scholarly journals Continuous Dependence on the Heat Source of 2D Large-Scale Primitive Equations in Oceanic Dynamics

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1961
Author(s):  
Yuanfei Li ◽  
Peng Zeng
Keyword(s):  
Heat Source ◽  
Continuous Dependence ◽  
Large Scale ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Primitive Equations ◽  
Weather And Climate ◽  
Inequality Technique ◽  
Initial Boundary Value

In this paper, we consider the initial-boundary value problem for the two-dimensional primitive equations of the large-scale oceanic dynamics. These models are often used to predict weather and climate change. Using the differential inequality technique, rigorous a priori bounds of solutions and the continuous dependence on the heat source are established. We show the application of symmetry in mathematical inequalities in practice.

scholarly journals Convergence Results on the Boundary Conditions for 2D Large-Scale Primitive Equations in Oceanic Dynamics

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Yuanfei Li
Keyword(s):  
Boundary Conditions ◽  
Large Scale ◽  
Weather Prediction ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convergence Result ◽  
Primitive Equations ◽  
Convergence Results ◽  
Initial Boundary Value

In this paper, the initial boundary value problem for the two-dimensional large-scale primitive equations of large-scale oceanic motion in geophysics is considered, which are fundamental models for weather prediction. By establishing rigorous a priori bounds with coefficients and deriving some useful inequalities, the convergence result for the boundary conditions is obtained.

Improved estimates for continuous data dependence in linear elastodynamics

1988 ◽  
Vol 103 (3) ◽  
pp. 535-559 ◽  
Author(s):  
R. J. Knops ◽  
L. E. Payne
Keyword(s):  
Continuous Dependence ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Body ◽  
Linear Elastodynamics ◽  
Sign Definiteness ◽  
Comprehensive Survey ◽  
Ill Posed ◽  
Initial Boundary Value

In a previous paper [6], the present authors established estimates for the continuous dependence of the solution on various data in the initial boundary value problem of linear elastodynamics on a bounded region of space. The main conclusion concerned continuous dependence on the body-force, but also it was shown how this result could be used to derive continuous dependence on the initial data, elasticities, boundary data and initial geometry. The method adopted was based upon logarithmic convexity arguments and hence led naturally to continuity in the sense of Hölder on compact sub-intervals of time. A special feature of the study entailed the lack of any sign-definiteness conditions on the elasticities which, of course, in the absence of any a priori constraint on the solution always gives rise to an ill-posed problem. (See, for instance, the comprehensive survey by Payne [10].)

scholarly journals On initial boundary value problem with Dirichlet integral conditions for a hyperbolic equation with the Bessel operator

2003 ◽  
Vol 2003 (10) ◽  
pp. 487-502
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Hyperbolic Equation ◽  
A Priori ◽  
Weighted Sobolev Spaces ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Bessel Operator ◽  
Integral Conditions ◽  
Initial Boundary Value

We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

scholarly journals On continuous dependence for the mixed problem of microstretch bodies

2017 ◽  
Vol 25 (1) ◽  
pp. 131-143
Author(s):  
M. Marin ◽  
I. Abbas ◽  
C. Cârstea
Keyword(s):  
Hilbert Space ◽  
Initial Data ◽  
Continuous Dependence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Operators ◽  
Evolutionary Equation ◽  
Trans Form ◽  
Semigroups Of Linear Operators ◽  
Initial Boundary Value

AbstractWe do a qualitative study on the mixed initial-boundary value problem in the elastodynamic theory of microstretch bodies. After we trans- form this problem in a temporally evolutionary equation on a Hilbert space, we will use some results from the theory of semigroups of linear operators in order to prove the continuous dependence of the solutions upon initial data and supply terms.

scholarly journals Qualitative results in thermoelasticity of type III for dipolar bodies

2021 ◽  
Vol 29 (1) ◽  
pp. 127-142
Author(s):  
M. Marin ◽  
S. Vlase ◽  
A. Öchsner
Keyword(s):  
Boundary Value Problem ◽  
Continuous Dependence ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Main Section ◽  
Type Iii ◽  
Basic Equations ◽  
Initial Boundary Value ◽  
Dipolar Structure

Abstract In our study we formulated the mixed initial boundary value problem corresponding to the thermoelasticity of type III for bodies with dipolar structure. In main section we approached four qualitative results regarding the solutions for this problem. In two of these (in the first two theorems) we obtained two results of uniqueness, proved in different ways. Also, we proven two results which show that the solutions of the considered problem depend continuously with respect to the supply terms. We use different procedures in the two theorems on continuous dependence, but we essentially rely on the auxiliary results from Section 3 and Gronwall-type inequalities. It is important to emphasize that all results are obtained by imposing on the basic equations and basic conditions, average constraints that are common in the mechanics of continuous solids.

Continuous Dependence on Data for Solutions of Fractional Extended Fisher–Kolmogorov Equation

2018 ◽  
Vol 19 (7-8) ◽  
pp. 735-739
Author(s):  
Pengyu Chen ◽  
Zhen Xin ◽  
Jiahui An
Keyword(s):  
Boundary Value Problem ◽  
General Class ◽  
Continuous Dependence ◽  
Mild Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Kolmogorov Equation ◽  
Initial Boundary Value ◽  
Continuous Dependence On Data

AbstractThis paper is concerned with the continuous dependence of mild solutions on initial values and orders for a general class of initial boundary-value problem to fractional extended Fisher–Kolmogorov equation. The results obtained in this paper can be considered as a contribution to this emerging field.

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.

Uniqueness and continuous dependence in the linear elastodynamic exterior and half-space problems

1986 ◽  
Vol 99 (2) ◽  
pp. 357-366 ◽  
Author(s):  
G. P. Galdi ◽  
R. J. Knops ◽  
S. Rionero
Keyword(s):  
Continuous Dependence ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Result ◽  
Similar Treatment ◽  
Lagrange Identity ◽  
Uniqueness Of The Solution ◽  
Initial Boundary Value ◽  
Continuous Dependence Results

A method based upon the Lagrange identity has been used by Brun [2] in the linear theories of thermoelasticity and viscoelasticity to establish uniqueness of the solution to the initial boundary value problem on bounded three-dimensional regions. A major feature of Brun's analysis is that it does not require any sign-definiteness assumptions on, for instance, the elasticities. The technique was extended by Knops and Payne [14] to derive certain continuous dependence results in linear elastodynamics, again for a bounded region. These authors had earlier recovered Brun's uniqueness result for linear elasticity [11] and derived other continuous dependence results based upon logarithmic convexity arguments [12, 14] (see also [13] for a similar treatment of thermoelasticity). Levine [18] later treated an abstract version of the Brun approach and applied it to a family of abstract linear operator equations. Among his results is a simplified proof that equipartition of the kinetic and potential energies occurs. Other applications of the Lagrange identity in proofs of uniqueness for bounded regions include those by Naghdi and Trapp [19] for a Cosserat surface, and by Green [9] for a theory of linear thermoelasticity that allows second sound.

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