scholarly journals Uniqueness in MHD in divergence form: Right nullvectors and well-posedness

2002 ◽  
Vol 43 (12) ◽  
pp. 6195-6208 ◽  
Author(s):  
Maurice H. P. M. van Putten
Keyword(s):  
Divergence Form ◽  
Well Posedness

scholarly journals The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

2019 ◽  
Vol 29 (10) ◽  
pp. 1819-1851 ◽  
Author(s):  
K. Disser ◽  
J. Rehberg
Keyword(s):  
Charge Carrier ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Divergence Form ◽  
Well Posedness ◽  
Elliptic Regularity ◽  
Surfaces And Interfaces ◽  
Interfacial Recombination ◽  
Semiconductor Equations ◽  
Three Space

We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.

scholarly journals Μερικές διαφορικές εξισώσεις και προβλήματα της επιστήμης των υλικών

2014 ◽  
Author(s):  
Ευτυχία Αργυροπούλου
Keyword(s):  
Hilbert Space ◽  
Main Part ◽  
Semigroup Theory ◽  
Divergence Form ◽  
Elliptic Type ◽  
Well Posedness ◽  
Initial Value ◽  
Time Harmonic ◽  
Periodic Unfolding Method ◽  
Unfolding Method

The main objective of this thesis is the homogenization of partial dierentialequations (mainly Maxwell'As equations) describing electromagneticphenomena in complex media. In particular, we study the homogenization ofMaxwell'As equations focusing on the periodic unfolding method in complexmedia under Drude-Born-Fedorov type, local in time, constitutive relations.Firstly, we formulate Maxwell'A s problem as an evolution initial value(Cauchy) problem in a Hilbert space supplemented with the constitutiverelations of a bianisotropic medium (the most general linear medium in electromagnetics).Further, we analyze the notion of homogenization and weapply it as examples to equations of elliptic type in divergence form and toMaxwell'As system in bianisotropic media.We present also the method of periodic unfolding in the case of an ellipticpartial dierential equation and in the main part of this work we considerthe problem of the well-posedness of the time-dependent Maxwell'As equationsin a Drude-Born-Fedorov type environment considering the elds to beelements of an appropriate Hilbert space. In order to prove the existence anduniqueness we apply the Faedo-Galerkin method and for the continuous dependencefrom the initial data we use semigroup theory for operators. Therest of the main part of the thesis deals with the homogenization of theconsidered problem, using the periodic unfolding method.In the last chapter, we examine the time-harmonic Maxwell problem ina bianisotropic cavity, which we study by transforming it to an eigenvalueproblem.

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.

scholarly journals The BV solution of the parabolic equation with degeneracy on the boundary

2016 ◽  
Vol 14 (1) ◽  
pp. 272-282
Author(s):  
Huashui Zhan ◽  
Shuping Chen
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Well Posedness ◽  
Test Functions ◽  
The Stability

AbstractConsider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

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