Unique solvability of the water waves problem in Sobolev spaces

2016 ◽  
Vol 57 (1) ◽  
pp. 97-123
Author(s):  
V. I. Nalimov
Keyword(s):  
Sobolev Spaces ◽  
Unique Solvability ◽  
Water Waves

scholarly journals W2,p-Solvability of the Dirichlet Problem for Elliptic Equations with Singular Data

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Loredana Caso ◽  
Roberta D’Ambrosio ◽  
Maria Transirico
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Unique Solvability ◽  
Elliptic Equations ◽  
Weighted Sobolev Spaces ◽  
Elliptic Partial Differential Equations ◽  
Planar Case ◽  
Singular Data

We give an overview on some results concerning the unique solvability of the Dirichlet problem inW2,p,p>1, for second-order linear elliptic partial differential equations in nondivergence form and with singular data in weighted Sobolev spaces. We also extend such results to the planar case.

scholarly journals Hadamard well-posedness of the gravity water waves system

2016 ◽  
Vol 13 (04) ◽  
pp. 791-820
Author(s):  
Quang-Huy Nguyen
Keyword(s):  
Initial Data ◽  
Unique Solvability ◽  
Water Waves ◽  
Well Posedness ◽  
Initial Surface ◽  
Free Surfaces ◽  
Low Regularity ◽  
Sobolev Embeddings ◽  
Pure Gravity ◽  
Invent Math

We consider the system of (pure) gravity water waves in any dimension and in a fluid domain with a general bottom geometry. The unique solvability of this problem was established by Alazard–Burq–Zuily [Invent. Math. 198(1) (2014) 71–163] at a low regularity level where the initial surface is [Formula: see text] in terms of Sobolev embeddings; this result allows the existence of free surfaces with unbounded curvature. Our result states that the solutions obtained in the above work depend continuously on initial data in the strong topology in which the solutions are constructed. This establishes a well-posedness result in the sense of Hadamard.

scholarly journals Boundary value problems for elliptic pseudodifferential operators II

1997 ◽  
Vol 127 (2) ◽  
pp. 395-405 ◽  
Author(s):  
Kazuaki Taira
Keyword(s):  
Probability Theory ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Markov Processes ◽  
Sobolev Spaces ◽  
Unique Solvability ◽  
Pseudodifferential Operators ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Intuitive Interpretation

The purpose of this paper is to study boundary value problems for elliptic pseudodifferential operators which originate from the problem of existence of Markov processes in probability theory, generalising some results of our previous work. Our approach has a great advantage of intuitive interpretation of sufficient conditions for the unique solvability of boundary value problems in terms of Markovian motion. In fact, we prove that if a Markovian particle moves incessantly both by jumps and continuously in the state space, not being trapped in the set where no reflection phenomenon occurs, then our boundary value problem is uniquely solvable in the framework of Sobolev spaces of LP style.

On uniqueness in the problem of gravity–capillary water waves above submerged bodies

2009 ◽  
Vol 465 (2106) ◽  
pp. 1743-1761 ◽  
Author(s):  
Oleg V. Motygin ◽  
Philip McIver
Keyword(s):  
Surface Tension ◽  
Unique Solvability ◽  
Water Waves ◽  
Linear Problem ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Trapped Modes ◽  
Capillary Water ◽  
Simple Bounds

In this paper, we consider the two-dimensional linear problem of wave–body interaction with surface tension effects being taken into account. We suggest a criterion for unique solvability of the problem based on symmetrization of boundary integral equations. The criterion allows us to develop an algorithm for detecting non-uniqueness (finding trapped modes) for given geometries of bodies; examples of numerical computation of trapped modes are given. We also prove a uniqueness theorem that provides simple bounds for the possible non-uniqueness parameters.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

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