effectiveness of the method. Recent developments are also reviewed and future potential is explored.

scholarly journals Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

2020 ◽  
Author(s):  
Bjoern Bringmann
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Local Existence ◽  
Approximate Solutions ◽  
Well Posedness ◽  
Dispersive Equations ◽  
Deterministic Theory ◽  
Nonlinear Smoothing ◽  
Iterative Decomposition

Abstract We study the derivative nonlinear wave equation $- \partial _{tt} u + \Delta u = |\nabla u|^2$ on $\mathbb{R}^{1 +3}$. The deterministic theory is determined by the Lorentz-critical regularity $s_L = 2$, and both local well-posedness above $s_L$ as well as ill-posedness below $s_L$ are known. In this paper, we show the local existence of solutions for randomized initial data at the super-critical regularities $s\geqslant 1.984$. In comparison to the previous literature in random dispersive equations, the main difficulty is the absence of a (probabilistic) nonlinear smoothing effect. To overcome this, we introduce an adaptive and iterative decomposition of approximate solutions into rough and smooth components. In addition, our argument relies on refined Strichartz estimates, a paraproduct decomposition, and the truncation method of de Bouard and Debussche.

scholarly journals ON GLOBAL ROUGH SOLUTIONS TO A NON-LINEAR SCHRÖDINGER SYSTEM

2009 ◽  
Vol 51 (3) ◽  
pp. 499-511 ◽  
Author(s):  
LI MA ◽  
XIANFA SONG ◽  
LIN ZHAO
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Nonlinear Schrödinger ◽  
Non Linear ◽  
Schrödinger Systems ◽  
Schrödinger System

AbstractThe non-linear Schrödinger systems arise from many important physical branches. In this paper, employing the ‘I-method’, we prove the global-in-time well-posedness for a coupled non-linear Schrödinger system in Hs(n) when n = 2, s > 4/7 and n = 3, s > 5/6, respectively, which extends the results of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao (Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math Res. Lett. 9, 2002, 659–682) to the system.

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