Hölder continuous dependence and uniqueness results for incremental elastodynamics

1986 ◽  
Vol 24 (5) ◽  
pp. 803-812 ◽  
Author(s):  
S. Chiriţä
Keyword(s):  
Continuous Dependence ◽  
Hölder Continuous ◽  
Uniqueness Results

scholarly journals Existence and Uniqueness Results for Two-Dimensional Stochastic Linearised Boussinesq Equation

2020 ◽  
Vol 6 (1) ◽  
pp. 20
Author(s):  
Sofije Hoxha ◽  
Fejzi Kolaneci
Keyword(s):  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Boussinesq Equation ◽  
Initial Condition ◽  
Two Dimensional ◽  
Boussinesq Problem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Drainable Porosity

The water flow in saturated zones of the soil is described by two-dimensional Boussinesq equation. This paper is devoted to investigating the linearised stochastic Boussinesq problem in the presence of randomness in hydraulic conductivity, drainable porosity, recharge, evapotranspiration, initial condition and boundary condition. We use the Sabolev spaces and Galerkin method. Under some suitable assumptions, we prove the existence and uniqueness results, as well as, the continuous dependence on the data for the solution of linearised stochastic Boussinesq problem. Keywords: linearised stochastic Boussinesq equation, Galerkin method, existence and uniqueness results, and continuous dependence on the data.

scholarly journals Stability in Kelvin–Voigt poroelasticity

2020 ◽  
Author(s):  
Brian Straughan
Keyword(s):  
Initial Data ◽  
Linear Theory ◽  
Continuous Dependence ◽  
Hölder Continuous ◽  
Logarithmic Convexity ◽  
Continuous Dependence Of Solutions ◽  
Elastic Coefficients

Abstract Hölder continuous dependence of solutions upon the initial data is established for the linear theory of Kelvin–Voigt poroelasticity requiring only symmetry conditions upon the elastic coefficients. A novel functional is introduced to which a logarithmic convexity technique is employed.

scholarly journals CONTINUOUS DEPENDENCE ON THE INITIAL DATA OF THE SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH FRACTIONAL BROWNIAN MOTIONS

2018 ◽  
Vol 54 (2) ◽  
pp. 193-209
Author(s):  
I. V. Kachan
Keyword(s):  
Initial Data ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Hölder Continuous ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Right Hand ◽  
The Right

In the present acticle we consider finite-dimensional stochastic differential equations with fractional Brownian motions having different Hurst indices larger than 1/3 and a drift. These heterogeneous components of the equations are combined into a single process. The solutions of the equations are understood in the integral sense, and the integrals in turn are Gubinelli’s rough path integrals [1] realizing the well-known approach of the rough paths theory [2]. The existence and uniqueness conditions of the solutions of these stochastic differential equations are specified. Such conditions are sufficient to obtain the results related the continuous dependence on the initial data. In this article, we have first proved a continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations under consideration for almost all their trajectories. The result obtained does not depend on the probabilistic properties of fractional Brownian motions, and therefore it can be easily generalized to the case of arbitrary Holder-continuous processes with an exponent greater than 1/3. In this case, the constant arising in the estimates appears to be exponentially dependent on the norms of fractional Brownian motions. Taking into account the last fact and the proved result, an expected logarithmic continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations con - si dered is subsequently derived. This is the major result of this article.

A Hölder continuous ODE related to traffic flow

2003 ◽  
Vol 133 (4) ◽  
pp. 759-772 ◽  
Author(s):  
Rinaldo M. Colombo ◽  
Andrea Marson
Keyword(s):  
Vector Field ◽  
Initial Data ◽  
Differential Equations ◽  
Traffic Flow ◽  
Conservation Law ◽  
Continuous Dependence ◽  
Well Posedness ◽  
Hölder Continuous ◽  
Scalar Conservation Law ◽  
Scalar Conservation

This paper is devoted to the proof of the well posedness of a class of ordinary differential equations (ODEs). The vector field depends on the solution to a scalar conservation law. Forward uniqueness of Filippov solutions is obtained, as well as their Hölder continuous dependence on the initial data of the ODE. Furthermore, we prove the continuous dependence in C0 of the solution to the ODE from the initial data of the conservation law in L1.This problem is motivated by a model of traffic flow.

scholarly journals Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

2016 ◽  
Vol 146 (4) ◽  
pp. 777-832 ◽  
Author(s):  
J. C. Meyer ◽  
D. J. Needham
Keyword(s):  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Continuous Dependence ◽  
Parabolic Partial Differential Equations ◽  
Well Posedness ◽  
Partial Differential ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Semilinear Parabolic

We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.

Hölder continuous dependence in nonlinear elastodynamics

1986 ◽  
Vol 102 (1-2) ◽  
pp. 49-60 ◽  
Author(s):  
Stan Chiriţă
Keyword(s):  
Boundary Value Problems ◽  
Continuous Dependence ◽  
Boundary Value ◽  
Initial Boundary ◽  
Classical Solutions ◽  
Initial Boundary Value Problems ◽  
Hölder Continuous ◽  
Nonlinear Elastodynamics ◽  
Body Forces ◽  
Initial Boundary Value

SynopsisIn this paper we establish conditions to prove that if classical solutions to the initial boundary value problems for nonlinear elastodynamics exist, then they depend Hölder continuously on their initialdata and body forces.

Gluing Solutions

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Stress Tensor ◽  
Compact Support ◽  
Smooth Solution ◽  
Continuous Solution ◽  
Total Momentum ◽  
Frame Of Reference ◽  
Galilean Transformation ◽  
Hölder Continuous ◽  
Pressure Form ◽  
Original Frame

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

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