scholarly journals Well-posedness of the peridynamic model with Lipschitz continuous pairwise force function

2013 ◽  
Vol 11 (4) ◽  
pp. 1039-1049 ◽  
Author(s):  
Etienne Emmrich ◽  
Dimitri Puhst
Keyword(s):  
Force Function ◽  
Well Posedness ◽  
Lipschitz Continuous ◽  
Peridynamic Model ◽  
Pairwise Force

scholarly journals Well-posedness for nonlinear SPDEs with strongly continuous perturbation

2020 ◽  
pp. 1-31
Author(s):  
Guy Vallet ◽  
Aleksandra Zimmermann
Keyword(s):  
Stochastic Perturbation ◽  
Dirichlet Boundary Conditions ◽  
Well Posedness ◽  
Lipschitz Continuous ◽  
Bounded Lipschitz Domain ◽  
Spatial Regularity ◽  
The Right ◽  
Diffusion Convection ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Continuous Perturbation

Abstract We consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝ d with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.

Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data

2004 ◽  
Vol 134 (5) ◽  
pp. 939-960 ◽  
Author(s):  
Song Jiang ◽  
Alexander Zlotnik
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Heat Conducting ◽  
Well Posedness ◽  
Global Weak Solutions ◽  
One Dimensional ◽  
Lipschitz Continuous ◽  
Viscous Heat ◽  
The Cauchy Problem ◽  
The One

We study the Cauchy problem for the one-dimensional equations of a viscous heat-conducting gas in the Lagrangian mass coordinates with the initial data in the Lebesgue spaces. We prove the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions.

Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organized criticality

2020 ◽  
pp. 2150029
Author(s):  
Marius Neuss
Keyword(s):  
Porous Media ◽  
Energy Functional ◽  
Physical Models ◽  
Well Posedness ◽  
Initial Value ◽  
Lipschitz Continuous ◽  
Self Organized ◽  
Porous Media Equations ◽  
Self Organized Criticality ◽  
Continuous Noise

We consider a class of generalized stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organized criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing nonlinearity, the arising energy functional is analyzed in detail.

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.

WELL-POSEDNESS AND FINITE ELEMENT APPROXIMABILITY OF TIME-HARMONIC ELECTROMAGNETIC BOUNDARY VALUE PROBLEMS INVOLVING BIANISOTROPIC MATERIALS AND METAMATERIALS

2009 ◽  
Vol 19 (12) ◽  
pp. 2299-2335 ◽  
Author(s):  
PAOLO FERNANDES ◽  
MIRCO RAFFETTO
Keyword(s):  
Finite Element ◽  
Original Problem ◽  
Boundary Value ◽  
Well Posedness ◽  
Problem Domain ◽  
Scattering Problems ◽  
Lipschitz Continuous ◽  
Time Harmonic ◽  
Microwave Components ◽  
Well Posed

A boundary value problem for the time harmonic Maxwell system is investigated through a variational formulation which is shown to be equivalent to it and well-posed if and only if the original problem is. Different bianisotropic materials and metamaterials filling subregions of the problem domain with Lipschitz continuous boundaries are allowed. Well-posedness and finite element approximability of the variational problem are proved by Lax–Milgram and Strang lemmas for a class of material configurations involving bianisotropic materials and metamaterials. Belonging to this class is not necessary, yet, for well-posedness and finite element approximability. Nevertheless, the material configurations of many radiation or scattering problems and many models of microwave components involving bianisotropic materials or metamaterials belong to the above class. Moreover, none of the other available tools commonly used to prove well-posedness seems to be able to cope with the material configurations left out by our treatment.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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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