scholarly journals Existence and Uniqueness of Weak Solutions for Novel Anisotropic Nonlinear Diffusion Equations Related to Image Analysis

2021 ◽  
Vol 2021 ◽  
pp. 1-18 ◽  
Author(s):  
Anas Tiarimti Alaoui ◽  
Mostafa Jourhmane
Keyword(s):  
Weak Solutions ◽  
Nonlinear Diffusion ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Monotonicity Condition ◽  
Restoration Technique

This paper establishes the existence and uniqueness of weak solutions for the initial-boundary value problem of anisotropic nonlinear diffusion partial differential equations related to image processing and analysis. An implicit iterative method combined with a variational approach has been applied to construct approximate solutions for this problem. Then, under some a priori estimates and a monotonicity condition, the existence of unique weak solutions for this problem has been proven. This work has been complemented by a consistent and stable approximation scheme showing its great significance as an image restoration technique.

scholarly journals Kawahara-Burgers Equation on a Strip

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
N. A. Larkin
Keyword(s):  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regular Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Initial Boundary Value ◽  
Channel Type

An initial-boundary value problem for the 2D Kawahara-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in theL2-norm.

scholarly journals The 2D Zakharov-Kuznetsov-Burgers equation on a strip

2016 ◽  
Vol 34 (1) ◽  
pp. 151-172 ◽  
Author(s):  
Nikolai Andreevitch Larkine
Keyword(s):  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regular Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Initial Boundary Value ◽  
Channel Type

An initial-boundary value problem for the 2D Zakharov-Kuznetsov-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in the $L^2$-norm.

GLOBAL WEAK SOLUTIONS OF AN INITIAL BOUNDARY VALUE PROBLEM FOR SCREW PINCHES IN PLASMA PHYSICS

2009 ◽  
Vol 19 (06) ◽  
pp. 833-875 ◽  
Author(s):  
JIANWEN ZHANG ◽  
SONG JIANG ◽  
FENG XIE
Keyword(s):  
Magnetic Field ◽  
Boundary Value Problem ◽  
Plasma Physics ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Field Equations ◽  
Initial Boundary Value

This paper is concerned with an initial-boundary value problem for screw pinches arisen from plasma physics. We prove the global existence of weak solutions to this physically very important problem. The main difficulties in the proof lie in the presence of 1/x-singularity in the equations at the origin and the additional nonlinear terms induced by the magnetic field. Solutions will be obtained as the limit of the approximate solutions in annular regions between two cylinders. Under certain growth assumption on the heat conductivity, we first derive a number of regularities of the approximate physical quantities in the fluid region, as well as a lot of uniform integrability in the entire spacetime domain. By virtue of these estimates we then argue in a similar manner as that in Ref. 20 to take the limit and show that the limiting functions are indeed a weak solution which satisfies the mass, momentum and magnetic field equations in the entire spacetime domain in the sense of distributions, but satisfies the energy equation only in the compact subsets of the fluid region. The analysis in this paper allows the possibility that energy is absorbed into the origin, i.e. the total energy be possibly lost in the limit as the inner radius goes to zero.

scholarly journals On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 136
Author(s):  
Mikhail Kamenskii ◽  
Valeri Obukhovskii ◽  
Garik Petrosyan ◽  
Jen-Chih Yao
Keyword(s):  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Vector Fields ◽  
A Priori Estimates ◽  
A Priori ◽  
Linear Part ◽  
Monotonicity Condition ◽  
Diffusion Type ◽  
Estimates Of Solutions ◽  
Linear Differential

We obtained results on the existence and uniqueness of a mild solution for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a Carathéodory-type multivalued nonlinearity satisfying some monotonicity condition in the phase variables. We used the Yosida approximations of the linear part of the inclusion, the method of a priori estimates of solutions, and the topological degree method for condensing vector fields. As an example, we considered the existence and uniqueness of a solution to the Cauchy problem for a system governed by a perturbed subdifferential inclusion of a fractional diffusion type.

scholarly journals Existence and Uniqueness of Weak Solutions for the Model Representing Motions of Curves Made of Elastic Materials

2021 ◽  
Vol 36 ◽  
pp. 44-56
Author(s):  
T. Aiki ◽  
◽  
C. Kosugi ◽  
Keyword(s):  
Mathematical Model ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Beam Equation ◽  
Elastic Materials ◽  
Existence Of Weak Solutions ◽  
Uniqueness And Existence ◽  
Initial Boundary Value

We consider the initial boundary value problem for the beam equation with the nonlinear strain. In our previous work this problem was proposed as a mathematical model for stretching and shrinking motions of the curve made of the elastic material on the plane. The aim of this paper is to establish uniqueness and existence of weak solutions. In particular, the uniqueness is proved by applying the approximate dual equation method.

scholarly journals Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain

2016 ◽  
Vol 16 (3) ◽  
pp. 195
Author(s):  
Ivo Fernandez Lopez ◽  
Gladson Octaviano Antunes ◽  
Maria Darci Godinho Da Silva ◽  
Luiz Adauto Da Justa Medeiros
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Title Page ◽  
Cylindrical Domain ◽  
Noncylindrical Domain

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>In this paper we investigate the existence and uniqueness of solution for a initial boundary value problem for the following nonlinear wave equation: </span></p><p><span>u′′</span> − ∆ u + | u | ˆρ = f in Q</p><div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>where </span><span>Q </span><span>represents a non-cylindrical domain of </span><span>R^{</span><span>n</span><span>+</span><span>1}</span><span>. The methodology, cf. Lions [3], consists of transforming this problem, by means of a perturbation depending on a parameter </span><span>ε &gt; </span><span>0, into another one defined in a cylindrical domain </span><span>Q </span><span>containing </span><span>Q</span><span>. By solving the cylindrical problem, we obtain estimates that depend on </span><span>ε</span><span>. These ones will enable a passage to the limit, when </span><span>ε </span><span>goes to zero, that will guarantee, later, a solution for the non-cylindrical problem. The nonlinearity </span><span>|</span><span>u_</span><span>ε</span><span>|^</span><span>ρ </span><span>introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar [8] plus a contradiction process. </span></p></div></div></div></div></div></div>

On the Dynamic Contact Problem for a Viscoelastic Plate

2010 ◽  
Author(s):  
Igor Bock
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dynamic Contact ◽  
Viscoelastic Plate ◽  
Weak Formulation ◽  
Priori Estimates ◽  
Elliptic Variational Inequalities ◽  
Initial Boundary Value

We deal with an initial-boundary value problem describing the perpendicular vibrations of an anisotropic viscoelastic plate free on its boundary and with a rigid inner obstacle. A weak formulation of the problem is in the form of the hyperbolic variational inequality. We solve the problem using the discretizing the time variable. The elliptic variational inequalities for every time level are uniquely solved. We derive the a priori estimates and the convergence of the sequence of segment line functions to a variational solution of the considered problem.

scholarly journals Parabolic Nonlinear Second Order Slip Reynolds Equation: Approximation and Existence

2007 ◽  
Vol 12 (1) ◽  
pp. 3-20
Author(s):  
K. Ait Hadi
Keyword(s):  
Reynolds Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Priori Estimates ◽  
Degenerate Parabolic ◽  
Initial Boundary Value ◽  
Model Existence ◽  
Nonlinear Degenerate

This work studies an initial boundary value problem for nonlinear degenerate parabolic equation issued from a lubrication slip model. Existence of solutions is established through a semi discrete scheme approximation combined with some a priori estimates.

XXII.—A Priori Estimates and Nonlinear Parabolic Equations of Arbitrary Order

1972 ◽  
Vol 69 (4) ◽  
pp. 287-293
Author(s):  
D. E. Edmunds ◽  
C. A. Stuart
Keyword(s):  
Parabolic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic Equations ◽  
Linear Parabolic Equation ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Initial Boundary Value

SynopsisIn this paper it is shown that the question of the existence of a classical solution of the first initial-boundary value problem for a non-linear parabolic equation may be reduced to the problem of the derivation of suitable a priori bounds.

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